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Let M be a compact K¨ahler manifold equipped with a pre-quantum line +bundle L. In [9], using T -symmetry, we constructed a polarization Pmix on M, which +generalizes real polarizations on toric manifolds. In this paper, we obtain the following +results for the quantum space Hmix associated to Pmix. First, Hmix consists of distri- +butional sections of L with supports inside µ−1(t∗ +Z). This gives Hmix = � +λ∈t∗ +Z Hmix,λ. +Second, the above decomposition of Hmix coincides with the weight decomposition for +the T -symmetry. Third, an isomorphism Hmix,λ ∼= H0(M �λ T, L �λ T ), for regular λ. +Namely, geometric quantization commutes with symplectic reduction. +1. Introduction +Let (M, ω) be a symplectic manifold equipped with a pre-quantum line bundle (L, ∇), +in particular F∇ = −iω. A polarization P on M is an integrable Lagrangian subbundle of +TM ⊗ C. Geometric quantization assigns a Hilbert space HP to these data. Namely, +(1.1) +HP = Γ(M, L) ∩ Ker(∇)|P, +where Γ(M, L) is the space of smooth sections of L 1. When (M, ω, J) is K¨ahler, we have +a K¨ahler polarization PJ = T 0,1 +J M, and HPJ = H0(M, L). If, in addition, M admits T- +symmetry, we constructed a polarization Pmix using T-symmetry in [9]. In this paper, we +study the quantum space Hmix associated to Pmix on M. Concretely, we have the following +assumption throughout this paper. +(∗) (M, ω, J) is a compact K¨ahler manifold of real dimension 2m equipped with an +effective Hamiltonian n-dimensional torus action ρ : T n → Diff(M, ω, J) by isome- +tries with moment map µ : M → t∗. Let (L, ∇, h) be a T n-invariant pre-quantum +line bundle on M. +Recall from [9], a (singular) polarization Pmix is constructed in this situation, which is, +Pmix = (PJ ∩ DC) ⊕ IC, +where DC = (Ker dµ) ⊗ C and IC = (Im dρ) ⊗ C. When n = m, i.e. M is a toric variety, +Pmix coincides with the singular real polarization defined by moment map and Hmix is the +space of Bohr-Sommerfeld states. +1In fact we need to allow distributional sections. +1 + +2 +LEUNG AND WANG +Recall that there is a natural way to embed the space of smooth sections into the space +of distributional sections using the Liouville measure volM = ωm +m! . That is, for any test +section τ ∈ Γc(M, L−1), +ι : Γ(M, L) → Γc(M, L−1)′, s �→ (ιs)(τ) = +� +M +⟨s, τ⟩ volM . +Then the quantum space Hmix can be described as: +Hmix = Γc(M, L−1)′ ∩ Ker(∇)|Pmix, +Our first result says that Hmix consists of distributional sections with supports inside +µ−1(t∗ +Z) and Hmix,λ is the λ-weight subspace of Hmix. +Theorem 1.1. (Theorem 3.2) Under the assumption (∗), +(1) given any δ ∈ Hmix, we have supp δ ⊂ � +λ∈t∗ +Z µ−1(λ). +This gives the following +decomposition +Hmix = +� +λ∈t∗ +Z +Hmix,λ, +where Hmix,λ = {δ ∈ Hmix | supp δ ⊂ µ−1(λ)}; +(2) for any λ ∈ t∗ +Z, Hmix,λ is a λ-weight subspace in Hmix. +Therefore the decomposition Hmix = � +λ∈t∗ +Z Hmix,λ is the weight decomposition with respect +to T n-action. +When n = m, this is a result for toric variety (see [1, 5]). Inspired by the works (see +[3]) of Guillemin and Sternberg that geometric quantizations commute with symplectic +reductions, we give a geometric description of Hmix,λ. Our main result (Theorem 1.5 or +Theorem 3.12) says that when λ is an integral regular value of µ, denoted as λ ∈ t∗ +Z,reg, we +have +Hmix,λ ∼= H0(Mλ, Lλ), +where (Mλ, Lλ) = (M �λ T, L �λ T) is the symplectic reduction of (M, L). Concretely, +Mλ = µ−1/T, we also denote the level set µ−1(λ) as Mλ. The restriction (Lλ, ∇) of pre- +quantum line (L, ∇) to Mλ can be descended to the quotient space Mλ denoted by (Lλ, ∇) +(see [3]). +Our second result states that, for any s ∈ H0(Mλ, Lλ), there is an associated distribu- +tional section δs ∈ Γc(Mλ, (Lλ)−1)′ such that ı(δs) lies in Hmix,λ, where ı : Γc(Mλ, (Lλ)−1)′ ֒→ +Γc(M, L−1)′ is the natural inclusion. +Definition 1.2. (Definition 3.4) For any λ ∈ t∗ +Z,reg and s ∈ H0(Mλ, Lλ), we define +the distributional section δs ∈ Γc(Mλ, (Lλ)−1)′ associated to s as follows: for any τ ∈ +Γc(Mλ, (Lλ)−1), +δs(τ) = +� +Mλ ⟨π∗s, τ⟩ volλ, + +GQ ASSOCIATED TO MIXED POLARIZATIONS ON K¨AHLER MANIFOLDS WITH T-SYMMETRY +3 +where volλ is the volume form on Mλ and π : Mλ → Mλ is the quotient map. +To see ı(δs) ∈ Hmix,λ, we need to investigate the interaction between the covariant +derivative on the space of smooth sections of L and the covariant derivative on the space +of distributional sections of Lλ. Our third result (Theorem 3.6) says that the following +diagram +Γ(M, L) +Γc(M, L−1)′ +Γc(M, L−1)′ +Γ(Mλ, Lλ) +Γc(Mλ, (Lλ)−1)′ +Γc(Mλ, (Lλ)−1)′, +volM +∇ξ +volλ +ı +∇ξ +ı +is a commutative diagram, for any ξ ∈ Γ(M, TM ⊗ C) satisfying ξ|Mλ ∈ Γ(Mλ, TMλ ⊗ C). +In order to show the above diagram commutes, we use the coisotropic embedding theorem +due to Weinstein [13] and further studied by Guillemin in [2] to relate the volume forms +volM and volλ. Then we obtain the following theorem: +Theorem 1.3. (Theorem 3.6) For any λ ∈ t∗ +Z,reg, δ ∈ Γc(Mλ, (Lλ)−1)′ and ξ ∈ Γ(M, TM ⊗ +C) satisfying ξ|Mλ ∈ Γ(Mλ, TMλ ⊗ C), we have +(1.2) +∇ξ(ı(δ)) = ı(∇ξδ), +where ı : Γc(Mλ, (Lλ)−1)′ ֒→ Γc(M, L−1)′ is the natural inclusion. +Proposition 1.4. (Proposition 3.7) For any λ ∈ t∗ +Z,reg and s ∈ H0(Mλ, Lλ), we have +ı(δs) ∈ Hmix,λ, +where ı : Γc(Mλ, (Lλ)−1)′ ֒→ Γc(M, L−1)′ is the natural inclusion. +This allows us to define a map κ : H0(Mλ, Lλ) → Hmix,λ by s �→ κ(s) = ı(δs). Finally +we show that κ is an isomorphism. +Theorem 1.5. (Theorem 3.12) For any λ ∈ t∗ +Z,reg, +κ : H0(Mλ, Lλ) → Hmix,λ +is an isomorphism. +We show the surjectivity of κ via the following steps. First, we show that any element +˜δ in Hmix,λ is locally a delta function along µ−1(λ), and does not involve any derivative of +delta functions. This implies ˜δ = ı(δ), for some δ ∈ Γc(Mλ, (Lλ)−1)′. +Second, we show that T n-invariant distributional sections of Lλ can be descended to +distributional sections of Lλ. That is, for any δ ∈ Γc(Mλ, (Lλ)−1)′ satisfying ∇ξ#δ = 0, +there exists a distributional section η ∈ Γc(Mλ, L−1 +λ )′ such that δ = π∗η (Lemma 3.8). +Third, we show that if ∇ζ(π∗η) = 0 for all ζ ∈ Γ(Mλ, Pmix), then η is ¯∂-closed (Theorem +3.11). By the regularity of elliptic operator ∆ = ¯∂∗ ¯∂, we have η is smooth (i.e. η = ι(s), +for some s ∈ H0(Mλ, Lλ)). Finally, we show that ˜δ = κ(s). + +4 +LEUNG AND WANG +1.1. Acknowledgements. We are grateful to Siye Wu for insightful comments and useful +discussions. D. Wang would like to thank Qingyuan Jiang, Yutung Yau and Ki Fung Chan +for many helpful discussions. This research was substantially supported by grants from the +Research Grants Council of the Hong Kong Special Administrative Region, China (Project +No. CUHK14301619 and CUHK14301721) and a direct grant from the Chinese University +of Hong Kong. +2. Preliminary +2.1. The Marsden-Weistein construction. In this subsection, we review the basic con- +cepts of Hamiltonian action and symplectic reduction in order to fix the notations in our +setting (for more details, the reader can refer to [3], [10]). +2.1.1. Hamiltonian action. Let (M, ω) be a compact symplectic manifold. For f ∈ C∞(M, R), +the Hamiltonian vector field Xf associated to f is determined by ıXf ω = −df. This gives +a Lie algebra homeomorphism +ψ : (C∞(M; R), {·, ·}) → (Vect(M, ω), [·, ·]) +defined by ψ(f) = Xf, where {, } is the Poisson bracket of two functions f, g ∈ C∞(M; R) +determined by {f, g} = ω(Xf, Xg). Let T n be a torus of real dimension n and ρ : T n → +Diff(M, ω) an action of T n on M which preserves ω. +Let t be the Lie algebra of T n. +Differentiating ρ at the identity element, we have +dρ : t → Vect(M, ω), ξ �→ ξ#, +where t is the Lie algebra of T n and ξ# is called the fundamental vector field associated to +ξ. The action of T n on M is said to be Hamiltonian if dρ factors through ψ. +Let ⟨, ⟩ : t∗ × t → R be the natural pairing between t∗ and t. For each point p ∈ M, we +can associate an element µ(p) ∈ t∗ by the formula +⟨µ(p), ξ⟩ = −µξ(p), ∀ξ ∈ t. +This gives us a moment mapping µ : M → t∗ which is a T n-equivariant map. +2.1.2. Symplectic reduction. We denote the set of regular values of µ by t∗ +reg, that is, +t∗ +reg = {λ ∈ t∗| λ is a regular value of µ}. +For any λ ∈ t∗ +reg, denote the level set µ−1(λ) by Mλ. +Then Mλ is a T n-invariant +coisotropic submanifold (i : Mλ ֒→ M) and the action of T n is locally free (see [10]). +For simplicity, we assume T n acts freely on Mλ. Then the projection mapping +π : Mλ → Mλ +is a principal T n-fibration. Moreover there exists a unique symplectic form ωλ on Mλ such +that π∗ωλ = i∗ω. Denote the volume form +1 +(m−n)!ωm−n +λ +on Mλ by volλ. Take a connection + +GQ ASSOCIATED TO MIXED POLARIZATIONS ON K¨AHLER MANIFOLDS WITH T-SYMMETRY +5 +α ∈ Ω1(Mλ, t) on Mλ, π∗ volλ ∧αn is a volume form on Mλ denoted by volλ (here αn is a +n-form on Mλ defined by α∧···∧α +v +, where v ∈ ∧nt is a T n-invariant top form on t). +(Mλ = µ−1(λ), volλ) +M +(Mλ = µ−1(λ)/T n, volλ). +i +π +2.2. Pre-quantum data. In this subsection, we first review the definition of T n-invariant +pre-quantum line bundles. Then we restate the result that the pre-quantum line bundle +can always be descended to the reduction space by Guillemin and Sternberg in our setting +(K¨ahler manifold equipped with T n-symmetry). +Definition 2.1. Let (M, ω, J) be a symplectic manifold, a pre-quantum line bundle (L, ∇, h) +on M is a complex line bundle L together with a Hermitian metric h and Hermitian con- +nection ∇, such that the curvature form F∇ = −iω. +The existence of a pre-quantum line bundle L on M is equivalent to [ ω +2π] being integral +(see [8]). When M is K¨ahler, L is an ample holomorphic line bundle. There is a canonical +representation of the Lie algebra t on space of smooth sections of L given by the operators +(2.1) +∇ξ# + iµξ, ξ ∈ t. +The pre-quantum line bundle is said to be T n-invariant if there exists a global action of +T n on L such that the induced action of t is given by (2.1). It is always possible if the +T n-action on M is Hamiltonian (see [8]). +Let tZ be the kernel of the exponential map exp : T n → t and t∗ +Z ⊂ t∗ be the dual lattice +of tZ. We denote the set of integral regular values of µ by t∗ +Z,reg, that is, t∗ +Z,reg = t∗ +reg ∩ t∗ +Z. +Guillemin and Sternberg in [3] showed that there are associated pre-quantum data on +the reduction space Mλ, for λ ∈ t∗ +Z,reg. +Theorem 2.2. [3, Theorem 3.2] There is a unique line bundle with connection (Lλ, ∇λ) +on Mλ such that +(2.2) +π∗Lλ = i∗L =: Lλ, and π∗∇λ = i∗∇. +Corollary 2.3. [3, Corollary 3.4] The curvature of the connection, ∇λ, is the symplectic +form ωλ. + +6 +LEUNG AND WANG +Therefore we have the following commuting diagram: +(Lλ, i∗∇) +(L, ∇) +t∗ +Z,reg +(Lλ, ∇λ) +(Mλ, volλ) +(M, volM) +t∗ +(Mλ, volλ) +π +i +µ +By abuse of notations, we denote both i∗∇ and ∇λ by ∇. In order to pull-back distribu- +tional sections from Mλ to Mλ later, we first recall how to push-forward sections of line +bundle Lλ. +Remark 2.4. Let π : P → B be a principal T n-bundle over B, E → B a line bundle over +B, and π∗E → P the pullback line bundle. Then we can define the dual map +π∗ : Γc(B, L−1)′ → Γc(P, (π∗L)−1)′, η �→ π∗η +by (π∗η)(τ) = η(π∗τ), for any τ ∈ Γc(P, L−1) +Throughout this paper, we fix a T n-invariant n-form dθ on T n such that +� +T n dθ = 1. +When we deal with the pull-back of distribution sections of Lλ, we mean in the sense of +Remark 2.4 with respect to dθ. +2.3. Complex structures on symplectic reduction spaces. In order to study the +relationship between geometric quantization associated to Pmix and symplectic reduction, +we recall the work on the existence of complex structures on symplectic reduction spaces +Mλ (see [3]). +Recall that the anti-holomorphic Lagrangian sub-bundle TM0,1 +J +⊂ TM ⊗ C is a K¨ahler +polarization denoted by PJ. We define F ⊂ TMλ ⊗ C by +(2.3) +Fp = (PJ)p ∩ (TMλ ⊗ C)p, +for any p ∈ Mλ. F can be descended to a bundle PJ,λ over the reduction space Mλ, which +is a positive-definite Lagrangian sub-bundle of TMλ ⊗ C. Under the assumption (∗), we +have (DC ∩ PJ)p = Fp, for any p ∈ Mλ. +Theorem 2.5. [3, Theorem 3.5] There is a positive-definite polarization PJ,λ canonically +associated with PJ on the reduction space Mλ. +By Definition 4.2 and Lemma 4.3 in [3], PJ,λ determined a complex structure Jλ on Mλ +such that +(2.4) +PJ,λ = TM0,1 +λ , +where TM0,1 +λ +is the anti-holomorphic sub-bundle of TMλ ⊗ C with respect to Jλ. + +GQ ASSOCIATED TO MIXED POLARIZATIONS ON K¨AHLER MANIFOLDS WITH T-SYMMETRY +7 +2.4. Polarizations on K¨ahler manifolds with T-symmetry. In [9], we constructed +polarizations Pmix on K¨ahler manifolds with T-symmetry. Throughout this sections, the +existence of pre-quantum line L in (∗) is not needed. +For any point p ∈ M, consider the map ρp : T n → M defined by ρp(g) = ρ(g)(p). Let +IR ⊂ TM be the singular distribution generated by fundamental vector fields in Im dρ, +that is (IR)p = Im dρp(e). Let DR = (Ker dµ) ⊂ TM be a distribution defined by the kernel +of dµ. Note that there is a K¨ahler polarization PJ = TM0,1 +J +associated to the complex +structure J on a K¨ahler manifold (M, ω, J). +Definition 2.6. [9, Definition] We define the singular distribution Pmix ⊂ TM ⊗ C by: +(2.5) +Pmix = (PJ ∩ DC) ⊕ IC, +where DC = DR ⊗ C and IC = IR ⊗ C are the complexification of DR and IR respectively. +Let Hp be the stabilizer of T n at point p ∈ M. Denote by ˇ +M the union of n-dimensional +orbits in M, that is, +ˇ +M = {p ∈ M| dim Hp = 0}, +which is an open dense subset in M. +Theorem 2.7. [9, Theorem 1.1] Under the assumption (∗), Pmix is a singular polarization +and smooth on ˇ +M. Moreover, rank(Pmix ∩ ¯Pmix ∩ TM)| ˇ +M = n. +According to Definition 4.6, Pmix is a singular real polarization on M, when n = m, +namely, M is toric manifold; Pmix is a singular mixed polarization on M, when 1 ≤ n < m. +3. Main results +We define the quantum space associated to the polarization Pmix = (PJ ∩ DC) ⊕ IC as +follows. Let (L, ∇, h) be the pre-quantum line bundle on M. We first recall the definition +of quantum space H associated to polarization P (see [14]). +Definition 3.1. The quantum space H associated to polarization P is the following sub- +space of Γc(M, L−1)′: +H = {δ ∈ Γc(M, L−1)′ | ∇ξδ = 0, ∀ ξ ∈ Γ(M, P)}, +where ∇ξ is the covariant derivative operator acting on the space of distributional sections +defined by equation (3.2). +In our setting, even through the polarization Pmix is singular, we continue to use the +above definition for the quantum space. We denote it by Hmix. When n = m, M is toric +variety and Pmix is a singular real polarization. The definition of Hmix coincides with the +definition of the quantum spaces associated to singular real polarizations studied in [1]. + +8 +LEUNG AND WANG +Moreover, for any λ ∈ t∗, we define the subspace of those sections with supports on µ−1(λ) +as: +Hmix,λ = {δ ∈ Hmix | supp δ ⊂ µ−1(λ)}. +3.1. Distributional sections in Hmix,λ associated to sections in H0(Mλ, Lλ). In this +subsection, we first confirm that for any distributional section δ ∈ Hmix we have (see +Theorem 3.2): +supp δ ⊂ +� +λ∈t∗ +Z +µ−1(λ). +After extending the T n-action from the space of smooth sections to the space of distri- +butional sections of L, we show that Hmix,λ is a λ-weight subspace of Hmix, for any λ ∈ t∗ +Z. +This gives the weight decomposition of Hmix, i.e. Hmix = � +λ∈t∗ +Z Hmix,λ. +Inspired by the work on geometric quantizations commute with symplectic reductions by +Guillemin and Sternberg in [3], we expect to establish the isomorphism between H0(Mλ, Lλ) +and Hmix,λ, where (Mλ, Lλ) is the symplectic reduction of M at a regular integral level λ. +At the end of this subsection, given any holomorphic section s ∈ H0(Mλ, Lλ), we de- +fine an associated distributional section δs ∈ Γc(Mλ, (Lλ)−1)′ (see Definition 3.4) with +respect to the volume form volλ. Then we show that distributional sections in Hmix,λ as- +sociated to sections in H0(Mλ, Lλ) (see Proposition 3.7). That is ı(δs) ∈ Hmix,λ, where +ı : Γc(Mλ, (Lλ)−1)′ ֒→ Γc(M, L−1)′ is the natural inclusion. +In order to study the quantum space Hmix, we recall how to extend covariant differenti- +ation to distributional sections of L (see [1]). First of all, there is a natural way to embed +the space of smooth sections into the space of distributional sections using the Liouville +measure volM = ωm +m! : +ι : Γ(M, L) → Γc(M, L−1)′ +s �→ (ιs)(τ) = +� +M +⟨s, τ⟩ volM . +Here ⟨, ⟩ : L×L−1 → C is the natural paring between L and L−1. Let ∇ be the connection +on L−1 such that d⟨s, τ⟩ = ⟨∇s, τ⟩+⟨s, ∇τ⟩. It is necessary to require that the operator ∇ +acting on the distributional sections ι(s) which come from any smooth section s coincides +with the operator ∇ acting on s, i.e. the following diagram +Γ(M, L) +Γc(M, L−1)′ +Γ(M, L) +Γc(M, L−1)′ +ι +∇ξ +∇ξ +ι + +GQ ASSOCIATED TO MIXED POLARIZATIONS ON K¨AHLER MANIFOLDS WITH T-SYMMETRY +9 +commutes, for any ξ ∈ Γ(M, TM ⊗ C). Let div ξ be the divergence of ξ with respect to +volM = ωm +m! , equivalently, Lξ(volM) = (div ξ) volM. It can be seen that: +0 = +� +M +Lξ(⟨s, τ⟩ volM) = +� +M +(Lξ⟨s, τ⟩) volM + +� +M +⟨s, τ⟩Lξ(volM) += +� +M +⟨∇ξs, τ⟩ volM + +� +M +⟨s, ∇ξτ⟩ volM + +� +M +⟨s, τ⟩(div ξ) volM . +This gives, for any smooth section s ∈ Γ(M, L) and smooth test section τ ∈ Γc(M, L−1), +(3.1) +(∇ξι(s))(τ) = +� +M +⟨∇ξs, τ⟩ volM = +� +M +⟨s, −((div ξ)τ + ∇τ)⟩ volM . +To determine ∇ξσ for a general distributional section σ not of the form ι(s), we built its +transpose by integrating the operator ∇ξ by parts. Namely, ∇ξ is characterized by its +transpose t∇ξ as follows: for any τ ∈ Γc(M, L−1), +(3.2) +(∇ξσ)(τ) = σ(t∇ξτ), with t∇ξτ = −(div ξτ + ∇ξτ). +Similarly we can extend the T n-action on space of smooth sections to space of distri- +butional sections such that the inclusion ι : Γ(M, L) ֒→ Γc(M, L−1)′ (with respect to the +Liouville volume form volM) is T n-equivariant. That is, for any ξ ∈ t, the following diagram +commute. +Γ(M, L) +Γc(M, L−1)′ +Γ(M, L) +Γc(M, L−1)′ +ξ· +volM +ξ· +volM +, i.e. ξ · (ι(s)) = ι(ξ · s). +Namely, for any δ ∈ Γc(M, L−1)′, τ ∈ Γc(M, L−1), and ξ ∈ t, ξ · δ is characterized by: +(3.3) +(ξ · δ)(τ) = δ(ξ · τ), with ξ · τ = ∇ξ#τ + iµξτ. +The T n-action on L preserve connection ∇, which implies that T n acts on Hmix. We obtain +the following results. +Theorem 3.2. Under the assumption (∗), +(1) given any δ ∈ Hmix, we have supp δ ⊂ � +λ∈t∗ +Z µ−1(λ). +This gives the following +decomposition +Hmix = +� +λ∈t∗ +Z +Hmix,λ, +where Hmix,λ = {δ ∈ Hmix | supp δ ⊂ µ−1(λ)}; +(2) for any λ ∈ t∗ +Z, Hmix,λ is a λ-weight subspace in Hmix. +Therefore the decomposition Hmix = � +λ∈t∗ +Z Hmix,λ is the weight decomposition with respect +to T n-action. + +10 +LEUNG AND WANG +Proof. (1) For a loop γb ⊂ T n specified by a vector b ∈ tZ, for any test function τ ∈ +Γc(M, L−1), parallel transporting τ(p) with respect to the connection ∇ around a loop +γb · p ⊂ M results in multiplication of τ(p) by e−2iπ⟨µ(p),b⟩, where ⟨, ⟩ : t∗ × t → R is the +natural pairing between t∗ and t. The reason is as follows. Recall given T 1-equivariant +line bundle L → M with equivariant curvature FA + µ, the holonomy around any T 1-orbit +at p ∈ M is given by e2πiµ(p). Applying this to our case, for a loop γb ⊂ T n specified by +b ∈ tZ, holonomies of (L−1, ∇) around the loops in M specified by b ∈ tZ define a smooth +function: +fb : M → C, p �→ fb(p) := e−2iπ⟨µ(p),b⟩. +Therefore, ∇b#τ = 0 implies fb · τ = τ. +By transporting this to the dual space, we +have fb · δ = δ for any δ ∈ Γc(M, L−1)′ satisfying ∇b#δ = 0. For δ ∈ Hmix, we have +∇ξ#δ = 0, ∀ξ ∈ t, in particular ∇b#δ = 0, ∀b ∈ tZ. This implies that fb is constant 1 on +supp δ, for any b ∈ tZ. Therefore we conclude that δ should be supported in the set where +µ takes integral value. That is, +supp δ ⊂ +� +λ∈t∗ +Z +µ−1(λ). +To prove (2), given any λ ∈ t∗ +Z and δ ∈ Hmix,λ, we need to show for any τ ∈ Γc(M, L−1) +and ξ ∈ t, +(ξ · δ)(τ) = i⟨λ, ξ⟩δ(τ), +where ⟨, ⟩ : t∗ × t → R is the natural pairing. Note that the Liouville volume form volM is +T n-invariant, div ξ# = 0. This implies, +(3.4) +(∇ξ#δ)(τ) = −δ(∇ξ#τ) = 0, and (ξ · δ)(τ) = −δ(ξ · τ). +Recall that ξ · τ = ∇ξ#τ + iµξτ. By equation (3.4), we have +(3.5) +(ξ · δ)(τk) = −δ(ξ · τk) = −δ(∇ξ#τk + iµξτk) = −δ(iµξτk). +Suppose τ = τk ∈ Γc(M, L−1) has weight k, i.e. ξ · τk = i⟨k, ξ⟩τk, by equation (3.4), one +has +(3.6) +(ξ · δ)(τk) = −δ(ξ · τk) = −δ(i⟨k, ξ⟩τk). +Combine equations (3.4) and (3.6), we obtain +(3.7) +δ(i(µξ − ⟨k, ξ⟩)τk) = 0, ∀ξ ∈ t. +For any k ̸= λ, there exists ξ ∈ t, such that ⟨λ, ξ⟩ ̸= ⟨k, ξ⟩. For such ξ, as µξ|Mλ = ⟨λ, ξ⟩, +µξ − ⟨k, ξ⟩ is no-where vanishing on a T n-invariant open neighbourhood of Mλ. One has: +(3.8) +δ(τk) = δ(i(µξ − ⟨k, ξ⟩) +1 +i(µξ − ⟨k, ξ⟩)τk). + +GQ ASSOCIATED TO MIXED POLARIZATIONS ON K¨AHLER MANIFOLDS WITH T-SYMMETRY 11 +Since the moment map is T n-invariant, +1 +i(µξ−⟨k,ξ⟩)τk still has weight k. Hence, by the above +discussion (i.e. in equation 3.7, replacing τk by +1 +i(µξ−⟨k,ξ⟩)τk), one has: +(3.9) +δ(τk) = 0. +Given the weight decomposition of τ = � +k τk (i.e. ξ · τk = i⟨k, ξ⟩τk), by linearity of δ +and equation (3.9), we obtain: +(ξ · δ)(τ) = i⟨λ, ξ⟩δ(τ⟨λ,ξ⟩) = i⟨λ, ξ⟩δ(τ). +Therefore we have: ξ · δ = i⟨λ, ξ⟩δ. +□ +The next corollary says that any element in Hmix,λ is locally a delta function along +µ−1(λ), and does not involve any derivative of delta functions. +Corollary 3.3. For any λ ∈ t∗ +Z, δ ∈ Hmix,λ, and any test section τ ∈ Γc(M, L−1) satisfying +τ|Mλ = 0, we have +(3.10) +δ(τ) = 0. +Proof. For any τ ∈ Γc(M, L−1) satisfying τ|Mλ = 0, let τ = � +k τk be its weight decompo- +sition, where τk = +� +eit∈T n(eit · τ)e−iktdt. By Theorem 3.2, δ has weight λ with respect to +T n-action. This implies, for k ̸= λ, +δ(τk) = 0. +Note that +τk(p) = +� +eit∈T n(e−it · τ)(p)eiktdt = +� +eit∈T n τ(e−it · p)eiktdt. +For any p ∈ Mλ and t ∈ t, eit · p ∈ Mλ since Mλ is T n-invariant. Therefore τk|Mλ = 0, +as τ|Mλ = 0. In particular, τλ|Mλ = 0. So there exists weight λ test section τ ′ such that +τλ = i(µξ − ⟨λ, ξ⟩)τ ′ +λ for some ξ ∈ t. Hence, for k = λ, by equation (3.7), we obtain +(3.11) +δ(τλ) = δ(i(µξ − ⟨λ, ξ⟩)τ ′ +λ) = 0. +Therefore, by the linearity of δ, we have δ(τ) = 0. +□ +In order to establish the isomorphism between H0(Mλ, Lλ) and Hmix,λ, where (Mλ, Lλ) = +(M, L)//λT is the symplectic reduction of M at a regular integral level λ. We first define +the distributional section δs ∈ Γc(Mλ, (Lλ)−1)′ on Mλ ⊂ M associated to s ∈ H0(Mλ, Lλ) +as follows. +Definition 3.4. For any λ ∈ t∗ +Z,reg and s ∈ H0(Mλ, Lλ), we define the distributional section +δs ∈ Γc(Mλ, (Lλ)−1)′ associated to s as follows: for any τ ∈ Γc(Mλ, (Lλ)−1), +(3.12) +δs(τ) = +� +Mλ ⟨π∗s, τ⟩ volλ . + +12 +LEUNG AND WANG +In fact, δs = ι(π∗s), under the embedding ι : Γ(Mλ, Lλ) +Γc(Mλ, (Lλ)−1)′ +volλ +defined +by σ �→ (ισ)(τ) = +� +M⟨σ, τ⟩ volλ with respect to volλ. Note that ı(δs) ∈ Γc(M, L−1)′, where +ı : Γc(Mλ, (Lλ)−1)′ ֒→ Γc(M, L−1)′ is a natural inclusion defined by: +(3.13) +(ı(δ))(τ) = δ(τ|Mλ), ∀δ ∈ Γc(Mλ, (Lλ)−1)′. +In order to show ı(δs) ∈ Hmix,λ, we first need to extend covariant derivative ∇ξ on the +space of smooth sections to the space of distributional sections of Lλ with respect to volλ +as before. That is, for any σ ∈ Γc(Mλ, (Lλ)−1)′, and ξ ∈ Γ(Mλ, TMλ ⊗ C), +(3.14) +(∇ξσ)(τ) = σ(t∇ξτ), with t∇ξτ = −(div ξτ + ∇ξτ), +where div2 ξ = Lξ volλ +volλ . +In particular, ı(δs) ∈ Γc(M, L−1)′. +we also need to study the relationship between +the covariant derivative on the space of distributional sections of Lλ and the space of +distributional sections of L. We expect the following diagram +Γ(M, L) +Γc(M, L−1)′ +Γc(M, L−1)′ +Γ(Mλ, Lλ) +Γc(Mλ, (Lλ)−1)′ +Γc(Mλ, (Lλ)−1)′, +volM +∇ξ +volλ +ı +∇ξ +ı +commute, for any ξ ∈ Γ(M, TM ⊗ C) satisfying ξ|Mλ ∈ Γ(Mλ, TMλ ⊗ C). In order to +show that the above diagram commute, we use the coisotropic embedding theorem due to +Weinstein [13] and further studied by Guillemin in [2] to (relate volM and volλ) show the +following lemma. +3.1.1. Restriction commutes with taking divergence. For any ξ ∈ Γ(M, TM ⊗C), we denote +the restriction of the divergence of ξ (with respect to volM) to Mλ by div1 +ξ and denote the +divergence of ξ|Mλ (with respect to volλ) by div2 +ξ i.e. +div1 +ξ = d(iξ volM) +volM +|Mλ, and div2 ξ = +d(iξ|Mλ volλ) +. +volλ. +Lemma 3.5. Under the assumption (∗), for any ξ ∈ Γ(M, Pmix) and λ ∈ t∗ +reg, we have +div1 ξ = div2 ξ, +as functions on Mλ. +Proof. Without loss of generality, we assume λ = 0 and n = 1. In order to show that +div1 ξ = div2 ξ, we shall first relate the volume form volM of M and the volume form volMλ +of Mλ. Taking a principal T 1-connection α ∈ Ω1(M0, t) on M0, choose a basis ξ1 of t and +denote the corresponding dual basis of t∗ by ξ∗ +1 with coordinate function t. In terms of ξ1, + +GQ ASSOCIATED TO MIXED POLARIZATIONS ON K¨AHLER MANIFOLDS WITH T-SYMMETRY 13 +we write α = ξ1 ⊗ α1, where α1 is a scalar valued form. By abuse of notations, we denote +α1 by α. Consider M0 as a submanifold of M0 × t∗ via the embedding +i0 : M0 → M0 × t∗, i0(p) = (p, 0). +The two-form +˜ω = π∗ω0 + d(tα) +is symplectic on a neighbourhood U of M0 in M0 × t∗ and satisfies i∗ +0˜ω = π∗ω0. +Note that (dt ∧ α)2 = 0 and (i∗ω)m = 0. +Then we restrict our attention to show +Lξt = 0, ∀ ξ ∈ Γ(M, Pmix). We extend the T 1-action on M0 to M0 ×t∗ in a trivial manner. +Then ˜ω is T 1-invariant and the action of T 1 on M0 × t∗ is Hamiltonian with moment map +µ0 : M0 × t∗ → t∗, (p, t) �→ t. +According to Theorem 2.2 of [2], in a neighborhood U of M0, the Hamiltonian T 1-spaces +(M, ω) and (M0 × t∗, ˜ω) are isomorphic (see Appendix). +This gives Lζt = 0, for any +ζ ∈ Γ(U, DC) and +volM = 1 +m!(i∗ω + tdα)m−1 ∧ α ∧ dt, +in a neighbourhood U of M0. As Pmix ⊂ DC, it is obvious that Γ(U, Pmix) ⊂ Γ(U, DC). +This gives us that, for any ξ ∈ Γ(U, Pmix), +(3.15) +Lξt = 0. +It follows that: +Lξ(i∗ω + tdα)m−1 = (m − 1)(Lξ(i∗ω) + tLξdα) ∧ (i∗ω + tdα)m−2, ∀ξ ∈ Γ(U, Pmix). +Therefore, we obtain: +1 +volM +d(iξ volM) = +1 +volM +Lξ volM = +1 +volM +1 +m!Lξ((i∗ω + tdα)m−1 ∧ α ∧ dt) += +1 +volM +1 +m!(Lξ(i∗ω + tdα)m−1 ∧ α ∧ dt + (i∗ω + tdα)m−1 ∧ Lξα ∧ dt) += (m − 1)(Lξi∗ω + tLξdα) ∧ (i∗ω + tdα)m−2 ∧ α + (i∗ω + tdα)m−1 ∧ Lξα) +(i∗ω + tdα)m−1 ∧ α +. +Recall that vol0 = +1 +(m−1)!(i∗ω)m−1 ∧ α. By abuse of notation, iξ vol0 means iξ|M0 vol0. Then +by a straight computation, +div2 ξ = +1 +vol0d(iξ vol0) = +1 +vol0Lξ vol0 += +1 +vol0 +1 +(m − 1)!Lξ((i∗ω)m−1 ∧ α) += (m − 1)(Lξi∗ω) ∧ (i∗ω)m−2 ∧ α + (i∗ω)m−1 ∧ Lξα) +(i∗ω)m−1 ∧ α +. + +14 +LEUNG AND WANG +Therefore, for ξ ∈ Γ(M, Pmix), we have: +div1 ξ = +� +1 +volM +d (iξ volM) +� +|M0 = +� +1 +volM +d (iξ volM) +� +|t=0 = div2 ξ. +□ +Then we obtain the following theorem: +Theorem 3.6. For any λ ∈ t∗ +Z,reg, δ ∈ Γc(Mλ, (Lλ)−1)′ and ξ ∈ Γ(M, TM ⊗ C) satisfying +ξ|Mλ ∈ Γ(Mλ, TMλ ⊗ C), we have +(3.16) +∇ξ(ı(δ)) = ı(∇ξδ), +where ı : Γc(Mλ, (Lλ)−1)′ ֒→ Γc(M, L−1)′ is the natural inclusion. +Proof. For any test section τ ∈ Γc(M, L−1), according to equation (3.2), one has +(3.17) +(∇ξ(ı(δ)))(τ) = ı(δ)(t∇ξτ) = δ(i∗(t∇ξτ)), +and +(3.18) +(ı(∇ξδ))(τ) = (∇ξδ)(i∗τ) = δ(t∇ξ(i∗τ)), +where i : Mλ ֒→ M is the inclusion. +To show equation (3.16), by (3.17) and (3.18), it is enough to prove that +(3.19) +i∗(t∇ξτ) =t ∇ξ(i∗τ). +According to equation (3.2), we have: +(3.20) +i∗(t∇ξτ) = −i∗ (div ξτ + ∇ξτ) , +where div ξ = iξ volM +vol M . Similarly, applying the equation (3.1) to L|Mλ, we have: +(3.21) +t∇ξ(i∗τ) = −((div2 ξ)(i∗τ) + ∇ξ(i∗τ)) +where div2 ξ = iξ volλ +volλ , i∗τ = τ|Mλ = τ by abuse of notation. Denote i∗(div ξ) by div1 ξ i.e. +div1 ξ = iξ volM +volM |Mλ. As i∗(∇ξτ) = ∇ξ(i∗τ) by abuse of notation ξ|Mλ = ξ, we have +(3.22) +− i∗ (div ξτ + ∇ξτ) = −(div1 ξ(i∗τ) + ∇ξ(i∗τ)). +By Lemma 3.5, +(3.23) +div1 ξ = div2 ξ. +Combining equations (3.20), (3.21), (3.22) with (3.23), one has +δ(i∗(t∇ξτ)) = δ(t∇ξ(i∗τ)). +Therefore we have: ∇ξ(ı(δ)) = ı(∇ξδ). +□ + +GQ ASSOCIATED TO MIXED POLARIZATIONS ON K¨AHLER MANIFOLDS WITH T-SYMMETRY 15 +Proposition 3.7. For any regular λ ∈ t∗ +Z,reg and s ∈ H0(Mλ, Lλ), we have: +ı(δs) ∈ Hmix,λ, +where ı : Γc(Mλ, (Lλ)−1)′ ֒→ Γc(M, L−1)′ is the natural inclusion. +Proof. By the definition of δs, we have ı(δs) ∈ Γc(M, L−1)′ and supp ı(δs) ⊂ µ−1(λ). It +remains to show that, for any ξ ∈ Γ(M, Pmix), +(3.24) +∇ξ(ı(δs)) = 0. +Note that for any ξ ∈ (M, Pmix), ξ|Mλ ∈ Γ(Mλ, TMλ ⊗ C). To check equation (3.24), by +Theorem 3.6, it is equivalent to prove, for any ξ ∈ Γ(Mλ, Pmix) +(3.25) +∇ξδs = 0. +Take any test section τ ∈ Γc(Mλ, (Lλ)−1), according to equation (3.2), we have: +(3.26) +(∇ξδs)(τ) = δs �t∇ξτ +� += −δs � +(div2 ξ)τ + ∇ξτ +� +, +where div2 ξ = iξ volλ +volλ . By definition of δs, it can be seen that: +(3.27) +− δs � +(div2 ξ)τ + ∇ξτ +� += − +� +Mλ +� +π∗s, (div2 ξ)τ + ∇ξτ +� +volλ . +Similarly, applying the equation (3.1) to L|Mλ, we have: +(3.28) +� +Mλ ⟨∇ξ(π∗s), τ⟩ volλ = − +� +Mλ +� +π∗s, (div2 ξ)τ + ∇ξτ +� +volλ . +Combining equations (3.26), (3.27), with (3.28), we have +(3.29) +(∇ξδs)(τ) = +� +Mλ ⟨∇ξ(π∗s), τ⟩ volλ . +Since s ∈ H0(Mλ, Lλ) is a holomorphic section, we have ∇s ∈ Γ(Mλ, T ∗M1,0 +λ +⊗ Lλ). For +any ξ ∈ Γ(M, Pmix) and q ∈ Mλ, as (Pmix)q ⊂ (DC)q = TqMλ ⊗ C, we have π∗(ξq) ∈ +Tπ(q)M0,1 +λ . +This implies ∇ξ(π∗s) = 0 on Mλ, for any ξ ∈ Γ(M, Pmix). +Then, for all +τ ∈ Γc(M, L−1), by equation (3.29), +(∇ξδs)(τ) = 0. +Therefore we have: ı(δs) ∈ Hmix,λ. +□ + +16 +LEUNG AND WANG +3.2. λ-weight quantum subspace Hmix,λ. In this subsection, we are going to show that +(see Theorem 3.12) for any regular λ ∈ t∗ +Z,reg, +κ : H0(Mλ, Lλ) → Hmix,λ +given by s �→ κ(s) = ı(δs) is an isomorphism. +Firstly, we show that T n-invariant distributional sections of Lλ can be descended to +distributional sections of Lλ. That is, for any δ ∈ Γc(Mλ, (Lλ)−1)′ satisfying ∇ξ#δ = 0, +there exists a distributional section η ∈ Γc(Mλ, L−1 +λ )′ such that δ = π∗η (Lemma 3.8). +Secondly, we show that if ∇ζ(π∗η) = 0 for all ζ ∈ Γ(Mλ, Pmix), then η is ¯∂-closed (Theorem +3.11). Finally, we show that H0(Mλ, Lλ) ∼= Hmix,λ (Theorem 3.12). +3.2.1. Descending distributional sections from Mλ to Mλ. For any λ ∈ t∗ +reg, let π : Mλ → +Mλ be the principal T n-bundle. Recall that (Lλ, ∇) can be descended to Mλ which we +denote as (Lλ, ∇). According to Remark 2.4, we have π∗ : Γ(Mλ, (Lλ)−1) → Γ(Mλ, L−1 +λ ) +and dually we have π∗ : Γc(Mλ, L−1 +λ )′ → Γc(Mλ, (Lλ)−1)′. +In fact, our above claim δ = π∗η holds true for any T n-principal bundle P → B. Let +π : P → B be a principal T n-bundle with a fiberwise T n-invariant volume form dθ such +that +� +P dθ = 1 ∈ C∞(B). Let (E, ∇) be a line bundle over B. We can push-forward +sections of π∗E to sections of E with respect to dθ. Furthermore we have: +Lemma 3.8. Taking δ ∈ Γc(P, (π∗E)−1)′, if ∇ξ#δ = 0 for any ξ ∈ t, then there exists a +distributional section η ∈ Γc(B, E−1)′ such that +δ = π∗η. +Proof. By partition of unity, it is enough to show that on any open subset U of B, for +δ ∈ Γc(π−1(U), (π∗E)−1)′, if ∇ξ#δ = 0 for any ξ ∈ t, there exists a distributional section +η ∈ Γc(U, E−1)′ such that δ = π∗η. That is, for any τ ∈ Γc(π−1(U), (π∗E)−1), +δ(τ) = η(π∗τ). +Fixing a local frame σ0 ∈ Γ(U, E) of E on an open subset U ⊂ B, let σ := π∗σ0 and +σ−1 be the corresponding local frames of π∗E and (π∗E)−1 respectively on π−1(U). With +respect to local frames σ and σ−1, the distributional section δ ∈ Γc(π−1(U), (π∗E)−1)′ +corresponds to the distributional function fδ ∈ Γc(π−1(U), C)′, where fδ is determined by: +(3.30) +fδ(gτ) = δ(gτσ−1), +for any text function gτ ∈ Γc(π−1(U), C). We restrict our attention to show that ∇ξ#δ = 0 +if and only if ξ#fδ = 0. Applying the equation (3.2) to line bundle π∗L and trivial bundle +over π−1(U) respectively, it can be seen that: +(3.31) +� +∇ξ#δ +� +(τ) = −δ +� +(div ξ#)τ + ∇ξ#τ +� +, + +GQ ASSOCIATED TO MIXED POLARIZATIONS ON K¨AHLER MANIFOLDS WITH T-SYMMETRY 17 +and +(3.32) +(ξ#fδ) (gτ) = fδ +� +− +� +div ξ#gτ + ξ#gτ +�� +. +Since ∇ξ#σ = ∇ξ#(π∗σ0) = 0, one has ∇ξ#σ−1 = 0 and +(3.33) +∇ξ#τ = ∇ξ# +� +gτσ−1� += +� +ξ#gτ +� +σ−1. +Combining equations ( 3.30), ( 3.31), (3.32), with (3.33), we obtain that +(3.34) +� +∇ξ#δ +� +(τ) = (ξ#fδ) (gτ) , +for any τ ∈ Γc (π−1(U), (π∗E)−1). +It turns out that ∇ξ#δ = 0 iff ξ#fδ = 0 for any +ξ ∈ t. Then by Lemma 3.9, there exists a distributional function fη ∈ Γc(U, C)′ such that +fδ = π∗(fη). Define η ∈ Γc(U, (π∗E)−1)′ to be distributional section associated to fη with +respect to the nowhere vanishing section σ−1 +0 , that is η(hτσ−1 +0 ) = fη(hτ). For any test +section τ ∈ Γc(π−1(U), π∗E), it can be check that: +δ(τ) = (π∗η)τ. +Therefore we have δ = π∗η. +□ +Lemma 3.9. Let π : P → B be the principal T n-bundle and let U be any open subset of +B. Let δ ∈ Γc(π−1(U), C)′ be a distributional function. If ξ#δ = 0 for any ξ ∈ t, there +exists a distributional function η ∈ Γc(U, C)′, such that δ = π∗η. Namely, +δ(g) = η(π∗g), ∀ g ∈ Γc(π−1(U), C). +Proof. For any δ ∈ Γc(π−1(U), C)′, there exist δǫ ∈ Γ(π−1(U), C) (see [6, 11]) such that +limǫ→0 δǫ = δ and +(3.35) +(ξ#δǫ)(g) = (ξ#δ)(gǫ), +for any g ∈ Γc(π−1(U), C). As ξ#δ = 0, we obtain ξ#δǫ = 0. Since δǫ is smooth, there +exists a smooth function ηǫ ∈ Γ(U, C), such that δǫ = π∗ηǫ ∈ Γ(π−1(U), C). It can be check +that +lim +ǫ→0 ηǫ(h) = lim +ǫ→0 δǫ(π∗h), +for any h ∈ Γc(U, C). Hence we have limǫ→0 ηǫ exists and denoted by η. It follows +δ = π∗η. +□ + +18 +LEUNG AND WANG +3.2.2. Pulling back commutes with taking divergence. Fix λ ∈ t∗ +Z,reg, let α ∈ Ω1(Mλ, t) be +a connection on the principal T n-bundle π : Mλ → Mλ. For any ζ ∈ Γ(Mλ, TMλ), the +horizontal lifting of ζ with respect to α is denoted by ˜ζ. Denote the divergence of ζ on Mλ +with respect to volλ by div ζ (i.e. div ζ = Lζ volλ +volλ ) and denote the divergence of ˜ζ on Mλ +with respect to volλ by div ˜ζ (i.e. div ˜ζ = +L˜ζ volλ +volλ ). +Lemma 3.10. Let div ζ and div ˜ζ be defined as above. Then we have +π∗(div ζ) = div ˜ζ, +as smooth functions on Mλ. +Proof. As T n is abelian, the horizontal lifting ˜ζ of ζ with respect to the connection one +form α is T n-invariant. That is +(3.36) +Lξ# ˜ζ = 0, +for all ξ ∈ t, where ξ# is the fundamental vector field associate to ξ. According to the +property of principal T n-connection and equation (3.37), we have +(3.37) +(L˜ζα)(ξ#) = L˜ζ(α(ξ#)) − α(L˜ζξ#) = 0. +Recall that volλ = π∗ volλ ∧αn. By equation (3.37), one has +(3.38) +L˜ζ volλ = (L˜ζ(π∗ volλ)) ∧ αn. +On the other hand, by Cartan formula and volλ being the volume form on B, we have: +(3.39) +L˜ζ(π∗ volλ) = d(i˜ζ(π∗ volλ)) = π∗(Lζ volλ), +Recall that +(3.40) +Lζ volλ = (div ζ) volλ, L˜ζ volλ = (div ˜ζ) volλ . +Combining equation (3.38), (3.39), with (3.40), one has +(div ˜ζ) volλ = L˜ζ volλ = π∗(Lζ volλ) ∧ αn += π∗(div ζ)π∗ volλ ∧αn += π∗(div ζ) volλ . +Therefore we obtain: π∗(div ζ) = div ˜ζ. +□ +Theorem 3.11. For any λ ∈ t∗ +Z,reg and distributional function η ∈ Γc(Mλ, C)′, if ∇ξ(π∗η) = +0, for any ξ ∈ Γ(Mλ, Pmix), then we have ∇ζη = 0, for all ζ ∈ Γ(Mλ, TM0,1 +λ ). + +GQ ASSOCIATED TO MIXED POLARIZATIONS ON K¨AHLER MANIFOLDS WITH T-SYMMETRY 19 +Proof. To prove this statement, fixing the connection one form α ∈ Ω1(Mλ, t) on principal +T n-bundle π : Mλ → Mλ, we denote the horizontal lifting of ζ with respect to the connec- +tion α by ˜ζ, for any ζ ∈ Γ(Mλ, TM0,1 +λ ). In order to show ∇ζη = 0, it is enough to show , +for any test function φ ∈ Dc(Mλ), +(∇ζη) (φ) = +� +∇˜ζ (π∗η) +� +(π∗φ) . +Let volλ and volλ be volume forms of Mλ and Mλ respectively as defined before. +In +particular, volλ = π∗ volλ ∧αn with respect to the principal T n-connection α ∈ Ω1(Mλ, t). +Applying equation (3.2) to the trivial bundle of Mλ and Mλ respectively, we obtain: +(3.41) +(∇ζη) (φ) = η (− (div ζ) φ − ∇ζφ) , +and +(3.42) +� +∇˜ζ (π∗η) +� +(π∗φ) = (π∗η) +� +− +� +div ˜ζ +� +π∗φ − ∇˜ζ (π∗φ) +� +, +where div ζ (div ˜ζ resp.) is the divergence of ζ (˜ζ resp.) with respect to volλ (volλ resp.). +According to the Remark 2.4, we have: +(3.43) +(π∗η) (π∗ (− (div ζ) φ − ∇ζφ)) = η (− (div ζ) φ − ∇ζφ) . +By Lemma 3.10, +(3.44) +π∗(div ζ) = div ˜ζ. +Note that π∗(ζ(φ)) = π∗ζ(π∗φ). By equation (3.44), one has +(3.45) +π∗ (− (div ζ) φ − ∇ζφ) = − +� +div ˜ζ +� +π∗φ − ∇˜ζ (π∗φ) . +Furthermore: +(3.46) +(π∗η) (π∗ (− (div ζ) φ − ∇ζφ)) = (π∗η) +� +− +� +div ˜ζ +� +π∗φ − ∇˜ζ (π∗φ) +� +. +Combining (3.41), (3.42), (3.43), with (3.46), we are able to conclude: +(3.47) +(∇ζη) (φ) = +� +∇˜ζ (π∗η) +� +(π∗φ) . +Then we restrict our attention to show ˜ζ ∈ Γ(Mλ, Pmix). As T n acts freely on Mλ, Mλ×t ∼= +IR|Mλ. Note that π∗(˜ζ) = ζ ∈ Γ(Mλ, TM0,1 +λ ) and α(˜ζ) = 0. Since TpMλ ⊗ C ⊂ (DC)p and +(Pmix)p = (DC ∩ TM0,1)p ⊕ (IC)p, for any p ∈ M0, we have ˜ζ ∈ Γ(Mλ, Pmix). According to +what we assume, we have ∇˜ζ (π∗η) = 0. Therefore, by equation (3.47), we have +(∇ζη) (φ) = +� +∇˜ζ (π∗η) +� +(π∗φ) = 0, ∀φ ∈ Dc(Mλ), ζ ∈ Γ(Mλ, TM0,1 +λ ). +□ + +20 +LEUNG AND WANG +3.2.3. Building the isomorphism H0(Mλ, Lλ) ∼= Hmix,λ. Recall given any s ∈ H0(Mλ, Lλ), +by Proposition 3.7, the associated distributional section ı(δs) belongs to Hmix,λ. Therefore +we can define a homomorphism +κ : H0(Mλ, Lλ) → Hmix,λ +given by s �→ κ(s) = ı(δs), where ı : Γc(Mλ, (Lλ)−1)′ ֒→ Γc(M, L−1)′ is the natural +inclusion. It can be checked that κ is injective. +Theorem 3.12. For any λ ∈ t∗ +Z,reg, κ : H0(Mλ, Lλ) → Hmix,λ is an isomorphism. +Proof. Given any ˜δ ∈ Hmix,λ, we need to construct s ∈ H0(Mλ, Lλ) such that ˜δ = κ(s). +Firstly we show that, there exists δ ∈ Γc(Mλ, (Lλ)−1)′ such that ˜δ = ı(δ) as follows: we +define the distributional section δ ∈ Γc(Mλ, (Lλ)−1)′ by: +δ(τ) = ˜δ(˜τ), +for any τ ∈ Γc(Mλ, (Lλ)−1), where ˜τ ∈ Γc(M, L−1) is any test section satisfying ˜τ|Mλ = τ +By Corollary 3.3, δ is well defined. Moreover, one has +(3.48) +˜δ = ı(δ). +That is, for any test section ˜τ ′ ∈ Γc(M, L−1), (ı(δ))(˜τ ′) = δ(˜τ ′|Mλ) = ˜δ(˜τ ′). Secondly we +show that there exists η ∈ Γc(Mλ, L−1 +λ )′ such that δ = π∗η, where π : Mλ → Mλ is the +projection. For any ˜δ ∈ Hmix,λ, since ξ# ∈ Γ(M, Pmix), we have ∇ξ#˜δ = 0, for any ξ ∈ t. +By Theorem 3.6, one has +(3.49) +0 = ∇ξ#˜δ = ∇ξ#(ı(δ)) = ı(∇ξ#δ), ∀ξ ∈ t. +By the injectivity of ı, we obtain, for any ξ ∈ t, +(3.50) +∇ξ#δ = 0. +According to Lemma 3.8, there exists a distributional section η ∈ Γc(Mλ, L−1 +λ )′, such that +(3.51) +δ = π∗η. +Next we show that there exists a holomorphic section s ∈ H0(Mλ, Lλ) such that η = ι(s) +under the inclusion map ι : Γ(Mλ, Lλ) → Γc(Mλ, L−1 +λ )′ with respect to volλ. +By the +definition of Pmix, for any ξ ∈ Γ(M, Pmix), we have ξ|Mλ ∈ Γ(Mλ, Pmix) ⊂ Γ(Mλ, TMλ⊗C). +By abuse of notation, we denote ξ|Mλ by ξ. According to Theorem 3.11 and equation (3.51), +we have +(3.52) +∇ξ˜δ = ∇ξ(ı(δ)) = ı(∇ξδ) = ı(∇ξ(π∗η)). +Since ˜δ ∈ Hmix,λ, ∇ξ˜δ = 0, for ξ ∈ Γ(M, Pmix). By the injectivity of ı and equation (3.52), +we obtain: +(3.53) +∇ξ(π∗η) = 0, ∀ξ ∈ Γ(Mλ, Pmix). + +GQ ASSOCIATED TO MIXED POLARIZATIONS ON K¨AHLER MANIFOLDS WITH T-SYMMETRY 21 +Then by Theorem 3.11, we have ∇ζη = 0, for any ζ ∈ Γ(Mλ, TM0,1 +λ ). +This implies +∇0,1η = 0. By the regularity of elliptic operator ∆ = ¯∂∗ ¯∂, η is smooth. Therefore there +exists a holomorphic section s ∈ H0(Mλ, Lλ) such that η = ι(s) under the inclusion map +ι : Γ(Mλ, Lλ) → Γc(Mλ, L−1 +λ )′ with respect to volλ. It is remain to show ˜δ = κ(s). +According to the above discussion, we have ˜δ = ı(π∗(ι(s))). Recall that κ(s) = ı(δs), +where δs with respect to volume form volλ is defined by +(3.54) +δs(τ) = +� +Mλ⟨π∗s, τ⟩ volλ, +for any test section τ ∈ Γc(Mλ, Lλ)′. By the injectivity of ı, to show ˜δ = κ(s), it is enough +to show: +(3.55) +π∗(ι(s)) = δs. +By remark 2.4, we have +(3.56) +� +Mλ⟨π∗s, τ⟩ volλ = (π∗s)(τ) = s(π∗τ) = +� +Mλ +⟨s, π∗τ⟩ volλ +And +(3.57) +π∗(ι(s))(τ) = (ι(s))(π∗τ) = +� +Mλ +⟨s, π∗τ⟩ volλ . +According to equations (3.54 ), (3.56), and (3.57), we have π∗(ι(s)) = δs. +□ +4. Appendix +4.1. Polarizations on symplectic manifolds. A step in the process of geometric quanti- +zation is to choose a polarization. We first recall the definitions polarizations on symplectic +manifolds (M, ω) (See [12, 14]). All polarizations discussed in this subsection are smooth. +Definition 4.1. A complex polarization on M is a complex sub-bundle of the complexified +tangent bundle TM ⊗ C satisfying the following conditions: +(1) P is involutive, i.e. if u, v ∈ Γ(M, P), then [u, v] ∈ Γ(M, P); +(2) for every x ∈ M, Px ⊆ TxM ⊗ C is Lagrangian; and +(3) rkR (P) := rank(P ∩ P ∩ TM) is constant. +Furthermore, P is called +· real polarization, if P = P, i.e. rkR (P) = m; +· K¨ahler polarization, if P ∩ P = 0, i.e. rkR (P) = 0; +· mixed polarization, if 0 < rank(P ∩ P ∩ TM) < m, i.e. 0 < rkR (P) < m. + +22 +LEUNG AND WANG +4.2. Singular polarizations on symplectic manifolds. In subsection, we review the +definitions of singular polarizations, smooth sections of singular polarizations which were +used in the proof of the main results (see [9]). +Definition 4.2. P ⊂ TM ⊗ C is a singular complex distribution on M if it satisfies: Pp is +a vector subspace of TpM ⊗ C, for all point p ∈ M. Such a P is called smooth on an open +subset ˇ +M ⊂ M if P| ˇ +M is a smooth sub-bundle of the tangent bundle T ˇ +M ⊗ C. +Remark 4.3. In this paper, we only consider such distributions with mild singularities in +the sense that they are only singular outside an open dense subset ˇ +M ⊂ M. Under our +setting, we define smooth sections of singular distributions and involutive distributions as +follows. +Definition 4.4. Let P be a singular complex distribution of TM ⊗C. For any open subset +U of M, the space of smooth sections of P on U is defined by the smooth section of TM ⊗C +with value in P, that is, +Γ(U, P) = {v ∈ Γ(U, TM ⊗ C) | vp ∈ (P)p, ∀p ∈ U}. +Definition 4.5. Let P be a singular complex distribution on M. P is involutive if it +satisfies: +[u, v] ∈ Γ(M, P), for any u, v ∈ Γ(M, P). +Definition 4.6. Let P be a singular complex distribution P on M and smooth on ˇ +M. +Such a P is called a singular polarization on M, if it satisfies the following conditions: +(a) P is involutive, i.e. if u, v ∈ Γ(M, P), then [u, v] ∈ Γ(M, P); +(b) for every x ∈ ˇ +M, Pp ⊆ TpM ⊗ C is Lagrangian; and +(c) the real rank rkR(P) := rank(P ∩ P ∩ TM)| ˇ +M is a constant. +Furthermore, such a singular P is called +· real polarization, if P| ˇ +M = P| ˇ +M, i.e. rkR(P| ˇ +M) = m; +· K¨ahler polarization, if P ˇ +M ∩ P| ˇ +M = 0 on ˇ +M, i.e. r (P| ˇ +M) = 0; +· mixed polarization, if 0 < rank(P ∩ P ∩ TM)| ˇ +M < m, i.e. 0 < rkR(P| ˇ +M) < m. +4.3. Coisotropic embedding theorem. We review the coisotropic embedding theorem +studied by Guillemin in [2], which was used in the proof of taking divergence. Let (M, ω) +be a symplectic manifold of dimensional 2m equipped with Hamiltonian T n-action with +moment map µ. Without loss of generality, we assume n = 1. Choose a principal T 1- +connection α ∈ Ω1(M0, t) on M0, where M0 = µ−1(0). Consider M0 as a submanifold of +M0 × R via the embedding +i : M0 → M0 × R, i(p) = (p, 0). + +GQ ASSOCIATED TO MIXED POLARIZATIONS ON K¨AHLER MANIFOLDS WITH T-SYMMETRY 23 +On the product space ˜ +M = M0 × (−ǫ, ǫ), the two-form +˜ω = π∗ω0 + d(tα), −ǫ < t < ǫ +is symplectic on ˜ +M and satisfies i∗˜ω = π∗ω0. Extending the T 1-action on M0 to M0×t∗ in a +trivial manner. Then ˜ω is T 1-invariant and that the action of T 1 on M0 ×t∗ is Hamiltonian +with moment map +µ0 : M0 × t∗ → t∗, (p, t) �→ t. +Theorem 4.7. [3, Theorem 2.2] In a neighborhood of M0, the Hamiltonian T n-spaces +(M, ω) and ( ˜ +M, ˜ω) are isomorphic. +4.4. Geometric quantization commute with symplectic reduction. In this subsec- +tion, we review the work on geometric quantization commute with symplectic reduction by +Guillemin and Sternberg in [3]. Let (L, ∇) and (Lλ, ∇λ) be the pre-quantum line bundle +on M and Mλ respectively as discussed before, for λ ∈ t∗ +Z,reg. Then the quantum space +HPJ associated to PJ is the space of J-holomorphic sections of L: +HPJ = {s ∈ Γ(M, L) | ¯∂Js = 0} = H0(M, L). +One can perform two processes on the pre-quantum line bundle (L, ∇); one is geometric +quantization, and the other is symplectic reduction. Guillemin and Sternberg in [3] showed +that these two processes commute with each other, that is, +(4.1) +(HPJ)λ ∼= HPJ,λ, +where (HPJ)λ (Jλ-holomorphic sections of Lλ) is the λ-weight subspace of HPJ and HPJ,λ +is the quantum space associated to reduced K¨ahler polarization PJ,λ, i.e. +(4.2) +HPJ,λ = {s ∈ Γ(Mλ, Lλ) | ¯∂Jλs = 0} = H0(Mλ, Lλ). +References +1. T. Baier, C. Florentino, J. M. Mour˜ao and J. P. Nunes, Toric K¨ahler metrics seen from infinity, +quantization and compact tropical amoebas, J. Diff. Geom., 89 (3), 411-454, 2011. +2. V. Guillemin, Moment Maps and Combinatorial Invariants of Hamiltonian T n- spaces, Progress in +Math., 122, Birkh¨auser, 1994. +3. V. Guillemin and S. Sternberg, Geometric Quantization and Multiplicities of Group Representations, +Inventiones mathematicae, 67.3 (1982): 515-538. +4. V. Guillemin and S. Sternberg, Symplectic Techniques in Physics, Cambridge University Press, Cam- +bridge University Press, Cambridge, 1984 +5. M. D. Hamilton, Locally toric manifolds and singular Bohr-Sommerfeld leaves, Mem. Amer. Math. +Soc. 207 (2010), no. 971, vi+60pp. +6. J. Horv´ath, Topological vector spaces and distributions, Courier Corporation, 2012. +7. A. A. Kirillov, Geometric quantization, in: Encyclopaedia of Mathematical Sciences, vol. 4 Dynamical +systems, Springer-Verlag, 1990, 137-172. + +24 +LEUNG AND WANG +8. B. Kostant, Quantization and unitary representations, In: Modern analysis and applications. Lecture +Notes in Math., Vol. 170, pp. 87-207. Berlin-Heidelberg-Mew York: Springer 1970. +9. N.C. Leung and D. Wang Geodesic rays in space of K¨ahler metrics with T-symmetry, arXiv preprint +arXiv: 2211.05324 (2022). +10. J. Marsden and A. Weinstein, Reduction of symplectic manifolds with symmetry. Report on Math. +Phys. 5,121-130 (1974). +11. W. Rudin, Functional analysis, Second edition. International Series in Pure and Applied Mathematics. +McGraw-Hill, Inc., New York, 1991. +12. D. Simms and N. Woodhouse, Lectures on geometric quantization, Lectures Notes in Physics, Vol. 53. +Berlin-Heidelberg-New York: Springer 1976. +13. A. Weinstein, Symplectic manifolds and their Lagrangian submanifolds, Advances in Math. 6 (1971), +329-346. +14. N. M. J. Woodhouse, Geometric quantization, Second Edition, Clarendon Press, Oxford, 1991. +The Institute of Mathematical Sciences and Department of Mathematics, The Chinese +University of Hong Kong, Shatin, Hong Kong +Email address: leung@math.cuhk.edu.hk +The Institute of Mathematical Sciences and Department of Mathematics, The Chinese +University of Hong Kong, Shatin, Hong Kong +Email address: dwang@math.cuhk.edu.hk + diff --git a/-NAzT4oBgHgl3EQfFfpP/content/tmp_files/load_file.txt b/-NAzT4oBgHgl3EQfFfpP/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..41dae0d512c1389fe8556f947a91b538f973f4f2 --- /dev/null +++ b/-NAzT4oBgHgl3EQfFfpP/content/tmp_files/load_file.txt @@ -0,0 +1,818 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf,len=817 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='01011v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='SG] 3 Jan 2023 GEOMETRIC QUANTIZATIONS ASSOCIATED TO MIXED POLARIZATIONS ON K¨AHLER MANIFOLDS WITH T-SYMMETRY NAICHUNG CONAN LEUNG, AND DAN WANG Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Let M be a compact K¨ahler manifold equipped with a pre-quantum line bundle L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' In [9], using T -symmetry, we constructed a polarization Pmix on M, which generalizes real polarizations on toric manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' In this paper, we obtain the following results for the quantum space Hmix associated to Pmix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' First, Hmix consists of distri- butional sections of L with supports inside µ−1(t∗ Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' This gives Hmix = � λ∈t∗ Z Hmix,λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Second, the above decomposition of Hmix coincides with the weight decomposition for the T -symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Third, an isomorphism Hmix,λ ∼= H0(M �λ T, L �λ T ), for regular λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Namely, geometric quantization commutes with symplectic reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Introduction Let (M, ω) be a symplectic manifold equipped with a pre-quantum line bundle (L, ∇), in particular F∇ = −iω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' A polarization P on M is an integrable Lagrangian subbundle of TM ⊗ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Geometric quantization assigns a Hilbert space HP to these data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Namely, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='1) HP = Γ(M, L) ∩ Ker(∇)|P, where Γ(M, L) is the space of smooth sections of L 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' When (M, ω, J) is K¨ahler, we have a K¨ahler polarization PJ = T 0,1 J M, and HPJ = H0(M, L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' If, in addition, M admits T- symmetry, we constructed a polarization Pmix using T-symmetry in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' In this paper, we study the quantum space Hmix associated to Pmix on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Concretely, we have the following assumption throughout this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' (∗) (M, ω, J) is a compact K¨ahler manifold of real dimension 2m equipped with an effective Hamiltonian n-dimensional torus action ρ : T n → Diff(M, ω, J) by isome- tries with moment map µ : M → t∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Let (L, ∇, h) be a T n-invariant pre-quantum line bundle on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Recall from [9], a (singular) polarization Pmix is constructed in this situation, which is, Pmix = (PJ ∩ DC) ⊕ IC, where DC = (Ker dµ) ⊗ C and IC = (Im dρ) ⊗ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' When n = m, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' M is a toric variety, Pmix coincides with the singular real polarization defined by moment map and Hmix is the space of Bohr-Sommerfeld states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' 1In fact we need to allow distributional sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' 1 2 LEUNG AND WANG Recall that there is a natural way to embed the space of smooth sections into the space of distributional sections using the Liouville measure volM = ωm m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' That is, for any test section τ ∈ Γc(M, L−1), ι : Γ(M, L) → Γc(M, L−1)′, s �→ (ιs)(τ) = � M ⟨s, τ⟩ volM .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Then the quantum space Hmix can be described as: Hmix = Γc(M, L−1)′ ∩ Ker(∇)|Pmix, Our first result says that Hmix consists of distributional sections with supports inside µ−1(t∗ Z) and Hmix,λ is the λ-weight subspace of Hmix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='2) Under the assumption (∗), (1) given any δ ∈ Hmix, we have supp δ ⊂ � λ∈t∗ Z µ−1(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' This gives the following decomposition Hmix = � λ∈t∗ Z Hmix,λ, where Hmix,λ = {δ ∈ Hmix | supp δ ⊂ µ−1(λ)};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' (2) for any λ ∈ t∗ Z, Hmix,λ is a λ-weight subspace in Hmix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Therefore the decomposition Hmix = � λ∈t∗ Z Hmix,λ is the weight decomposition with respect to T n-action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' When n = m, this is a result for toric variety (see [1, 5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Inspired by the works (see [3]) of Guillemin and Sternberg that geometric quantizations commute with symplectic reductions, we give a geometric description of Hmix,λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Our main result (Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='5 or Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='12) says that when λ is an integral regular value of µ, denoted as λ ∈ t∗ Z,reg, we have Hmix,λ ∼= H0(Mλ, Lλ), where (Mλ, Lλ) = (M �λ T, L �λ T) is the symplectic reduction of (M, L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Concretely, Mλ = µ−1/T, we also denote the level set µ−1(λ) as Mλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' The restriction (Lλ, ∇) of pre- quantum line (L, ∇) to Mλ can be descended to the quotient space Mλ denoted by (Lλ, ∇) (see [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Our second result states that, for any s ∈ H0(Mλ, Lλ), there is an associated distribu- tional section δs ∈ Γc(Mλ, (Lλ)−1)′ such that ı(δs) lies in Hmix,λ, where ı : Γc(Mλ, (Lλ)−1)′ ֒→ Γc(M, L−1)′ is the natural inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' (Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='4) For any λ ∈ t∗ Z,reg and s ∈ H0(Mλ, Lλ), we define the distributional section δs ∈ Γc(Mλ, (Lλ)−1)′ associated to s as follows: for any τ ∈ Γc(Mλ, (Lλ)−1), δs(τ) = � Mλ ⟨π∗s, τ⟩ volλ, GQ ASSOCIATED TO MIXED POLARIZATIONS ON K¨AHLER MANIFOLDS WITH T-SYMMETRY 3 where volλ is the volume form on Mλ and π : Mλ → Mλ is the quotient map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' To see ı(δs) ∈ Hmix,λ, we need to investigate the interaction between the covariant derivative on the space of smooth sections of L and the covariant derivative on the space of distributional sections of Lλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Our third result (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='6) says that the following diagram Γ(M, L) Γc(M, L−1)′ Γc(M, L−1)′ Γ(Mλ, Lλ) Γc(Mλ, (Lλ)−1)′ Γc(Mλ, (Lλ)−1)′, volM ∇ξ volλ ı ∇ξ ı is a commutative diagram, for any ξ ∈ Γ(M, TM ⊗ C) satisfying ξ|Mλ ∈ Γ(Mλ, TMλ ⊗ C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' In order to show the above diagram commutes, we use the coisotropic embedding theorem due to Weinstein [13] and further studied by Guillemin in [2] to relate the volume forms volM and volλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Then we obtain the following theorem: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='6) For any λ ∈ t∗ Z,reg, δ ∈ Γc(Mλ, (Lλ)−1)′ and ξ ∈ Γ(M, TM ⊗ C) satisfying ξ|Mλ ∈ Γ(Mλ, TMλ ⊗ C), we have (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='2) ∇ξ(ı(δ)) = ı(∇ξδ), where ı : Γc(Mλ, (Lλ)−1)′ ֒→ Γc(M, L−1)′ is the natural inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' (Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='7) For any λ ∈ t∗ Z,reg and s ∈ H0(Mλ, Lλ), we have ı(δs) ∈ Hmix,λ, where ı : Γc(Mλ, (Lλ)−1)′ ֒→ Γc(M, L−1)′ is the natural inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' This allows us to define a map κ : H0(Mλ, Lλ) → Hmix,λ by s �→ κ(s) = ı(δs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Finally we show that κ is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='12) For any λ ∈ t∗ Z,reg, κ : H0(Mλ, Lλ) → Hmix,λ is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' We show the surjectivity of κ via the following steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' First, we show that any element ˜δ in Hmix,λ is locally a delta function along µ−1(λ), and does not involve any derivative of delta functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' This implies ˜δ = ı(δ), for some δ ∈ Γc(Mλ, (Lλ)−1)′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Second, we show that T n-invariant distributional sections of Lλ can be descended to distributional sections of Lλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' That is, for any δ ∈ Γc(Mλ, (Lλ)−1)′ satisfying ∇ξ#δ = 0, there exists a distributional section η ∈ Γc(Mλ, L−1 λ )′ such that δ = π∗η (Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Third, we show that if ∇ζ(π∗η) = 0 for all ζ ∈ Γ(Mλ, Pmix), then η is ¯∂-closed (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' By the regularity of elliptic operator ∆ = ¯∂∗ ¯∂, we have η is smooth (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' η = ι(s), for some s ∈ H0(Mλ, Lλ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Finally, we show that ˜δ = κ(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' 4 LEUNG AND WANG 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' We are grateful to Siye Wu for insightful comments and useful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Wang would like to thank Qingyuan Jiang, Yutung Yau and Ki Fung Chan for many helpful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' This research was substantially supported by grants from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' CUHK14301619 and CUHK14301721) and a direct grant from the Chinese University of Hong Kong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Preliminary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' The Marsden-Weistein construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' In this subsection, we review the basic con- cepts of Hamiltonian action and symplectic reduction in order to fix the notations in our setting (for more details, the reader can refer to [3], [10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Hamiltonian action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Let (M, ω) be a compact symplectic manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' For f ∈ C∞(M, R), the Hamiltonian vector field Xf associated to f is determined by ıXf ω = −df.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' This gives a Lie algebra homeomorphism ψ : (C∞(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' R), {·, ·}) → (Vect(M, ω), [·, ·]) defined by ψ(f) = Xf, where {, } is the Poisson bracket of two functions f, g ∈ C∞(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' R) determined by {f, g} = ω(Xf, Xg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Let T n be a torus of real dimension n and ρ : T n → Diff(M, ω) an action of T n on M which preserves ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Let t be the Lie algebra of T n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Differentiating ρ at the identity element, we have dρ : t → Vect(M, ω), ξ �→ ξ#, where t is the Lie algebra of T n and ξ# is called the fundamental vector field associated to ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' The action of T n on M is said to be Hamiltonian if dρ factors through ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Let ⟨, ⟩ : t∗ × t → R be the natural pairing between t∗ and t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' For each point p ∈ M, we can associate an element µ(p) ∈ t∗ by the formula ⟨µ(p), ξ⟩ = −µξ(p), ∀ξ ∈ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' This gives us a moment mapping µ : M → t∗ which is a T n-equivariant map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Symplectic reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' We denote the set of regular values of µ by t∗ reg, that is, t∗ reg = {λ ∈ t∗| λ is a regular value of µ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' For any λ ∈ t∗ reg, denote the level set µ−1(λ) by Mλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Then Mλ is a T n-invariant coisotropic submanifold (i : Mλ ֒→ M) and the action of T n is locally free (see [10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' For simplicity, we assume T n acts freely on Mλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Then the projection mapping π : Mλ → Mλ is a principal T n-fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Moreover there exists a unique symplectic form ωλ on Mλ such that π∗ωλ = i∗ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Denote the volume form 1 (m−n)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='ωm−n λ on Mλ by volλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Take a connection GQ ASSOCIATED TO MIXED POLARIZATIONS ON K¨AHLER MANIFOLDS WITH T-SYMMETRY 5 α ∈ Ω1(Mλ, t) on Mλ, π∗ volλ ∧αn is a volume form on Mλ denoted by volλ (here αn is a n-form on Mλ defined by α∧···∧α v , where v ∈ ∧nt is a T n-invariant top form on t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' (Mλ = µ−1(λ), volλ) M (Mλ = µ−1(λ)/T n, volλ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' i π 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Pre-quantum data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' In this subsection, we first review the definition of T n-invariant pre-quantum line bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Then we restate the result that the pre-quantum line bundle can always be descended to the reduction space by Guillemin and Sternberg in our setting (K¨ahler manifold equipped with T n-symmetry).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Let (M, ω, J) be a symplectic manifold, a pre-quantum line bundle (L, ∇, h) on M is a complex line bundle L together with a Hermitian metric h and Hermitian con- nection ∇, such that the curvature form F∇ = −iω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' The existence of a pre-quantum line bundle L on M is equivalent to [ ω 2π] being integral (see [8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' When M is K¨ahler, L is an ample holomorphic line bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' There is a canonical representation of the Lie algebra t on space of smooth sections of L given by the operators (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='1) ∇ξ# + iµξ, ξ ∈ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' The pre-quantum line bundle is said to be T n-invariant if there exists a global action of T n on L such that the induced action of t is given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' It is always possible if the T n-action on M is Hamiltonian (see [8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Let tZ be the kernel of the exponential map exp : T n → t and t∗ Z ⊂ t∗ be the dual lattice of tZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' We denote the set of integral regular values of µ by t∗ Z,reg, that is, t∗ Z,reg = t∗ reg ∩ t∗ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Guillemin and Sternberg in [3] showed that there are associated pre-quantum data on the reduction space Mλ, for λ ∈ t∗ Z,reg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' [3, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='2] There is a unique line bundle with connection (Lλ, ∇λ) on Mλ such that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='2) π∗Lλ = i∗L =: Lλ, and π∗∇λ = i∗∇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' [3, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='4] The curvature of the connection, ∇λ, is the symplectic form ωλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' 6 LEUNG AND WANG Therefore we have the following commuting diagram: (Lλ, i∗∇) (L, ∇) t∗ Z,reg (Lλ, ∇λ) (Mλ, volλ) (M, volM) t∗ (Mλ, volλ) π i µ By abuse of notations, we denote both i∗∇ and ∇λ by ∇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' In order to pull-back distribu- tional sections from Mλ to Mλ later, we first recall how to push-forward sections of line bundle Lλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Let π : P → B be a principal T n-bundle over B, E → B a line bundle over B, and π∗E → P the pullback line bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Then we can define the dual map π∗ : Γc(B, L−1)′ → Γc(P, (π∗L)−1)′, η �→ π∗η by (π∗η)(τ) = η(π∗τ), for any τ ∈ Γc(P, L−1) Throughout this paper, we fix a T n-invariant n-form dθ on T n such that � T n dθ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' When we deal with the pull-back of distribution sections of Lλ, we mean in the sense of Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='4 with respect to dθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Complex structures on symplectic reduction spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' In order to study the relationship between geometric quantization associated to Pmix and symplectic reduction, we recall the work on the existence of complex structures on symplectic reduction spaces Mλ (see [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Recall that the anti-holomorphic Lagrangian sub-bundle TM0,1 J ⊂ TM ⊗ C is a K¨ahler polarization denoted by PJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' We define F ⊂ TMλ ⊗ C by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='3) Fp = (PJ)p ∩ (TMλ ⊗ C)p, for any p ∈ Mλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' F can be descended to a bundle PJ,λ over the reduction space Mλ, which is a positive-definite Lagrangian sub-bundle of TMλ ⊗ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Under the assumption (∗), we have (DC ∩ PJ)p = Fp, for any p ∈ Mλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' [3, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='5] There is a positive-definite polarization PJ,λ canonically associated with PJ on the reduction space Mλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' By Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='2 and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='3 in [3], PJ,λ determined a complex structure Jλ on Mλ such that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='4) PJ,λ = TM0,1 λ , where TM0,1 λ is the anti-holomorphic sub-bundle of TMλ ⊗ C with respect to Jλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' GQ ASSOCIATED TO MIXED POLARIZATIONS ON K¨AHLER MANIFOLDS WITH T-SYMMETRY 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Polarizations on K¨ahler manifolds with T-symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' In [9], we constructed polarizations Pmix on K¨ahler manifolds with T-symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Throughout this sections, the existence of pre-quantum line L in (∗) is not needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' For any point p ∈ M, consider the map ρp : T n → M defined by ρp(g) = ρ(g)(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Let IR ⊂ TM be the singular distribution generated by fundamental vector fields in Im dρ, that is (IR)p = Im dρp(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Let DR = (Ker dµ) ⊂ TM be a distribution defined by the kernel of dµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Note that there is a K¨ahler polarization PJ = TM0,1 J associated to the complex structure J on a K¨ahler manifold (M, ω, J).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' [9, Definition] We define the singular distribution Pmix ⊂ TM ⊗ C by: (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='5) Pmix = (PJ ∩ DC) ⊕ IC, where DC = DR ⊗ C and IC = IR ⊗ C are the complexification of DR and IR respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Let Hp be the stabilizer of T n at point p ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Denote by ˇ M the union of n-dimensional orbits in M, that is, ˇ M = {p ∈ M| dim Hp = 0}, which is an open dense subset in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' [9, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='1] Under the assumption (∗), Pmix is a singular polarization and smooth on ˇ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Moreover, rank(Pmix ∩ ¯Pmix ∩ TM)| ˇ M = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' According to Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='6, Pmix is a singular real polarization on M, when n = m, namely, M is toric manifold;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Pmix is a singular mixed polarization on M, when 1 ≤ n < m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Main results We define the quantum space associated to the polarization Pmix = (PJ ∩ DC) ⊕ IC as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Let (L, ∇, h) be the pre-quantum line bundle on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' We first recall the definition of quantum space H associated to polarization P (see [14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' The quantum space H associated to polarization P is the following sub- space of Γc(M, L−1)′: H = {δ ∈ Γc(M, L−1)′ | ∇ξδ = 0, ∀ ξ ∈ Γ(M, P)}, where ∇ξ is the covariant derivative operator acting on the space of distributional sections defined by equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' In our setting, even through the polarization Pmix is singular, we continue to use the above definition for the quantum space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' We denote it by Hmix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' When n = m, M is toric variety and Pmix is a singular real polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' The definition of Hmix coincides with the definition of the quantum spaces associated to singular real polarizations studied in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' 8 LEUNG AND WANG Moreover, for any λ ∈ t∗, we define the subspace of those sections with supports on µ−1(λ) as: Hmix,λ = {δ ∈ Hmix | supp δ ⊂ µ−1(λ)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Distributional sections in Hmix,λ associated to sections in H0(Mλ, Lλ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' In this subsection, we first confirm that for any distributional section δ ∈ Hmix we have (see Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='2): supp δ ⊂ � λ∈t∗ Z µ−1(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' After extending the T n-action from the space of smooth sections to the space of distri- butional sections of L, we show that Hmix,λ is a λ-weight subspace of Hmix, for any λ ∈ t∗ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' This gives the weight decomposition of Hmix, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Hmix = � λ∈t∗ Z Hmix,λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Inspired by the work on geometric quantizations commute with symplectic reductions by Guillemin and Sternberg in [3], we expect to establish the isomorphism between H0(Mλ, Lλ) and Hmix,λ, where (Mλ, Lλ) is the symplectic reduction of M at a regular integral level λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' At the end of this subsection, given any holomorphic section s ∈ H0(Mλ, Lλ), we de- fine an associated distributional section δs ∈ Γc(Mλ, (Lλ)−1)′ (see Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='4) with respect to the volume form volλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Then we show that distributional sections in Hmix,λ as- sociated to sections in H0(Mλ, Lλ) (see Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' That is ı(δs) ∈ Hmix,λ, where ı : Γc(Mλ, (Lλ)−1)′ ֒→ Γc(M, L−1)′ is the natural inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' In order to study the quantum space Hmix, we recall how to extend covariant differenti- ation to distributional sections of L (see [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' First of all, there is a natural way to embed the space of smooth sections into the space of distributional sections using the Liouville measure volM = ωm m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' : ι : Γ(M, L) → Γc(M, L−1)′ s �→ (ιs)(τ) = � M ⟨s, τ⟩ volM .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Here ⟨, ⟩ : L×L−1 → C is the natural paring between L and L−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Let ∇ be the connection on L−1 such that d⟨s, τ⟩ = ⟨∇s, τ⟩+⟨s, ∇τ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' It is necessary to require that the operator ∇ acting on the distributional sections ι(s) which come from any smooth section s coincides with the operator ∇ acting on s, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' the following diagram Γ(M, L) Γc(M, L−1)′ Γ(M, L) Γc(M, L−1)′ ι ∇ξ ∇ξ ι GQ ASSOCIATED TO MIXED POLARIZATIONS ON K¨AHLER MANIFOLDS WITH T-SYMMETRY 9 commutes, for any ξ ∈ Γ(M, TM ⊗ C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Let div ξ be the divergence of ξ with respect to volM = ωm m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' , equivalently, Lξ(volM) = (div ξ) volM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' It can be seen that: 0 = � M Lξ(⟨s, τ⟩ volM) = � M (Lξ⟨s, τ⟩) volM + � M ⟨s, τ⟩Lξ(volM) = � M ⟨∇ξs, τ⟩ volM + � M ⟨s, ∇ξτ⟩ volM + � M ⟨s, τ⟩(div ξ) volM .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' This gives, for any smooth section s ∈ Γ(M, L) and smooth test section τ ∈ Γc(M, L−1), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='1) (∇ξι(s))(τ) = � M ⟨∇ξs, τ⟩ volM = � M ⟨s, −((div ξ)τ + ∇τ)⟩ volM .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' To determine ∇ξσ for a general distributional section σ not of the form ι(s), we built its transpose by integrating the operator ∇ξ by parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Namely, ∇ξ is characterized by its transpose t∇ξ as follows: for any τ ∈ Γc(M, L−1), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='2) (∇ξσ)(τ) = σ(t∇ξτ), with t∇ξτ = −(div ξτ + ∇ξτ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Similarly we can extend the T n-action on space of smooth sections to space of distri- butional sections such that the inclusion ι : Γ(M, L) ֒→ Γc(M, L−1)′ (with respect to the Liouville volume form volM) is T n-equivariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' That is, for any ξ ∈ t, the following diagram commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Γ(M, L) Γc(M, L−1)′ Γ(M, L) Γc(M, L−1)′ ξ· volM ξ· volM , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' ξ · (ι(s)) = ι(ξ · s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Namely, for any δ ∈ Γc(M, L−1)′, τ ∈ Γc(M, L−1), and ξ ∈ t, ξ · δ is characterized by: (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='3) (ξ · δ)(τ) = δ(ξ · τ), with ξ · τ = ∇ξ#τ + iµξτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' The T n-action on L preserve connection ∇, which implies that T n acts on Hmix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' We obtain the following results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Under the assumption (∗), (1) given any δ ∈ Hmix, we have supp δ ⊂ � λ∈t∗ Z µ−1(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' This gives the following decomposition Hmix = � λ∈t∗ Z Hmix,λ, where Hmix,λ = {δ ∈ Hmix | supp δ ⊂ µ−1(λ)};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' (2) for any λ ∈ t∗ Z, Hmix,λ is a λ-weight subspace in Hmix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Therefore the decomposition Hmix = � λ∈t∗ Z Hmix,λ is the weight decomposition with respect to T n-action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' 10 LEUNG AND WANG Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' (1) For a loop γb ⊂ T n specified by a vector b ∈ tZ, for any test function τ ∈ Γc(M, L−1), parallel transporting τ(p) with respect to the connection ∇ around a loop γb · p ⊂ M results in multiplication of τ(p) by e−2iπ⟨µ(p),b⟩, where ⟨, ⟩ : t∗ × t → R is the natural pairing between t∗ and t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' The reason is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Recall given T 1-equivariant line bundle L → M with equivariant curvature FA + µ, the holonomy around any T 1-orbit at p ∈ M is given by e2πiµ(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Applying this to our case, for a loop γb ⊂ T n specified by b ∈ tZ, holonomies of (L−1, ∇) around the loops in M specified by b ∈ tZ define a smooth function: fb : M → C, p �→ fb(p) := e−2iπ⟨µ(p),b⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Therefore, ∇b#τ = 0 implies fb · τ = τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' By transporting this to the dual space, we have fb · δ = δ for any δ ∈ Γc(M, L−1)′ satisfying ∇b#δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' For δ ∈ Hmix, we have ∇ξ#δ = 0, ∀ξ ∈ t, in particular ∇b#δ = 0, ∀b ∈ tZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' This implies that fb is constant 1 on supp δ, for any b ∈ tZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Therefore we conclude that δ should be supported in the set where µ takes integral value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' That is, supp δ ⊂ � λ∈t∗ Z µ−1(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' To prove (2), given any λ ∈ t∗ Z and δ ∈ Hmix,λ, we need to show for any τ ∈ Γc(M, L−1) and ξ ∈ t, (ξ · δ)(τ) = i⟨λ, ξ⟩δ(τ), where ⟨, ⟩ : t∗ × t → R is the natural pairing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Note that the Liouville volume form volM is T n-invariant, div ξ# = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' This implies, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='4) (∇ξ#δ)(τ) = −δ(∇ξ#τ) = 0, and (ξ · δ)(τ) = −δ(ξ · τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Recall that ξ · τ = ∇ξ#τ + iµξτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' By equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='4), we have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='5) (ξ · δ)(τk) = −δ(ξ · τk) = −δ(∇ξ#τk + iµξτk) = −δ(iµξτk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Suppose τ = τk ∈ Γc(M, L−1) has weight k, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' ξ · τk = i⟨k, ξ⟩τk, by equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='4), one has (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='6) (ξ · δ)(τk) = −δ(ξ · τk) = −δ(i⟨k, ξ⟩τk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Combine equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='4) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='6), we obtain (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='7) δ(i(µξ − ⟨k, ξ⟩)τk) = 0, ∀ξ ∈ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' For any k ̸= λ, there exists ξ ∈ t, such that ⟨λ, ξ⟩ ̸= ⟨k, ξ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' For such ξ, as µξ|Mλ = ⟨λ, ξ⟩, µξ − ⟨k, ξ⟩ is no-where vanishing on a T n-invariant open neighbourhood of Mλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' One has: (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='8) δ(τk) = δ(i(µξ − ⟨k, ξ⟩) 1 i(µξ − ⟨k, ξ⟩)τk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' GQ ASSOCIATED TO MIXED POLARIZATIONS ON K¨AHLER MANIFOLDS WITH T-SYMMETRY 11 Since the moment map is T n-invariant, 1 i(µξ−⟨k,ξ⟩)τk still has weight k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Hence, by the above discussion (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' in equation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='7, replacing τk by 1 i(µξ−⟨k,ξ⟩)τk), one has: (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='9) δ(τk) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Given the weight decomposition of τ = � k τk (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' ξ · τk = i⟨k, ξ⟩τk), by linearity of δ and equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='9), we obtain: (ξ · δ)(τ) = i⟨λ, ξ⟩δ(τ⟨λ,ξ⟩) = i⟨λ, ξ⟩δ(τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Therefore we have: ξ · δ = i⟨λ, ξ⟩δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' □ The next corollary says that any element in Hmix,λ is locally a delta function along µ−1(λ), and does not involve any derivative of delta functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' For any λ ∈ t∗ Z, δ ∈ Hmix,λ, and any test section τ ∈ Γc(M, L−1) satisfying τ|Mλ = 0, we have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='10) δ(τ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' For any τ ∈ Γc(M, L−1) satisfying τ|Mλ = 0, let τ = � k τk be its weight decompo- sition, where τk = � eit∈T n(eit · τ)e−iktdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='2, δ has weight λ with respect to T n-action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' This implies, for k ̸= λ, δ(τk) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Note that τk(p) = � eit∈T n(e−it · τ)(p)eiktdt = � eit∈T n τ(e−it · p)eiktdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' For any p ∈ Mλ and t ∈ t, eit · p ∈ Mλ since Mλ is T n-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Therefore τk|Mλ = 0, as τ|Mλ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' In particular, τλ|Mλ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' So there exists weight λ test section τ ′ such that τλ = i(µξ − ⟨λ, ξ⟩)τ ′ λ for some ξ ∈ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Hence, for k = λ, by equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='7), we obtain (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='11) δ(τλ) = δ(i(µξ − ⟨λ, ξ⟩)τ ′ λ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Therefore, by the linearity of δ, we have δ(τ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' □ In order to establish the isomorphism between H0(Mλ, Lλ) and Hmix,λ, where (Mλ, Lλ) = (M, L)//λT is the symplectic reduction of M at a regular integral level λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' We first define the distributional section δs ∈ Γc(Mλ, (Lλ)−1)′ on Mλ ⊂ M associated to s ∈ H0(Mλ, Lλ) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' For any λ ∈ t∗ Z,reg and s ∈ H0(Mλ, Lλ), we define the distributional section δs ∈ Γc(Mλ, (Lλ)−1)′ associated to s as follows: for any τ ∈ Γc(Mλ, (Lλ)−1), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='12) δs(τ) = � Mλ ⟨π∗s, τ⟩ volλ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' 12 LEUNG AND WANG In fact, δs = ι(π∗s), under the embedding ι : Γ(Mλ, Lλ) Γc(Mλ, (Lλ)−1)′ volλ defined by σ �→ (ισ)(τ) = � M⟨σ, τ⟩ volλ with respect to volλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Note that ı(δs) ∈ Γc(M, L−1)′, where ı : Γc(Mλ, (Lλ)−1)′ ֒→ Γc(M, L−1)′ is a natural inclusion defined by: (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='13) (ı(δ))(τ) = δ(τ|Mλ), ∀δ ∈ Γc(Mλ, (Lλ)−1)′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' In order to show ı(δs) ∈ Hmix,λ, we first need to extend covariant derivative ∇ξ on the space of smooth sections to the space of distributional sections of Lλ with respect to volλ as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' That is, for any σ ∈ Γc(Mλ, (Lλ)−1)′, and ξ ∈ Γ(Mλ, TMλ ⊗ C), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='14) (∇ξσ)(τ) = σ(t∇ξτ), with t∇ξτ = −(div ξτ + ∇ξτ), where div2 ξ = Lξ volλ volλ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' In particular, ı(δs) ∈ Γc(M, L−1)′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' we also need to study the relationship between the covariant derivative on the space of distributional sections of Lλ and the space of distributional sections of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' We expect the following diagram Γ(M, L) Γc(M, L−1)′ Γc(M, L−1)′ Γ(Mλ, Lλ) Γc(Mλ, (Lλ)−1)′ Γc(Mλ, (Lλ)−1)′, volM ∇ξ volλ ı ∇ξ ı commute, for any ξ ∈ Γ(M, TM ⊗ C) satisfying ξ|Mλ ∈ Γ(Mλ, TMλ ⊗ C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' In order to show that the above diagram commute, we use the coisotropic embedding theorem due to Weinstein [13] and further studied by Guillemin in [2] to (relate volM and volλ) show the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Restriction commutes with taking divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' For any ξ ∈ Γ(M, TM ⊗C), we denote the restriction of the divergence of ξ (with respect to volM) to Mλ by div1 ξ and denote the divergence of ξ|Mλ (with respect to volλ) by div2 ξ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' div1 ξ = d(iξ volM) volM |Mλ, and div2 ξ = d(iξ|Mλ volλ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' volλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Under the assumption (∗), for any ξ ∈ Γ(M, Pmix) and λ ∈ t∗ reg, we have div1 ξ = div2 ξ, as functions on Mλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Without loss of generality, we assume λ = 0 and n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' In order to show that div1 ξ = div2 ξ, we shall first relate the volume form volM of M and the volume form volMλ of Mλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Taking a principal T 1-connection α ∈ Ω1(M0, t) on M0, choose a basis ξ1 of t and denote the corresponding dual basis of t∗ by ξ∗ 1 with coordinate function t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' In terms of ξ1, GQ ASSOCIATED TO MIXED POLARIZATIONS ON K¨AHLER MANIFOLDS WITH T-SYMMETRY 13 we write α = ξ1 ⊗ α1, where α1 is a scalar valued form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' By abuse of notations, we denote α1 by α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Consider M0 as a submanifold of M0 × t∗ via the embedding i0 : M0 → M0 × t∗, i0(p) = (p, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' The two-form ˜ω = π∗ω0 + d(tα) is symplectic on a neighbourhood U of M0 in M0 × t∗ and satisfies i∗ 0˜ω = π∗ω0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Note that (dt ∧ α)2 = 0 and (i∗ω)m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Then we restrict our attention to show Lξt = 0, ∀ ξ ∈ Γ(M, Pmix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' We extend the T 1-action on M0 to M0 ×t∗ in a trivial manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Then ˜ω is T 1-invariant and the action of T 1 on M0 × t∗ is Hamiltonian with moment map µ0 : M0 × t∗ → t∗, (p, t) �→ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' According to Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='2 of [2], in a neighborhood U of M0, the Hamiltonian T 1-spaces (M, ω) and (M0 × t∗, ˜ω) are isomorphic (see Appendix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' This gives Lζt = 0, for any ζ ∈ Γ(U, DC) and volM = 1 m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' (i∗ω + tdα)m−1 ∧ α ∧ dt, in a neighbourhood U of M0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' As Pmix ⊂ DC, it is obvious that Γ(U, Pmix) ⊂ Γ(U, DC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' This gives us that, for any ξ ∈ Γ(U, Pmix), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='15) Lξt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' It follows that: Lξ(i∗ω + tdα)m−1 = (m − 1)(Lξ(i∗ω) + tLξdα) ∧ (i∗ω + tdα)m−2, ∀ξ ∈ Γ(U, Pmix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Therefore, we obtain: 1 volM d(iξ volM) = 1 volM Lξ volM = 1 volM 1 m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='Lξ((i∗ω + tdα)m−1 ∧ α ∧ dt) = 1 volM 1 m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' (Lξ(i∗ω + tdα)m−1 ∧ α ∧ dt + (i∗ω + tdα)m−1 ∧ Lξα ∧ dt) = (m − 1)(Lξi∗ω + tLξdα) ∧ (i∗ω + tdα)m−2 ∧ α + (i∗ω + tdα)m−1 ∧ Lξα) (i∗ω + tdα)m−1 ∧ α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Recall that vol0 = 1 (m−1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' (i∗ω)m−1 ∧ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' By abuse of notation, iξ vol0 means iξ|M0 vol0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Then by a straight computation, div2 ξ = 1 vol0d(iξ vol0) = 1 vol0Lξ vol0 = 1 vol0 1 (m − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='Lξ((i∗ω)m−1 ∧ α) = (m − 1)(Lξi∗ω) ∧ (i∗ω)m−2 ∧ α + (i∗ω)m−1 ∧ Lξα) (i∗ω)m−1 ∧ α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' 14 LEUNG AND WANG Therefore, for ξ ∈ Γ(M, Pmix), we have: div1 ξ = � 1 volM d (iξ volM) � |M0 = � 1 volM d (iξ volM) � |t=0 = div2 ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' □ Then we obtain the following theorem: Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' For any λ ∈ t∗ Z,reg, δ ∈ Γc(Mλ, (Lλ)−1)′ and ξ ∈ Γ(M, TM ⊗ C) satisfying ξ|Mλ ∈ Γ(Mλ, TMλ ⊗ C), we have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='16) ∇ξ(ı(δ)) = ı(∇ξδ), where ı : Γc(Mλ, (Lλ)−1)′ ֒→ Γc(M, L−1)′ is the natural inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' For any test section τ ∈ Γc(M, L−1), according to equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='2), one has (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='17) (∇ξ(ı(δ)))(τ) = ı(δ)(t∇ξτ) = δ(i∗(t∇ξτ)), and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='18) (ı(∇ξδ))(τ) = (∇ξδ)(i∗τ) = δ(t∇ξ(i∗τ)), where i : Mλ ֒→ M is the inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' To show equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='16), by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='17) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='18), it is enough to prove that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='19) i∗(t∇ξτ) =t ∇ξ(i∗τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' According to equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='2), we have: (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='20) i∗(t∇ξτ) = −i∗ (div ξτ + ∇ξτ) , where div ξ = iξ volM vol M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Similarly, applying the equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='1) to L|Mλ, we have: (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='21) t∇ξ(i∗τ) = −((div2 ξ)(i∗τ) + ∇ξ(i∗τ)) where div2 ξ = iξ volλ volλ , i∗τ = τ|Mλ = τ by abuse of notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Denote i∗(div ξ) by div1 ξ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' div1 ξ = iξ volM volM |Mλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' As i∗(∇ξτ) = ∇ξ(i∗τ) by abuse of notation ξ|Mλ = ξ, we have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='22) − i∗ (div ξτ + ∇ξτ) = −(div1 ξ(i∗τ) + ∇ξ(i∗τ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='5, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='23) div1 ξ = div2 ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Combining equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='20), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='21), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='22) with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='23), one has δ(i∗(t∇ξτ)) = δ(t∇ξ(i∗τ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Therefore we have: ∇ξ(ı(δ)) = ı(∇ξδ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' □ GQ ASSOCIATED TO MIXED POLARIZATIONS ON K¨AHLER MANIFOLDS WITH T-SYMMETRY 15 Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' For any regular λ ∈ t∗ Z,reg and s ∈ H0(Mλ, Lλ), we have: ı(δs) ∈ Hmix,λ, where ı : Γc(Mλ, (Lλ)−1)′ ֒→ Γc(M, L−1)′ is the natural inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' By the definition of δs, we have ı(δs) ∈ Γc(M, L−1)′ and supp ı(δs) ⊂ µ−1(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' It remains to show that, for any ξ ∈ Γ(M, Pmix), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='24) ∇ξ(ı(δs)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Note that for any ξ ∈ (M, Pmix), ξ|Mλ ∈ Γ(Mλ, TMλ ⊗ C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' To check equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='24), by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='6, it is equivalent to prove, for any ξ ∈ Γ(Mλ, Pmix) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='25) ∇ξδs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Take any test section τ ∈ Γc(Mλ, (Lλ)−1), according to equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='2), we have: (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='26) (∇ξδs)(τ) = δs �t∇ξτ � = −δs � (div2 ξ)τ + ∇ξτ � , where div2 ξ = iξ volλ volλ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' By definition of δs, it can be seen that: (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='27) − δs � (div2 ξ)τ + ∇ξτ � = − � Mλ � π∗s, (div2 ξ)τ + ∇ξτ � volλ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Similarly, applying the equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='1) to L|Mλ, we have: (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='28) � Mλ ⟨∇ξ(π∗s), τ⟩ volλ = − � Mλ � π∗s, (div2 ξ)τ + ∇ξτ � volλ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Combining equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='26), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='27), with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='28), we have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='29) (∇ξδs)(τ) = � Mλ ⟨∇ξ(π∗s), τ⟩ volλ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Since s ∈ H0(Mλ, Lλ) is a holomorphic section, we have ∇s ∈ Γ(Mλ, T ∗M1,0 λ ⊗ Lλ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' For any ξ ∈ Γ(M, Pmix) and q ∈ Mλ, as (Pmix)q ⊂ (DC)q = TqMλ ⊗ C, we have π∗(ξq) ∈ Tπ(q)M0,1 λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' This implies ∇ξ(π∗s) = 0 on Mλ, for any ξ ∈ Γ(M, Pmix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Then, for all τ ∈ Γc(M, L−1), by equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='29), (∇ξδs)(τ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Therefore we have: ı(δs) ∈ Hmix,λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' □ 16 LEUNG AND WANG 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' λ-weight quantum subspace Hmix,λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' In this subsection, we are going to show that (see Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='12) for any regular λ ∈ t∗ Z,reg, κ : H0(Mλ, Lλ) → Hmix,λ given by s �→ κ(s) = ı(δs) is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Firstly, we show that T n-invariant distributional sections of Lλ can be descended to distributional sections of Lλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' That is, for any δ ∈ Γc(Mλ, (Lλ)−1)′ satisfying ∇ξ#δ = 0, there exists a distributional section η ∈ Γc(Mλ, L−1 λ )′ such that δ = π∗η (Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Secondly, we show that if ∇ζ(π∗η) = 0 for all ζ ∈ Γ(Mλ, Pmix), then η is ¯∂-closed (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Finally, we show that H0(Mλ, Lλ) ∼= Hmix,λ (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Descending distributional sections from Mλ to Mλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' For any λ ∈ t∗ reg, let π : Mλ → Mλ be the principal T n-bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Recall that (Lλ, ∇) can be descended to Mλ which we denote as (Lλ, ∇).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' According to Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='4, we have π∗ : Γ(Mλ, (Lλ)−1) → Γ(Mλ, L−1 λ ) and dually we have π∗ : Γc(Mλ, L−1 λ )′ → Γc(Mλ, (Lλ)−1)′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' In fact, our above claim δ = π∗η holds true for any T n-principal bundle P → B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Let π : P → B be a principal T n-bundle with a fiberwise T n-invariant volume form dθ such that � P dθ = 1 ∈ C∞(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Let (E, ∇) be a line bundle over B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' We can push-forward sections of π∗E to sections of E with respect to dθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Furthermore we have: Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Taking δ ∈ Γc(P, (π∗E)−1)′, if ∇ξ#δ = 0 for any ξ ∈ t, then there exists a distributional section η ∈ Γc(B, E−1)′ such that δ = π∗η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' By partition of unity, it is enough to show that on any open subset U of B, for δ ∈ Γc(π−1(U), (π∗E)−1)′, if ∇ξ#δ = 0 for any ξ ∈ t, there exists a distributional section η ∈ Γc(U, E−1)′ such that δ = π∗η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' That is, for any τ ∈ Γc(π−1(U), (π∗E)−1), δ(τ) = η(π∗τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Fixing a local frame σ0 ∈ Γ(U, E) of E on an open subset U ⊂ B, let σ := π∗σ0 and σ−1 be the corresponding local frames of π∗E and (π∗E)−1 respectively on π−1(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' With respect to local frames σ and σ−1, the distributional section δ ∈ Γc(π−1(U), (π∗E)−1)′ corresponds to the distributional function fδ ∈ Γc(π−1(U), C)′, where fδ is determined by: (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='30) fδ(gτ) = δ(gτσ−1), for any text function gτ ∈ Γc(π−1(U), C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' We restrict our attention to show that ∇ξ#δ = 0 if and only if ξ#fδ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Applying the equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='2) to line bundle π∗L and trivial bundle over π−1(U) respectively, it can be seen that: (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='31) � ∇ξ#δ � (τ) = −δ � (div ξ#)τ + ∇ξ#τ � , GQ ASSOCIATED TO MIXED POLARIZATIONS ON K¨AHLER MANIFOLDS WITH T-SYMMETRY 17 and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='32) (ξ#fδ) (gτ) = fδ � − � div ξ#gτ + ξ#gτ �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Since ∇ξ#σ = ∇ξ#(π∗σ0) = 0, one has ∇ξ#σ−1 = 0 and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='33) ∇ξ#τ = ∇ξ# � gτσ−1� = � ξ#gτ � σ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Combining equations ( 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='30), ( 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='31), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='32), with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='33), we obtain that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='34) � ∇ξ#δ � (τ) = (ξ#fδ) (gτ) , for any τ ∈ Γc (π−1(U), (π∗E)−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' It turns out that ∇ξ#δ = 0 iff ξ#fδ = 0 for any ξ ∈ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Then by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='9, there exists a distributional function fη ∈ Γc(U, C)′ such that fδ = π∗(fη).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Define η ∈ Γc(U, (π∗E)−1)′ to be distributional section associated to fη with respect to the nowhere vanishing section σ−1 0 , that is η(hτσ−1 0 ) = fη(hτ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' For any test section τ ∈ Γc(π−1(U), π∗E), it can be check that: δ(τ) = (π∗η)τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Therefore we have δ = π∗η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Let π : P → B be the principal T n-bundle and let U be any open subset of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Let δ ∈ Γc(π−1(U), C)′ be a distributional function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' If ξ#δ = 0 for any ξ ∈ t, there exists a distributional function η ∈ Γc(U, C)′, such that δ = π∗η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Namely, δ(g) = η(π∗g), ∀ g ∈ Γc(π−1(U), C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' For any δ ∈ Γc(π−1(U), C)′, there exist δǫ ∈ Γ(π−1(U), C) (see [6, 11]) such that limǫ→0 δǫ = δ and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='35) (ξ#δǫ)(g) = (ξ#δ)(gǫ), for any g ∈ Γc(π−1(U), C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' As ξ#δ = 0, we obtain ξ#δǫ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Since δǫ is smooth, there exists a smooth function ηǫ ∈ Γ(U, C), such that δǫ = π∗ηǫ ∈ Γ(π−1(U), C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' It can be check that lim ǫ→0 ηǫ(h) = lim ǫ→0 δǫ(π∗h), for any h ∈ Γc(U, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Hence we have limǫ→0 ηǫ exists and denoted by η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' It follows δ = π∗η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' □ 18 LEUNG AND WANG 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Pulling back commutes with taking divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Fix λ ∈ t∗ Z,reg, let α ∈ Ω1(Mλ, t) be a connection on the principal T n-bundle π : Mλ → Mλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' For any ζ ∈ Γ(Mλ, TMλ), the horizontal lifting of ζ with respect to α is denoted by ˜ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Denote the divergence of ζ on Mλ with respect to volλ by div ζ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' div ζ = Lζ volλ volλ ) and denote the divergence of ˜ζ on Mλ with respect to volλ by div ˜ζ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' div ˜ζ = L˜ζ volλ volλ ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Let div ζ and div ˜ζ be defined as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Then we have π∗(div ζ) = div ˜ζ, as smooth functions on Mλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' As T n is abelian, the horizontal lifting ˜ζ of ζ with respect to the connection one form α is T n-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' That is (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='36) Lξ# ˜ζ = 0, for all ξ ∈ t, where ξ# is the fundamental vector field associate to ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' According to the property of principal T n-connection and equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='37), we have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='37) (L˜ζα)(ξ#) = L˜ζ(α(ξ#)) − α(L˜ζξ#) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Recall that volλ = π∗ volλ ∧αn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' By equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='37), one has (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='38) L˜ζ volλ = (L˜ζ(π∗ volλ)) ∧ αn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' On the other hand, by Cartan formula and volλ being the volume form on B, we have: (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='39) L˜ζ(π∗ volλ) = d(i˜ζ(π∗ volλ)) = π∗(Lζ volλ), Recall that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='40) Lζ volλ = (div ζ) volλ, L˜ζ volλ = (div ˜ζ) volλ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Combining equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='38), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='39), with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='40), one has (div ˜ζ) volλ = L˜ζ volλ = π∗(Lζ volλ) ∧ αn = π∗(div ζ)π∗ volλ ∧αn = π∗(div ζ) volλ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Therefore we obtain: π∗(div ζ) = div ˜ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' □ Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' For any λ ∈ t∗ Z,reg and distributional function η ∈ Γc(Mλ, C)′, if ∇ξ(π∗η) = 0, for any ξ ∈ Γ(Mλ, Pmix), then we have ∇ζη = 0, for all ζ ∈ Γ(Mλ, TM0,1 λ ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' GQ ASSOCIATED TO MIXED POLARIZATIONS ON K¨AHLER MANIFOLDS WITH T-SYMMETRY 19 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' To prove this statement, fixing the connection one form α ∈ Ω1(Mλ, t) on principal T n-bundle π : Mλ → Mλ, we denote the horizontal lifting of ζ with respect to the connec- tion α by ˜ζ, for any ζ ∈ Γ(Mλ, TM0,1 λ ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' In order to show ∇ζη = 0, it is enough to show , for any test function φ ∈ Dc(Mλ), (∇ζη) (φ) = � ∇˜ζ (π∗η) � (π∗φ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Let volλ and volλ be volume forms of Mλ and Mλ respectively as defined before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' In particular, volλ = π∗ volλ ∧αn with respect to the principal T n-connection α ∈ Ω1(Mλ, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Applying equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='2) to the trivial bundle of Mλ and Mλ respectively, we obtain: (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='41) (∇ζη) (φ) = η (− (div ζ) φ − ∇ζφ) , and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='42) � ∇˜ζ (π∗η) � (π∗φ) = (π∗η) � − � div ˜ζ � π∗φ − ∇˜ζ (π∗φ) � , where div ζ (div ˜ζ resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=') is the divergence of ζ (˜ζ resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=') with respect to volλ (volλ resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' According to the Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='4, we have: (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='43) (π∗η) (π∗ (− (div ζ) φ − ∇ζφ)) = η (− (div ζ) φ − ∇ζφ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='10, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='44) π∗(div ζ) = div ˜ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Note that π∗(ζ(φ)) = π∗ζ(π∗φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' By equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='44), one has (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='45) π∗ (− (div ζ) φ − ∇ζφ) = − � div ˜ζ � π∗φ − ∇˜ζ (π∗φ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Furthermore: (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='46) (π∗η) (π∗ (− (div ζ) φ − ∇ζφ)) = (π∗η) � − � div ˜ζ � π∗φ − ∇˜ζ (π∗φ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Combining (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='41), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='42), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='43), with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='46), we are able to conclude: (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='47) (∇ζη) (φ) = � ∇˜ζ (π∗η) � (π∗φ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Then we restrict our attention to show ˜ζ ∈ Γ(Mλ, Pmix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' As T n acts freely on Mλ, Mλ×t ∼= IR|Mλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Note that π∗(˜ζ) = ζ ∈ Γ(Mλ, TM0,1 λ ) and α(˜ζ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Since TpMλ ⊗ C ⊂ (DC)p and (Pmix)p = (DC ∩ TM0,1)p ⊕ (IC)p, for any p ∈ M0, we have ˜ζ ∈ Γ(Mλ, Pmix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' According to what we assume, we have ∇˜ζ (π∗η) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Therefore, by equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='47), we have (∇ζη) (φ) = � ∇˜ζ (π∗η) � (π∗φ) = 0, ∀φ ∈ Dc(Mλ), ζ ∈ Γ(Mλ, TM0,1 λ ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' □ 20 LEUNG AND WANG 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Building the isomorphism H0(Mλ, Lλ) ∼= Hmix,λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Recall given any s ∈ H0(Mλ, Lλ), by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='7, the associated distributional section ı(δs) belongs to Hmix,λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Therefore we can define a homomorphism κ : H0(Mλ, Lλ) → Hmix,λ given by s �→ κ(s) = ı(δs), where ı : Γc(Mλ, (Lλ)−1)′ ֒→ Γc(M, L−1)′ is the natural inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' It can be checked that κ is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' For any λ ∈ t∗ Z,reg, κ : H0(Mλ, Lλ) → Hmix,λ is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Given any ˜δ ∈ Hmix,λ, we need to construct s ∈ H0(Mλ, Lλ) such that ˜δ = κ(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Firstly we show that, there exists δ ∈ Γc(Mλ, (Lλ)−1)′ such that ˜δ = ı(δ) as follows: we define the distributional section δ ∈ Γc(Mλ, (Lλ)−1)′ by: δ(τ) = ˜δ(˜τ), for any τ ∈ Γc(Mλ, (Lλ)−1), where ˜τ ∈ Γc(M, L−1) is any test section satisfying ˜τ|Mλ = τ By Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='3, δ is well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Moreover, one has (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='48) ˜δ = ı(δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' That is, for any test section ˜τ ′ ∈ Γc(M, L−1), (ı(δ))(˜τ ′) = δ(˜τ ′|Mλ) = ˜δ(˜τ ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Secondly we show that there exists η ∈ Γc(Mλ, L−1 λ )′ such that δ = π∗η, where π : Mλ → Mλ is the projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' For any ˜δ ∈ Hmix,λ, since ξ# ∈ Γ(M, Pmix), we have ∇ξ#˜δ = 0, for any ξ ∈ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='6, one has (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='49) 0 = ∇ξ#˜δ = ∇ξ#(ı(δ)) = ı(∇ξ#δ), ∀ξ ∈ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' By the injectivity of ı, we obtain, for any ξ ∈ t, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='50) ∇ξ#δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' According to Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='8, there exists a distributional section η ∈ Γc(Mλ, L−1 λ )′, such that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='51) δ = π∗η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Next we show that there exists a holomorphic section s ∈ H0(Mλ, Lλ) such that η = ι(s) under the inclusion map ι : Γ(Mλ, Lλ) → Γc(Mλ, L−1 λ )′ with respect to volλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' By the definition of Pmix, for any ξ ∈ Γ(M, Pmix), we have ξ|Mλ ∈ Γ(Mλ, Pmix) ⊂ Γ(Mλ, TMλ⊗C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' By abuse of notation, we denote ξ|Mλ by ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' According to Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='11 and equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='51), we have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='52) ∇ξ˜δ = ∇ξ(ı(δ)) = ı(∇ξδ) = ı(∇ξ(π∗η)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Since ˜δ ∈ Hmix,λ, ∇ξ˜δ = 0, for ξ ∈ Γ(M, Pmix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' By the injectivity of ı and equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='52), we obtain: (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='53) ∇ξ(π∗η) = 0, ∀ξ ∈ Γ(Mλ, Pmix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' GQ ASSOCIATED TO MIXED POLARIZATIONS ON K¨AHLER MANIFOLDS WITH T-SYMMETRY 21 Then by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='11, we have ∇ζη = 0, for any ζ ∈ Γ(Mλ, TM0,1 λ ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' This implies ∇0,1η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' By the regularity of elliptic operator ∆ = ¯∂∗ ¯∂, η is smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Therefore there exists a holomorphic section s ∈ H0(Mλ, Lλ) such that η = ι(s) under the inclusion map ι : Γ(Mλ, Lλ) → Γc(Mλ, L−1 λ )′ with respect to volλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' It is remain to show ˜δ = κ(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' According to the above discussion, we have ˜δ = ı(π∗(ι(s))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Recall that κ(s) = ı(δs), where δs with respect to volume form volλ is defined by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='54) δs(τ) = � Mλ⟨π∗s, τ⟩ volλ, for any test section τ ∈ Γc(Mλ, Lλ)′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' By the injectivity of ı, to show ˜δ = κ(s), it is enough to show: (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='55) π∗(ι(s)) = δs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' By remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='4, we have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='56) � Mλ⟨π∗s, τ⟩ volλ = (π∗s)(τ) = s(π∗τ) = � Mλ ⟨s, π∗τ⟩ volλ And (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='57) π∗(ι(s))(τ) = (ι(s))(π∗τ) = � Mλ ⟨s, π∗τ⟩ volλ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' According to equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='54 ), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='56), and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='57), we have π∗(ι(s)) = δs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Appendix 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Polarizations on symplectic manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' A step in the process of geometric quanti- zation is to choose a polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' We first recall the definitions polarizations on symplectic manifolds (M, ω) (See [12, 14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' All polarizations discussed in this subsection are smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' A complex polarization on M is a complex sub-bundle of the complexified tangent bundle TM ⊗ C satisfying the following conditions: (1) P is involutive, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' if u, v ∈ Γ(M, P), then [u, v] ∈ Γ(M, P);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' (2) for every x ∈ M, Px ⊆ TxM ⊗ C is Lagrangian;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' and (3) rkR (P) := rank(P ∩ P ∩ TM) is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Furthermore, P is called real polarization, if P = P, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' rkR (P) = m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' K¨ahler polarization, if P ∩ P = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' rkR (P) = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' mixed polarization, if 0 < rank(P ∩ P ∩ TM) < m, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' 0 < rkR (P) < m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' 22 LEUNG AND WANG 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Singular polarizations on symplectic manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' In subsection, we review the definitions of singular polarizations, smooth sections of singular polarizations which were used in the proof of the main results (see [9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' P ⊂ TM ⊗ C is a singular complex distribution on M if it satisfies: Pp is a vector subspace of TpM ⊗ C, for all point p ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Such a P is called smooth on an open subset ˇ M ⊂ M if P| ˇ M is a smooth sub-bundle of the tangent bundle T ˇ M ⊗ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' In this paper, we only consider such distributions with mild singularities in the sense that they are only singular outside an open dense subset ˇ M ⊂ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Under our setting, we define smooth sections of singular distributions and involutive distributions as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Let P be a singular complex distribution of TM ⊗C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' For any open subset U of M, the space of smooth sections of P on U is defined by the smooth section of TM ⊗C with value in P, that is, Γ(U, P) = {v ∈ Γ(U, TM ⊗ C) | vp ∈ (P)p, ∀p ∈ U}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Let P be a singular complex distribution on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' P is involutive if it satisfies: [u, v] ∈ Γ(M, P), for any u, v ∈ Γ(M, P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Let P be a singular complex distribution P on M and smooth on ˇ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Such a P is called a singular polarization on M, if it satisfies the following conditions: (a) P is involutive, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' if u, v ∈ Γ(M, P), then [u, v] ∈ Γ(M, P);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' (b) for every x ∈ ˇ M, Pp ⊆ TpM ⊗ C is Lagrangian;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' and (c) the real rank rkR(P) := rank(P ∩ P ∩ TM)| ˇ M is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Furthermore, such a singular P is called real polarization, if P| ˇ M = P| ˇ M, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' rkR(P| ˇ M) = m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' K¨ahler polarization, if P ˇ M ∩ P| ˇ M = 0 on ˇ M, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' r (P| ˇ M) = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' mixed polarization, if 0 < rank(P ∩ P ∩ TM)| ˇ M < m, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' 0 < rkR(P| ˇ M) < m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Coisotropic embedding theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' We review the coisotropic embedding theorem studied by Guillemin in [2], which was used in the proof of taking divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Let (M, ω) be a symplectic manifold of dimensional 2m equipped with Hamiltonian T n-action with moment map µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Without loss of generality, we assume n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Choose a principal T 1- connection α ∈ Ω1(M0, t) on M0, where M0 = µ−1(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Consider M0 as a submanifold of M0 × R via the embedding i : M0 → M0 × R, i(p) = (p, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' GQ ASSOCIATED TO MIXED POLARIZATIONS ON K¨AHLER MANIFOLDS WITH T-SYMMETRY 23 On the product space ˜ M = M0 × (−ǫ, ǫ), the two-form ˜ω = π∗ω0 + d(tα), −ǫ < t < ǫ is symplectic on ˜ M and satisfies i∗˜ω = π∗ω0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Extending the T 1-action on M0 to M0×t∗ in a trivial manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Then ˜ω is T 1-invariant and that the action of T 1 on M0 ×t∗ is Hamiltonian with moment map µ0 : M0 × t∗ → t∗, (p, t) �→ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' [3, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='2] In a neighborhood of M0, the Hamiltonian T n-spaces (M, ω) and ( ˜ M, ˜ω) are isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Geometric quantization commute with symplectic reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' In this subsec- tion, we review the work on geometric quantization commute with symplectic reduction by Guillemin and Sternberg in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Let (L, ∇) and (Lλ, ∇λ) be the pre-quantum line bundle on M and Mλ respectively as discussed before, for λ ∈ t∗ Z,reg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Then the quantum space HPJ associated to PJ is the space of J-holomorphic sections of L: HPJ = {s ∈ Γ(M, L) | ¯∂Js = 0} = H0(M, L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' One can perform two processes on the pre-quantum line bundle (L, ∇);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' one is geometric quantization, and the other is symplectic reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Guillemin and Sternberg in [3] showed that these two processes commute with each other, that is, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='1) (HPJ)λ ∼= HPJ,λ, where (HPJ)λ (Jλ-holomorphic sections of Lλ) is the λ-weight subspace of HPJ and HPJ,λ is the quantum space associated to reduced K¨ahler polarization PJ,λ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='2) HPJ,λ = {s ∈ Γ(Mλ, Lλ) | ¯∂Jλs = 0} = H0(Mλ, Lλ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content=' Baier, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} 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Kong Email address: leung@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='cuhk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='hk The Institute of Mathematical Sciences and Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong Email address: dwang@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='cuhk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} +page_content='hk' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfFfpP/content/2301.01011v1.pdf'} diff --git a/-NE1T4oBgHgl3EQf8QVm/content/tmp_files/2301.03543v1.pdf.txt b/-NE1T4oBgHgl3EQf8QVm/content/tmp_files/2301.03543v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..187a1c2a6158f0cd693b83b57be30a5ca3e34acc --- /dev/null +++ b/-NE1T4oBgHgl3EQf8QVm/content/tmp_files/2301.03543v1.pdf.txt @@ -0,0 +1,1778 @@ +1 +Efficient Mutation Testing via Pre-Trained +Language Models +Ahmed Khanfir , Renzo Degiovanni , Mike Papadakis and Yves Le Traon +SnT, University of Luxembourg, Luxembourg +Abstract—Mutation testing is an established fault-based testing technique. It operates by seeding faults into the programs under test +and asking developers to write tests that reveal these faults. These tests have the potential to reveal a large number of faults – those +that couple with the seeded ones – and thus are deemed important. To this end, mutation testing should seed faults that are both +“natural” in a sense easily understood by developers and strong (have high chances to reveal faults). To achieve this we propose using +pre-trained generative language models (i.e. CodeBERT) that have the ability to produce developer-like code that operates similarly, +but not exactly, as the target code. This means that the models have the ability to seed natural faults, thereby offering opportunities to +perform mutation testing. We realise this idea by implementing µBERT, a mutation testing technique that performs mutation testing +using CodeBert and empirically evaluated it using 689 faulty program versions. Our results show that the fault revelation ability of +µBERT is higher than that of a state-of-the-art mutation testing (PiTest), yielding tests that have up to 17% higher fault detection +potential than that of PiTest. Moreover, we observe that µBERT can complement PiTest, being able to detect 47 bugs missed by PiTest, +while at the same time, PiTest can find 13 bugs missed by µBERT. +Index Terms—Fault Injection, Mutation Testing, Pre-Trained Language Models +! +1 +INTRODUCTION +Mutation testing aims at seeding faults using simple syntac- +tic transformations [19]. These transformations, also known +as mutation operators are typically constructed based on +syntactic rules crafted based on the grammar of the target +programming language [8], i.e. replacing an arithmetic op- +erator with another such as a + by a -. Unfortunately, such +techniques generate mutants (seeded faults), many of which +are “unatural”, i.e., non-conforming to the way developers +code, thereby perceived as unrealistic by developers [11]. +At the same time, the syntactic-based fault seeding fails to +capture the semantics of the code snippets that they apply, +leading to numerous trivial or low utility faults [46]. +To deal with the above issue we propose forming natural +mutations by using big code. Thus, we aim at introducing +modifications that follow the implicit rules, norms and cod- +ing conventions followed by programmers, by leveraging +the capabilities of pre-trained language models to capture +the underlying distribution of code and its writing, as +learned by the pre-training process on big code. +To this end, we rely on CodeBERT [22], an NL-PL +bimodal language model that has been trained on over +6.4 million programs. More precisely, we use its Masking +Modelling Language (MLM) functionality, which given a +code sequence with a masked token, predicts alternative +replacements to that token, that is best matching the se- +quence context. This is important, since the predictions +do not follow fixed predefined patterns as is the case of +conventional mutation testing, but are instead adapted to +fit best the target code. For instance, given a sequence +• +A. Khanfir, R. Degiovanni, M. Papadakis, Y. Le Traon are with the +University of Luxembourg, Luxembourg. +int a = 1;, we pass a masked version of it as int a = +;, then CodeBERT by default proposes 5 predictions +sorted by likelihood score: 0, 1, b, 2, and 10. Being the most +likely fitting tokens to the code context, our intuition is that +replacing the masked token with these predictions would +induce “natural” mutants. +Precisely, we introduce µBERT, a mutation testing ap- +proach that uses a pre-trained language model (CodeBERT) +to generate mutants by masking and replacing tokens with +the aim of forming natural mutants. µBERT iterates through +the program statements and modifies their token. In par- +ticular, µBERT proceeds as follows: (1) it selects and masks +one token at a time; (2) feeds CodeBERT with the masked +sequence and obtains the predictions; (3) creates mutants by +replacing the masked token with the predicted ones; and (4) +discards non-compilable, duplicate and equivalent mutants +(mutants syntactically equal to original code). +Recent research [32] has shown that some real faults +are only captured by using complex patterns, i.e. patterns +that require more than one token mutation. To account for +such cases, µBERT is equipped with additive mutations, i.e., +mutations that add code (instead of deleting or altering). For +example, consider a boolean expression e1 (typically present +in if, do, while and return statements), which is mutated +by µBERT by adding a new condition e2, thereby generating +a new condition e1||e2 (or e1&&e2), which is then masked +and completed by CodeBERT. For instance, given a condi- +tion if(a == b), µBERT produces a new condition if(a +== b || a > 0) that is masked and produces if(a == +b || b > 0). +We implement µBERT, and evaluate its ability to serve +the main purposes of mutation testing, i.e. guiding the +testing towards finding faults. We thus, evaluate it using +689 faults from Defects4J and asses µBERT effectiveness and +arXiv:2301.03543v1 [cs.SE] 9 Jan 2023 + +2 +cost-efficiency to reveal1 them. Our results show that µBERT +is very effective in terms of fault revelation, finding on +average 84% of the faults. This implies that µBERT mutants +cover efficiently faulty behaviours caused by real bugs. +More importantly, the approach is noticeably more effec- +tive and cost-efficient than a traditional mutation testing +technique, namely PiTest [17], that we use as a baseline +in our evaluation. Precisely, we consider three different +configurations for PiTest that uses different sets of mutation +operators (DEFAULT, ALL and RV). In fact, test suites that +kill all mutants of µBERT find on average between 5.5% to +33% more faults than those generated to kill all mutants +introduced by PiTest. Moreover, even when analysing the +same number of mutants, µBERT induces test suites that find +on average 6% to 16% more faults than PiTest. These results +are promising and endorse the usage of µBERT over the +considered mutation testing technique, as a test generation +and assessment criterion. +We also study the impact of the condition-seeding-based +mutations in the fault detection capability of µBERT. We +observe that test-suites designed to kill both kinds of µBERT +mutants – induced by 1) direct CodeBERT predictions and +2) a combination of conditions-seeding with CodeBERT +predictions – find on average over 9% more bugs than the +ones designed to kill direct CodeBERT prediction mutants +only (1). +Overall, our main contributions are: +• We introduce µBERT, the first mutation testing ap- +proach that uses pre-trained language models. It lever- +ages the model’s code knowledge captured during its +pretraining on large code corpora and its ability to +capture the program context, to produce “natural” mu- +tants. +• We propose new additive mutations which operate +by seeding new conditions in the existing conditional +expressions of the target code, then masking and re- +placing their tokens with the model predictions. +• We provide empirical evidence that µBERT mutants can +guide testing towards higher fault detection capabili- +ties, outperforming those achieved by SOA techniques +(i.e. PiTest), in terms of effectiveness and cost-efficiency. +In our empirical study, we validate also the advantage +of employing the new additive mutation patterns, w.r.t +improving the effectiveness and cost-efficiency in writ- +ing test suites with higher fault revelation capability. +2 +BACKGROUND +2.1 +Mutation Testing +Mutation analysis [47] is a test adequacy criterion repre- +senting test requirements by the mean of mutants, which +are obtained by performing slight syntactic modifications +to the original program. For instance, an expression like +x > 0 can be mutated to x < 0 by replacing the relational +operator > with <. These mutants are then used to assess the +effectiveness and thoroughness of a test suite in detecting +their corresponding code modification. +1. Tests are written/generated to kill (reveal) the mutants. A bug is +revealed by a mutation testing approach, if the written tests to kill its +mutants also reveal the bug. +A test case detects a mutant if it is capable of producing +distinguishable observable outputs between the mutant and +the original program. A mutant is said to be killed if it is +detected by a test case or a test suite; otherwise, it is called +live or survived. Some mutants cannot be killed as they are +functionally equivalent to the original program. The mutation +score measures the test suite adequacy and is computed +as the ratio of killed mutants over the total number of +generated mutants. +2.2 +Generative Language Models +Advances in deep learning approaches gave birth to new +language models for code generation +[1], [4], [15], [22]. +These models are trained on large corpora counting multiple +projects, thereby acquiring a decent knowledge of code, +enabling them to predict accurately source code to devel- +opers. Among these pre-trained models, CodeBERT [22], a +language model that has been recently introduced and made +openly accessible for researchers by Microsoft. +CodeBERT is an NL-PL bimodal pre-trained language +model (Natural Language Programming Language) that +supports multiple applications such as code search, code +documentation generation, etc. Same as most large pre- +trained models, i.e. BERT [20], CodeBERT’s developing +adopts a Multilayer Transformer [55] architecture. It has +been trained on a large corpus collected from over 6.4 +million projects available on GitHub, counting 6 different +programming languages, including Java. The model was +trained in a cross-modal fashion, through bimodal NL-PL +data, where the input data is formed by pairs of source code +and its related documentation, as well-as unimodal data, +including either natural language or programming language +sequences per input. This way, it enables the model to offer +both – PL and NL-PL – functionalities. The training targets +a hybrid objective function, that is based on replaced token +detection. +µBERT incorporates the Masked Language Modeling +(MLM) functionality [2] of CodeBERT in its workflow, to +generate “natural” mutants. The CodeBERT MLM pipeline +takes as input a code sequence of maximum 512 tokens, +including among them one masked as , whose +value will be predicted by the model based on the context +captured from the remaining tokens. CodeBERT provides +by default 5 predictions per token, among which we use the +inaccurate and compilable predicted codes as mutants. +3 +APPROACH +We propose µBERT, a generative language-model-based mu- +tation testing approach, which is described step by step +in Figure 1. Given an input source code, µBERT leverages +CodeBERT’s knowledge of code and its capability in captur- +ing the program’s context to produce “natural” mutations, +i.e. that are similar to eventual developer mistakes.To do so, +µBERT proceeds as follows in six steps: +1) First, it extracts relevant locations (AST 2 nodes) where +to mutate +2) Second, it masks the identified node-tokens, creating +one masked version per selected token. +2. AST: Abstract Syntax Tree. + +3 + if (a != b) +return a != d || b>0; +Source-code +AST nodes +locations +1. AST Nodes +selection +Y +Y +3. Masked code 
 +prediction +Model +Predictions +Predict° +4. Conditions seeding +5. Injection & +Compilation +check +Injected faults +if (a != b) +return a != d; +if (a != b && b>i) +return a != d; + a = b + c; +return a d; +2. AST Nodes +Masking +a = b + c; +return a == d; +a = + c; +return a == d; +Fig. 1: µBERT Workflow: (1) it parses the Java code given as input, and extracts the expressions to mutate; (2) it creates +simple-replacement mutants by masking the tokens of interest and invoking CodeBERT; (3) it generates the mutants by +replacing the masked token with CodeBERT predictions; (4) it generates complex mutants via a) conditions-seeding, b) +tokens masking then c) replacing by CodeBERT predictions; and finally, (5) it discards not compiling and syntactically +identical mutants. +3) Then, it invokes CodeBERT to predict replacements for +these masked tokens. +4) In addition to the mutants produced in Step (3), µBERT +also implements some condition-seeding additive mu- +tations that modify more than one token. Precisely, it +modifies the conditional expressions in the control flow +(typically present in if, do, while and return state- +ments) by extending the original condition with a new +one, combined with the logical operator && or ||. Then, +the new conditional expression is mutated by following +the same steps (2) and (3) – masking and replacing the +masked tokens by the CodeBERT predictions. +5) Finally, the approach discards duplicate predictions +or those inducing similar code to the original one, +or not compiling, and outputs the remaining ones as +mutants, from diverse locations of the target code. More +precisely, it iterates through the statements in random +order and outputs in every iteration one mutant per +line, until achieving the desired number of mutants or +all mutants are outputted. +3.1 +AST Nodes Selection +µBERT parses the AST of the input source code and selects +the lines that are more likely to carry the program’s specifi- +cation implementation, excluding the import statements and +the declaration ones, e.g. the statements declaring a class, a +method, an attribute, etc. This way, the approach focuses the +mutation on the business-logic portion of the program and +excludes the lines that are probably of lower impact on the +program behaviour. It proceeds then, by selecting from each +of these statements, the relevant nodes to mutate, i.e. the +operators, the operands, the method calls and variables, etc., +and excluding the language-specific ones, like the separators +and the flow controls, i.e. semicolons, brackets, if, else, +etc. Table 1 summarises the type of targeted AST nodes +by µBERT, with corresponding example expressions and +induced mutants. We refer to these as the conventional +mutations provided by µBERT, denoted by µBERTconv in +our evaluation, previously introduced in the preliminary +version of the approach [18]. +3.2 +Token Masking +In this step, we mask the selected nodes one by one, pro- +ducing a masked version from the original source code for +each node of interest. This means that every masked version +contains the original code with one missing node, replaced +by the placeholder . +This way, µBERT can generate several mutants in the +same program location. For instance, for an assignment ex- +pression like res = a + b, µBERT will create (potentially +25) mutants from the following masked sequences: +• = a + b +• res = a + b +• res = + b +• res = a b +• res = a + +3.3 +CodeBERT-MLM prediction +µBERT invokes CodeBERT to predict replacements for the +masked nodes. To do so, it tokenizes every masked version +into a tokens vector then crops it to a subset one that fits the +maximum size allowed by the model (512) and counts the +masked token with the surrounding code-tokens. Next, our +approach feeds these vectors to CodeBERT MLM to predict +the most probable replacements of the masked token. Our +intuition is that the larger the code portion accompanying +the mask placeholder, the better CodeBERT would be able +to capture the code context, and consequently, the more +meaningful its predictions would be. This step ends with +the generation of five predictions per masked token. +3.4 +Condition seeding +µBERT generates second-order mutants by combining con- +dition seeding with CodeBERT prediction capabilities. To do +so, our approach modifies the conditions in control flow and + +ava4 +TABLE 1: Example of µBERT conventional mutations, available in the preliminary version of the approach [18], denoted by +µBERTconv. +Ast node +Expression +Masked Expression +Mutant Example +literals +res + 10 +res + +res + 0 +identifiers +res + 10 + + 10 +a + 10 +binary expressions +a && b +a b +a || b +unary expressions +--a +a +++a +assignments +sum += current +sum = current +sum -= current +object fields +node.next +node. +node.prev +method calls +list.add(node) +list.(node) +list.push(node) +array access +arr[index + 1] +arr[] +arr[index] +static type references +Math.random() * 10 +.random() * 10 +Random.random() * 10 +return statements, including if, do, while and return +conditional expressions. For every one of these statements, +it starts by extending the original condition by a new one, +separated with the logical operator && or ||, in both orders +(original condition first or the other way around) and with +or without negation (!). +Next, all substitute conditions are put one by one in place +in the original code, forming multiple condition-seeded +code versions, that we pass as input to Step (2), in which +their tokens are masked and then (3) passed each to Code- +BERT to predict the best substitute of their corresponding +masked tokens. +The seeded conditions are created in two ways: +3.4.1 +Using existing conditions in the same class +To mutate a given condition – if, do, while and return +conditional expressions –, we collect all other conditions +existing in the same class, then combine each one of them +with the target condition, using logical operators. +Precisely, let Expt a conditional expression to mutate +and SE = {Exp0, ..., Expn} the set of other conditional +expressions appearing in the same class, excluding the null- +check ones (i.e. var == null). The alternative replacement +conditions generated for Expt are the combinations of: +• Expt op neg Expi and +• Expi op neg Expt, +where op is a binary logical operator taking the values in +{&&,||}, neg is either the negation operator ! or nothing +and Expi is a condition from SE. +3.4.2 +Using existing variables in the same class +When the target if conditional expression to mutate con- +tains variables (including fields), we create new additional +conditions by combining these variables with others of the +same type from the same class. Then we combine each one +of the newly created conditions with the original one, using +logical operators. +Precisely, let Expt be a conditional expression to mutate +containing a set of variables Svt. For every variable vart in +Svt, we load Sv = {var0, ..., varn} the set of other variables +appearing in the same class and of the same type T as vart, +then we generate the following new conditions: +• Expt op (vart relop vari) and +• (vart relop vari) op Expt, +where op is a binary logical operator taking the values in +{&&,||}, relop is a relational operator applicable on the +type T and vari is a variable from Sv. +3.5 +Mutant filtering +In this step, our approach starts by discarding accurate and +duplicate predictions; the redundant predictions and the +ones that are exactly the same as the original code. Then, it +iterates through the statements and selects in every iteration +one compilable prediction by line, while discarding not +compilable ones. Once all first-order mutants are selected +(issued by one single token replacement), our approach +proceeds by selecting second-order ones (issued by the com- +bination of condition seeding and one token replacement) in +the same iterative manner. µBERT continues iterating until +achieving the desired number of mutants or all mutants are +outputted. +4 +RESEARCH QUESTIONS +We start our analysis by investigating the advantage +brought by the additive mutations (a.k.a. conditions seeding +ones) w.r.t. the fault detection capabilities of test suites +designed to kill µBERT’s mutants. Thus, we ask: +RQ1 (µBERT Additive mutations) What is the added value of +the additive mutations on the fault detection capabili- +ties of test suites designed to kill µBERT’s mutants? +To answer this question, we generate two sets of mutants +using µBERT: 1) the first set using all possible mutations that +we denote as µBERT and 2) a second one using only the +conventional µBERT’ mutations – part of our preliminary +implementation [18], excluding the additive ones – that we +denote as µBERTconv. Then we evaluate the fault detection +ability of test suites selected to kill the mutants from each +set. +The answer of this question provides evidence that the +additive mutations increase the fault detection capability of +µBERT. Yet, to assess its general performance we compare +it to state-of-the-art (SOA) mutation testing, particularly +PiTest [17], and thus, we ask: +RQ2 (Fault detection) How does µBERT compare with state- +of-the-art mutation testing, in terms of fault detection? +To answer this question we generate mutants using the +latest version of PiTest [17], on the same target projects as +for RQ1. As we are interested in comparing the approaches +and not the implementations of the tools, we exclude the +subjects on which PiTest did not run correctly or did not +generate any mutant. This way we ensure having a fair base +of comparison by counting exactly the same study subjects +for both approaches (further details are given in Section 5). +Then, we compare the fault detection capability of test suites + +5 +selected to kill the same number of mutants produced by +each approach. +Finally, we qualitatively analyse some of the mutants +generated with µBERT and ask: +RQ3 (Qualitative analysis) Does µBERT generate different +mutants than traditional mutation testing operators? +To answer this question, we showcase the mutants gen- +erated by µBERT that help in detecting faults not found +by PiTest. Additionally, we discuss the program-context- +capturing importance in µBERT’s functioning, by rerunning +it with a reduced size of the masked codes passed to the +model, and comparing examples of yielded mutants with +those obtained in our original setup. +5 +EXPERIMENTAL SETUP +5.1 +Dataset & Benchmark +To evaluate µBERT’s fault detection, we use real bugs from a +popular dataset in the software engineering research area +– Defects4J [29] v2.0.0. In this benchmark, every subject +bug is provided with a buggy version of the source code, +its corresponding fixed version, and equipped with a test +suite that passes on the fixed version and fails with at least +one test on the buggy one. The dataset includes over 800 +bugs from which, we exclude the ones presenting issues, i.e. +with wrong revision ids, not compiling or with execution +issues, or having failing tests on the fixed version, at the +reporting time. Next, we run µBERT and PiTest on the +corresponding classes impacted by the bug from the fixed +versions of the remaining bugs and exclude the ones where +no tool generated any mutant, ending up with 689 bugs +covered by µBERT and 457 covered by PitTest. As we’re +interested in comparing the approaches and not the tools’ +implementations, and to exclude eventual threats related to +the environment (i.e. supported java and juint versions by +each technique, etc.) or the limitations and shortages of the +dataset, we establish every comparison study on a dataset +counting only bugs covered by all considered approaches: +689 bugs to answer RQ1 and 457 to answer RQ2 and RQ3. +5.2 +Experimental Procedure +To assess the complementary and added value in terms of +fault revelation of the condition-seeding-based mutations +(answer to RQ1), we run our approach with and without +those additional mutations – that we name respectively +µBERT and µBERTconv–, and thus, generating all possible +mutants on our dataset programs’ fixed versions. Next, +we compare the average effectiveness of the test suites +generated to kill the mutants of each set; induced by µBERT +and µBERTconv. +Once the added value of the proposed condition- +seeding-based mutations is validated, we compare its per- +formance to S.O.A. mutation testing (answer to RQ2 and +RQ3). We use PiTest [17], a stable and mature Java mutation +testing tool, because it has been more effective at finding +faults than other tools [33] and it is among the most com- +monly used by researchers and practitioners [47], [52], as of +today. The tool proposes different configurations to adapt +the produced mutations and their general cost to the target +users, by excluding or including mutators. Among these +configurations we used the three following: +• Pit-all (ALL) which counts all available mutation oper- +ators available in the current version3. +• Pit-default (DEFAULTS) whose mutators are selected to +form a stable and cost-efficient subset of operators by +producing less but more relevant mutants. +• Pit-rv-all (ALL) which is a version4 that includes the +mutators of Pit-all and extra experimental [7] ones that +are made available for research studies. +To compare the different approaches, we evaluate their +effectiveness and cost-efficiency in achieving one of the +main purposes of mutation testing, i.e., to guide the testing +towards higher fault detection capabilities. For this reason, +we simulate a mutation testing use-case scenario, where +a developer/tester selects mutants and writes tests to kill +them [13], [34]. +We run every approach on the fixed versions and test +suites provided by Defects4J, then collect the mutants and +their test execution results; whether the mutant is killed +(breaks at least one test of the test suite) and if yes by which +tests. Next, we suppose that the not killed mutants are +equivalent or irrelevant, explaining why no tests have been +written to kill them by the developers. Then, we simulate the +scenario of a developer testing the fixed version, in a state +where 1) it did not have any test 2) thus all mutants did not +have killing tests and 3) the developer had no knowledge +of which mutants are equivalent or not. This way, we can +reproduce the developer flow of +1) selecting and analysing one mutant, +2) to either (a) discard it from the mutant set if it is +equivalent (not killed in the actual test suite) or (b) write +a test to kill it (by selecting one of the actual killing tests +of the mutant), +3) then discarding all killed mutants by that test and +4) iterating similarly over the remaining mutants until all +of them are analysed. +We say that a bug is found by a mutation testing technique +if the resulting test suite – formed by the written (selected) +tests by the developer – contains at least one test that reveals +it; a test that breaks when executed on the buggy version. +We express the testing cost in terms of mutants analysed, +and hence, we consider the effort required to find a bug as +the number of mutants analysed until the first bug-revealing +test is written. To set a common basis of comparison be- +tween the approaches, accounting for the different number +of generated mutants, we run the simulations until the same +maximum effort is reached (maximum number of mutants +to analyse), which we set to the least cost required to kill +all the mutants by one of the compared approaches. During +our evaluation study, we use the same mutation selection +strategy for all compared approaches, iterating through the +lines in random order and selecting 1 arbitrary mutant per +line per iteration. To reduce the process randomness impact +3. Version +1.9.4 +available +in +PitTest’s +[6] +GitHub +repository +(branch=master, +repo=https://github.com/hcoles/pitest.git, +rev- +id=17e1eecf) +4. Version +1.7.4 +available +in +PitTest’s +[6] +GitHub +repository +(branch=master, +repo=https://github.com/hcoles/pitest.git, +rev- +id=2ec1178a) + +6 +on our results (in the selection of mutants and tests), we +run every simulation 100 times, then average their results +for every target-bug and considered approach. Finally, we +aggregate these averages computed on all target bugs and +normalise them as global percentages of achieved fault +detection by spent effort, in terms of mutants analysed. +Finally, to answer RQ3, we select example mutants that +enabled µBERT to find bugs exclusively (not found by any +of PiTest versions), from the results of RQ2. Then we discuss +the added value of µBERT mutations through the analysis of +the mutants’ behavioural difference from the fixed version +and similarity with the buggy one. +5.3 +Implementation +We implemented µBERT’s approach as described in Sec- +tion 3: we have used Spoon [51] and Jdt [21] libraries to +parse and extract the business logic related AST nodes and +apply condition-seeding mutators. To predict the masked +tokens we have used the implementation proposed by +CodeBERT-nt [3], [31], using CodeBERT Masked Language +Modeling (MLM) functionality [2], [22]. +We provide the implementation of our approach and the +reproduction package of its evaluation at https://github. +com/Ahmedfir/mBERTa. +6 +EXPERIMENTAL RESULTS +6.1 +RQ1: µBERT Additive mutations +To answer this question we compare the fault detection +effectiveness of test suites written to kill mutants generated +by µBERT with and without additive mutations, noted re- +spectively µBERT and µBERTconv. Figure 2 depicts the fault +detection improvement when extending µBERT mutations +by the additive ones. In fact, µBERT fault detection increased +on average by over 9% compared to the one achieved by +µBERTconv, achieving 84.64% on average. We can also see +that besides outliers, the majority of bugs are found in 100% +of the times. Moreover, when examining the bugs separately, +we find that µBERT finds 20 more bugs than µBERTconv +(with fault detection > 0%), and 70 more when considering +bugs found with fault detection percentages above 90%. +This confirms that the additive patterns induce relevant +mutants ensuring the detection of some bugs always or in +most of the cases, as well as representing better new types +of faults, which were not detectable otherwise. +To check the significance of the fault detection advantage +brought by the additive patterns, we performed a statistical +test (Wilcoxon paired test) on the data of Figure 2a to vali- +date the hypothesis ”the fault detection yielded by µBERT +is greater than the one by µBERTconv ”. The very small +obtained p-values of 5.92e-21 (≪ 0.05) showed that the dif- +ferences are significant, indicating the low probability of this +fault detection amelioration to be happening by chance. The +difference size confirms also the same advantage, with ˆA12 +values of 0.5827 (> 0.5), indicating that µBERT induces test- +suites with higher fault detection capability in the majority +of the cases. +Next, we compare the fault detection performance of +µBERT and µBERTconv when analysing the same number +of mutants, and illustrate in Figure 3 their average fault +BERT +BERTconv +tool +0 +20 +40 +60 +80 +100 +Fault detection % +84.64% +75.30% +(a) Effectiveness: mean fault-detection per subject. +0 +20 +40 +60 +80 +100 +Effort % (number of analysed mutants) +0 +20 +40 +60 +80 +Fault detection % +tool +BERT +BERTconv +(b) Cost-efficiency: fault detection by the number of mutants +analysed. +Fig. 2: Fault-detection performance improvement when us- +ing additive patterns. Comparison between µBERT and +µBERTconv, w.r.t. the fault-detection of test suites written to +kill all generated mutants. +detection effectiveness and cost-efficiency in terms of anal- +ysed mutants. The box-plots of the Subfigure 3a show that +even when spending the same effort as µBERTconv, µBERT +keeps a similar advantage of on average 6.05% higher fault +detection, achieving a maximum of 81.35%. From the line- +plots of the Subfigure 3b, we can see that both approaches +achieve a comparable fault detection (≈ 70%) at (≤≈ 40%) +of the maximum costs. At higher costs, µBERTconv’s curve +increases slowly until achieving a plateau at ≈ 60% of +the effort, whereas µBERT’s curve keeps increasing to- +wards higher fault detection ratios even when achieving the +≈ 100% of the fixed maximum effort. +To validate these findings we re-conducted the same +statistical tests on the data of Subfigure 3a and found that +µBERT outperforms significantly µBERTconv with negligible +p-values of 1.15e-19 and ˆA12 values of 0.5711. + +7 +BERT +BERTconv +tool +0 +20 +40 +60 +80 +100 +Fault detection % +81.35% +75.30% +(a) Effectiveness: mean fault-detection per subject. +0 +20 +40 +60 +80 +100 +Effort % (number of analysed mutants) +0 +10 +20 +30 +40 +50 +60 +70 +80 +Fault detection % +tool +BERT +BERTconv +(b) Cost-efficiency: fault detection by the number of mutants +analysed. +Fig. 3: Fault-detection comparison between µBERT and +µBERTconv, with the same effort: where the maximum effort +is limited to the minimum effort required to analyse all mutants +of any of them, which is µBERTconv in most of the cases. +6.2 +RQ2: Fault Detection comparison with PiTest +To answer this research question we reduce our dataset to +the bugs covered by µBERT and the 3 considered versions of +PitTest approaches: ”Pit-default” which contains the default +mutation operators of PiTest, ”Pit-all” containing all PiTest +operators including the default ones and ”Pit-rv-all” which +contains experimental operators [7] in addition to the ”Pit- +all” ones. Then, we perform the same study as in RQ1, +where we compare the considered approaches’ effectiveness +and cost-efficiency based on the fault detection capability of +test suites written to kill their generated mutants. To have a +fair base of comparison, we compare the approaches under +the same effort in analysing mutants, which is equal to +the least average effort required to kill all mutants of one +of the approaches (which is the one of Pit-default in the +majority of the cases). As we are interested in comparing +the mutation testing approaches and not mutant selection +strategies, we run the simulation with the same one-mutant- +BERT +Pit-all +Pit-default +Pit-rv-all +tool +0 +20 +40 +60 +80 +100 +Fault detection % +66.43% +60.87% +49.90% +56.33% +(a) Effectiveness: mean fault-detection per subject. +0 +20 +40 +60 +80 +100 +Effort % (number of analysed mutants) +0 +10 +20 +30 +40 +50 +60 +Fault detection % +tool +BERT +Pit-all +Pit-default +Pit-rv-all +(b) Cost-efficiency: fault detection by the number of mutants +analysed. +Fig. 4: Fault-detection comparison between µBERT and +PiTest, with the same effort: where the maximum effort is +limited to the minimum effort required to analyse all mutants +of any of them, which is Pit-default in most of the cases. +per-line random sampling of mutants for all techniques (see +Subsection 5.2). +Figure 4b shows that with small effort (≤≈ 5%) all +approaches yield comparable fault detection scores (≈ 10%). +However, the difference becomes more noticeable when +spending more effort, with µBERT outperforming all ver- +sions of PiTest; achieving on average 16.53% higher fault +detection scores than Pit-default, 10.10% higher than Pit-rv- +all and 5.56% higher than Pit-all (see Figure 4a). +To validate these results, we performed the same statis- +tical tests as in RQ1, checking the hypothesis that ”µBERT +yields better fault detection capabilities than the other ap- +proaches”. We illustrate in the first row of Tables 2a and 2b +the corresponding computed Wilcoxon paired test p-values +and Vargha and Delaney ˆA12 values. Our results show that +µBERT has a significant advantage over the considered SOA +approaches with p-values under 0.05. Additionally, µBERT +scores ˆA12 values above 0.5 which confirms that guiding + +8 +TABLE 2: Paired (per subject bug) statistical tests of the +average fault detection of test suites written to kill the same +number of mutants generated by each approach (data of +Figure 4a). +(a) Wilcoxon paired test p-values computed on every dataset +subject, comparing each approach (A1) from the first column +to the other approaches (A2). p-values smaller than 0.05 in- +dicate that (A1) yields an average fault detection significantly +higher than that of (A2). +p-values +Pit-rv-all +Pit-default +Pit-all +µBERT +7.78e-11 +1.18e-12 +3.32e-02 +Pit-all +1.54e-22 +8.87e-06 +– +Pit-default +9.55e-01 +– +– +(b) Vargha and Delaney ˆA12 values computed on every dataset +subject, comparing each approach (A1) from the first column +to the other approaches (A2). ˆA12 values higher than 0.5 +indicate that (A1) yields an average fault detection higher than +that of (A2) in the majority of the cases. +ˆA12 +Pit-rv-all +Pit-default +Pit-all +µBERT +0.6488 +0.5514 +0.5066 +Pit-all +0.7210 +0.4956 +– +Pit-default +0.5449 +– +– +the testing by µBERT mutants instead of those generated by +SOA techniques yields comparable or higher fault detection +ratios, in the majority of the cases. Indeed, the ˆA12 differ- +ence between Pit-all and µBERT is small (0.5066), indicating +that both approaches perform similarly or better on some +studied subjects and worst on others. +We notice also from the sub-figure 4b that Pit-default +achieves a plateau at around 60% of the effort while the +other tools keep increasing, showing that they are able to +achieve higher fault detection capabilities, at a higher cost. +This is very noticeable when we compare the sub-figures +(a) and (b) of Figure 4 with the figure 2, where the average +fault detection of µBERT is way lower than what it achieves +in RQ1 – around 66% instead of 84%. This is a direct +consequence of the fact that Pit default produces fewer +mutants than the other approaches, limiting considerably +the maximum effort of the mutation campaigns and thus +the fault detection ratios, in the majority of the cases. Indeed, +as illustrated in Figure 5, all approaches score higher fault +detection percentages when spending more effort, achieving +on average ≈65% for Pit-all, ≈66% for Pit-rv-all and ≈83% +for µBERT. We explain the small decrease of 1.72% in the +mean fault detection achieved by µBERT in comparison +with RQ1 (82,92% in RQ2 instead of 84.64% in RQ1) by the +difference in the considered dataset for each RQ. +In Table 3, we illustrate the ˆA12 and p-values computed +on data of the boxplots in Sub-figure 5a. The results confirm +that µBERT outperforms significantly SOA mutation testing +w.r.t the fault detection capability of test suites written to all +kill mutants generated by each approach. +Next, we turned our interest to the set of particular bugs +that every approach can and cannot reveal when spending +the same effort. Hence, we map each bug with its revealing +tool, from the simulation results of Figure 4a and illustrate +their corresponding Venn diagrams in Figure 6. +BERT +Pit-all +Pit-default +Pit-rv-all +tool +0 +20 +40 +60 +80 +100 +Fault detection % +82.92% +65.49% +49.90% +66.35% +(a) Effectiveness: mean fault-detection per subject. +0 +20 +40 +60 +80 +100 +Effort % (number of analysed mutants) +0 +20 +40 +60 +80 +Fault detection % +tool +BERT +Pit-all +Pit-default +Pit-rv-all +(b) Cost-efficiency: fault detection by the number of mutants +analysed. +Fig. 5: Comparison between µBERT and PiTest, relative to +the fault-detection of test suites written to kill all generated +mutants. +From the disjoint sets in Sub-figure 6a, we notice a +clear advantage in using µBERT over the considered SOA +baselines, as it finds most of the bugs they find in addition to +finding exclusively 47 bugs when spending the same effort. +More precisely, µBERT finds 52, 77 and 52 more bugs than +Pit-all, Pit-default and Pit-rv-all, respectively, whereas they +find each 13, 10 and 13 bugs that µBERT missed. +This endorses the fact that µBERT introduces mutants +that represent more real bugs than SOA mutation tech- +niques, and encourages the investigation of the eventual +complementary between the approaches. This observation +is more noticeable when considering the overlapping be- +tween bugs found by each approach in at least 90% of the +simulations (Sub-figure 6b). We notice that the approaches +perform comparably, with a particular distinction of Pit-all +and Pit-default results which find exclusively 19 and 21 bugs +with these high fault detection percentages instead of 0, as +observed in Sub-figure 6a. Nevertheless, µBERT conserves +the same advantage over the considered baselines in this + +9 +TABLE 3: Paired (per subject bug) statistical tests of the +average fault detection of test suites written to kill all the +mutants generated by each approach (data of Figure 5a). +(a) Wilcoxon paired test p-values computed on every dataset +subject, comparing each approach (A1) from the first column +to the other approaches (A2). p-values smaller than 0.05 in- +dicate that (A1) yields an average fault detection significantly +higher than that of (A2). +p-values +Pit-rv-all +Pit-default +Pit-all +µBERT +2.49e-13 +2.14e-33 +1.47e-14 +Pit-all +4.71e-01 +2.76e-23 +– +Pit-default +1.00e+00 +– +– +(b) Vargha and Delaney ˆA12 values computed on every dataset +subject, comparing each approach (A1) from the first column +to the other approaches (A2). ˆA12 values higher than 0.5 +indicate that (A1) yields an average fault detection higher than +that of (A2) in the majority of the cases. +ˆA12 +Pit-rv-all +Pit-default +Pit-all +µBERT +0.6028 +0.7123 +0.6061 +Pit-all +0.5077 +0.6400 +– +Pit-default +0.3676 +– +– +regard, finding exclusively 42 bugs more. It finds also 50, 63 +and 69 more bugs than respectively Pit-all, Pit-default and +Pit-rv-all, whereas they find each 59, 58 and 27 bugs that +µBERT missed. +6.3 +RQ3: Qualitative Analysis of µBERT Mutants +To answer this research question we investigate the mutants +generated by µBERT, which induced test suites able to find +bugs that were not detected otherwise, i.e. by the considered +SOA approaches (see Figure 6). Meaning that the mutants +break similar tests as the target real buggy version. +As a simple bug example (requiring only one change +to +fix +it), +we +consider +Lang-49 +from +Defects4J +and +we investigate mutants that have been generated by +µBERT and helped in generating tests that reveal it. This +bug impacts the results of the method reduce() from +the class org.apache.commons.lang.math.Fraction, +which returns a new reduced fraction instance, if possible, +or the same instance, otherwise. The bug is caused by a +miss-implementation of a specific corner case, which con- +sists of calling the method on a fraction instance that has +0 as numerator. In Table 4, we illustrate example mutants +generated by µBERT that helped in revealing this bug. Every +mutant is represented by a diff between the fixed and the +mutated version by µBERT. +As can be seen, µBERT can generate mutants that can be +induced by applying conventional pattern-based mutations, +i.e., Mutant 1 replaces a relational operator (==) with an- +other (>) and Mutant 2 replaces an integer operand (0) with +another one (1). +In addition, it proposes more complex mutations that +are unlikely achievable without any knowledge of either the +AST or the context of the considered program. For instance, +it can generate Mutant 4 by changing a conditional return +statement with (this) the current instance of Fraction, +which matches the return type of the method. Similarly, to +47 +0 +1 +0 +1 +0 +3 +0 +3 +3 +23 +0 +1 +10 +354 +Pit-all +Pit-default +Pit-rv-all +BERT +(a) Faults discovered at least once per 100 runs +(Fault detection > 0%). +42 +2 +3 +21 +3 +0 +2 +19 +10 +3 +8 +15 +14 +22 +114 +Pit-all +Pit-default +Pit-rv-all +BERT +(b) Faults discovered in over 90% of the runs +(Fault detection≥ 90%). +Fig. 6: Number of faults discovered by test-suites written to +kill mutants generated by µBERT and PiTest versions when +analysing the same number of mutants (same effort). +generate Mutant 5, it replaces (this) the current instance of +the class Fraction by an existent instance of the same type +(ONE), making the statement returning either the object ONE +or the object ZERO. +To produce more complex mutants, µBERT applies a +condition seeding followed by token-masking and Code- +BERT prediction, such as adding || (numerator == +other.numerator) to the original condition of a return +statement, inducing Mutant 8, or adding || !(numerator +== Integer.MIN_VALUE) to the original condition of an +if statement, inducing Mutant 3. +To investigate further the impact of the code context +captured by the model on the generated mutants, we have +rerun µBERT on 5 subjects from our dataset, with a max- +imum number of surrounding tokens equal to 10 (instead +of 512). Then, we compared manually the induced mutants +with those generated by our default setup, in the same +locations. From our results, we observed a noticeable de- +crease in the number of compilable predictions, indicating +the general performance decrease of the model when it lacks +information about the code context. Particularly, we notice + +10 +TABLE 4: Example of mutants generated by µBERT that +helped find the bug Lang-49 from Defects4J. +Mutant 1: replacing binary operator +@@ org . apache . commons . lang . math . Fraction +: +466 @@ +− i f +( numerator == 0) { ++ i f +( numerator > 0) { +Mutant 2: replacing literal implementation +@@ org . apache . commons . lang . math . Fraction +: +466 @@ +− i f +( numerator == 0) { ++ i f +( numerator == 1) { +Mutant 3: adding a condition to an if statement +@@ org . apache . commons . lang . math . Fraction +: +466 @@ +− i f +( numerator == 0) { ++ i f +( ( numerator == 0) ++ +| | +! ( numerator==Integer .MIN VALUE) ) +{ +Mutant 4: replacing a condition +@@ org . apache . commons . lang . math . Fraction +: +467 @@ +− return +equals (ZERO) +? +t h i s : ZERO; ++ return +t h i s ; +Mutant 5: replacing this access by another object +@@ org . apache . commons . lang . math . Fraction +: +467 @@ +− return +equals (ZERO) +? +t h i s : ZERO; ++ return +equals (ZERO) +? ONE: ZERO; +Mutant 6: replacing method argument +@@ org . apache . commons . lang . math . Fraction +: +469 @@ +int gcd = greatestCommonDivisor ( +− Math . abs ( numerator ) , +denominator ) ; ++ Math . abs ( numerator ) , +1 ) ; +Mutant 7: replacing a variable +@@ org . apache . commons . lang . math . Fraction +: +473 @@ +− return +Fraction . getFraction ( numerator / gcd , ++ return +Fraction . getFraction ( numerator / 3 , +denominator / gcd ) ; +Mutant 8: adding a condition to a return statement +@@ org . apache . commons . lang . math . Fraction +: +840 @@ +return +( getNumerator ( ) == other . getNumerator ( ) +− +&& getDenominator ( ) == other . getDenominator ( ) ) ; ++ +&& getDenominator ( ) == other . getDenominator ( ) ) ) ++ +| | +( numerator == other . numerator ) ; +that it is not able to produce program-specific mutants, i.e. +by changing an object by another or a method call with +another. In Table 5, we illustrate some example mutants that +helped find each of the studied subjects (breaking same tests +as the original bug), which µBERT failed to generate when +the maximum number of surrounding tokens is limited to +10. +7 +THREATS TO VALIDITY +One external threat to validity concerns the generalisation +of our findings and results in the empirical evaluation. To +reduce this threat, we used a large number of real bugs +from popular open-source projects with their associated +developer test-suites, provided by an established and in- +dependently built benchmark (i.e. Defects4J [29]). Never- +theless, we acknowledge that the results may be different +considering projects in different domains. +Other threats may arise from our way of assessing the +fault detection capability of mutation testing approaches, +based on their capability of guiding the testing via a devel- +oper/tester simulation in which we assume that the current +test suites are complete and the not killed mutants are +equivalent. Although we acknowledge that this may not +be the case in real-world scenarios, we believe that this +process is sufficient to evaluate our approach, particularly +considering the fact the test suites provided by Defects4J +are relatively strong. Additionally, to mitigate any com- +parison threat between the considered approaches, we use +consistently and similarly the same test-suites, setups and +simulation assumptions in all our study. +The choice of our comparison baseline may form other +threats to the validity of our findings. While different fault- +seeding approaches have been proposed recently, PiTest +remains among the most mature and stable mutation test- +ing tools for Java programs, thus, forming an appropriate +comparison baseline to evaluate our work. Furthermore, we +compared our results with those obtained by mutants from +different configurations proposed by PiTest, enlarging our +study to the different audiences targeted by this latter. We +acknowledge however that the results may change when +considering other techniques and consider the evaluation +of the effectiveness and cost-efficiency of different mutation +testing techniques as out of the scope of this paper. +Other construct threats may arise from considering the +number of mutants analysed as metric to measure the effort +required to find a fault. In addition to the fact that this metric +has been widely used by the literature [9], [34], [47], we +believe that it is intuitive and representative of the global +manual effort of the tester in analysing the mutants, dis- +carding them or writing tests to kill them. While being the +standard in the literature, we acknowledge that this measure +does not account for the cost difference between mutants, +attributing the same cost to all mutants. This is simply +because we do not know the specific effort required to +analyse every specific mutant or to write every specific test. +Additionally, our cost-efficiency results may be impacted +by costs that are not captured with this metric, such as +the execution or the developing effort of either CodeBERT, +the key component of µBERT, or the set of patterns and +execution enhancements over the different releases of PiTest. +Nevertheless, we tried to mitigate any major threats that +can be induced by the implementation of the different tools, +i.e. we reduce the dataset subjects to those on which every +approach generated at least one mutant and consider any +implementation difference between the approaches as out +of the current scope. +8 +RELATED WORK +Since the 1970s, mutation testing has been the main focus +of multiple research works [57]. Their findings have proven +that artificial faults can be useful in multiple software en- +gineering applications, such as testing [47], debugging [37], +[48], assessing fault tolerance [42], risk analysis [16], [56] and +dependability evaluation [10]. +Despite this long history of research, the generation +of relevant mutants remains an open question. Most of +the related research has focused on the design of fault + +11 +TABLE 5: Example of mutants generated by µBERT that helped in finding bugs from Defects4J and could not be generated +when limiting the maximum number of surrounding tokens to 10. +Mutant 1 (JacksonCore-4) : replacing a method call +@@ com . fasterxml . jackson . core . u t i l . TextBuffer +: +515 @@ +− unshare ( 1 ) ; ++ expand ( 1 ) ; +Mutant 2 (Closure-26) : replacing an object +@@ com . google . j a v a s c r i p t . jscomp . ProcessCommonJSModules +: +89 @@ +− . replaceAll ( Pattern . quote ( F i l e . separator ) , MODULE NAME SEPARATOR) ++ . replaceAll ( Pattern . quote ( filename ) , MODULE NAME SEPARATOR) +Mutant 3 (Closure-35) : replacing a method call +@@ com . google . j a v a s c r i p t . jscomp . TypeInference +: +1092 @@ +− scope = traverseChildren (n , +scope ) ; ++ scope = traverse (n , +scope ) ; +Mutant 4 (Lang-27) : replacing a method call +@@ org . apache . commons . lang3 . math . NumberUtils +: +526 @@ +− i f +( ! ( f . i s I n f i n i t e ( ) +| | +( f . floatValue ( ) == 0.0 F && ! allZeros ) ) ) +{ ++ i f +( ! ( f . i s I n f i n i t e ( ) +| | +( f . round ( ) == 0.0 F && ! allZeros ) ) ) +{ +/ / +a l s o +” f . f l o a t V a l u e ( ) ” +to ” f . s c a l e ( ) ” +Mutant 5 (Math-64) : replacing an object +@@ org . apache . commons . lang . math . Fraction +: +852 @@ +− for +( i nt +j = k ; +j < jacobian . length ; ++ j ) { ++ for +( i nt +j = k ; +j < beta . length ; ++ j ) { +Mutant 6 (Lang-27) : replacing an object +@@ org . apache . commons . lang3 . math . NumberUtils +: +526 @@ +− i f +( ! ( f . i s I n f i n i t e ( ) +| | +( f . floatValue ( ) == 0.0 F && ! allZeros ) ) ) +{ ++ i f +( ! ( f . i s I n f i n i t e ( ) +| | +( f . round ( ) == 0.0 F && ! zero ) ) ) +{ +patterns (mutation operators) which are usually defined +based on the target language grammar [8], [47] then refined +through empirical studies [33], [40], [44] aiming at reducing +the redundancy and noise among their generated mutants. +The continuous advances in this sense were followed by +a constant emergence of pattern-based mutation testing +tools and releases [17], [35], [39], among which some are +becoming popular and widely adopted by researchers and +practitioners, such as PiTest [17], from which we consider +three configurations as our comparison baseline. +Recent research has focused their interest on improving +the representativeness of artificial faults aiming at reducing +the mutation space to real-like faults. For instance, instead of +basing the mutation operators’ design on the programming +language grammar, Brown et al. [12] proposed inferring +them from real bug fixes. Similarly, Tufano et al. [54] pro- +posed a neural machine translation technique that learns +how to inject faults from real bug fixes. Along the same +line, Patra et al. [50] proposed a semantic-aware learning +approach, that learns and then adapts fault patterns to the +project of interest. Their results are promising, however, +the fact that these techniques depend on the availability +of numerous, diverse, comprehensive and untangled fix +commits [27] of not coupled faults [43], which is often hard +to fulfil in practice, may hinder their performance. Acknowl- +edging for the injection location [13], [42], Khanfir et al. [32] +combined the usage of information retrieved from bug +reports with inverted automated-program-repair patterns to +replicate real faults fixable by the original fix-patterns. Their +results showed that they can generate faults that mimic real +ones, however, their approach remains dependent and lim- +ited to the presence of good bug reports. Overall, designing +the mutation operators based on the known faults space +yields more diverse mutants that represent more fault types. +However, these extended operator sets tend to increase the +number of generated mutants and consequently the general +cost of the mutation campaign i.e. the fault patterns pro- +posed by Brown et al. and Khanfir et al. counted also most of +the conventional mutators in addition to new ones. Unlike +these techniques, µBERT leverages pre-trained models to +introduce mutants based on code knowledge instead of the +faults one. As code is more available than faults, it offers a +more flexible and complete knowledge base than faults, i.e. +it perms to overcome the limitations and efforts required 1) +to collect clean bug-fixing commits, 2) to capture the faulty +behaviour and 3) design fault patterns, be it manually or via +machine learning techniques. +Aiming at reducing the number of generated mutants, +researchers have proposed different strategies to generate +relevant mutants. For instance, studies that show that mu- +tant strength resides in not only its inducing pattern but also +where it is injected [13], [42], motivated the importance of +selecting relevant locations to mutate. In this regard, Sun +et al. [53] suggest mutating multiple places within diverse +program execution paths. Gong et al. [26] also propose the +mutation in diverse locations of the program extracted from +graph analysis. Similarly, Mirshokraie et al. [41] compute +complexity metrics from program executions to extract loca- + +12 +tions with good observability to mutate. Other approaches +restrict the fault injection on specific locations of the pro- +gram, such as the code impacted by the last commits [38], +[58] for better usability in continuous integration, or target- +ing locations related to a given bug-report [32] to target a +specific feature or behaviour, etc. More recent advances have +resulted in powerful techniques for cost-effectively selecting +mutants, i.e., by avoiding the analysis of redundant mutants +(basically, equivalent and subsumed ones) [24], [25], [28]. In +particular, the work of Garg et al. [24] utilises the knowledge +of mutants’ surrounding context, embedded into the vector +space, to predict whether a mutant is likely subsuming +or not. In this work, we do not target any specific code +part or any narrow use case, but instead, perform fault +injection in a brute-force way similarly to mutation testing, +by iterating every program statement and masking every +involved token. +Multiple studies have been focused on the relationship +between artificial and real faults [47]. The results of the stud- +ies conducted by Ojdanic et al. [45], Papadakis et al. [49], +Just et al. [30] and Andrews et al. [9] showed that there +is a correlation between tests broken by a bug and tests +killing mutants. Meaning that artificial faults can be used +as alternatives to real faults in controlled studies. Moreover, +the findings of Chekam et al. [14], Frankl et al. [23] and +Li et al. [36] show that guiding testing by mutants leads +to significantly higher fault revelation capability than the +ones of other test adequacy criteria. Based on these findings, +we assess our approach based on the relation between the +injected and real faults, in terms of breaking tests. More +precisely, we conduct a fault detection effectiveness and +cost-efficiency study to evaluate our approach’s mutants in +guiding testing and compare it to state-of-the-art techniques. +Furthermore, we discuss the diversity and readability of +µBERT mutants through real examples. +The closest related work is a preliminary implementation +of µBERT that was recently presented in the 2022 mutation +workshop [18]. This implementation, denoted as µBERTconv +in our evaluation, includes the conventional mutations (to +mask and replace tokens by the model predictiosn), but it +does not include the condition-seeding additive mutations +that provide major benefits for fault detection. Moreover, +µBERTconv was evaluated only on 40 bugs from Defects4J, +and compared only to an early version of PiTest (similar +to Pit-rv-all). In this work, we perform an extensive exper- +imental evaluation including 689 bugs from Defects4J and +compare µBERT effectiveness with three different configura- +tions from PiTest. Moreover, we show that µBERT finds on +average more bugs than µBERTconv without requiring more +effort. +9 +CONCLUSION +We presented µBERT; a pre-trained language model based +fault injection approach. µBERT provides researchers and +practitioners with easy-to-understand “natural” mutantsto +help them in writing tests of higher fault revelation capabil- +ities. +Unlike state-of-the-art approaches, it does neither re- +quire nor depend on any kind of faults knowledge or +language grammar but instead on the actual code definition +and distribution, as written by developers in numerous +projects. This facilitates its developing, maintainability, inte- +gration and extension to different programming languages. +In fact, it reduces the overhead of learning how to mutate, +be it via creating and selecting patterns or collecting good +bug-fixes and learning from their patches. +In a nutshell, µBERT takes as input a given program and +replaces different pieces of its code base with predictions +made by a pretrained generative language model, produc- +ing multiple likely-to-occur mutations. The approach targets +diverse business code locations and injects either simple +one-token replacement mutants or more complex ones by +extending the control-flow conditions. This provides proba- +ble developer-like faults impacting different functionalities +of the program with higher relevance and lower cost to +developers. This is further endorsed by our results where +µBERT induces high fault detection test suites at low effort, +outperforming state-of-the-art techniques (PiTest), in this +regard. +We have made our implementation and results avail- +able [5] to enable reproducibility and support future re- +search. +ACKNOWLEDGMENT +This work was supported by the Luxembourg National +Research Fund (FNR) projects C20/IS/14761415/TestFlakes +and TestFast, ref. 12630949. +REFERENCES +[1] +Amazon +codewhisperer. +https://aws.amazon.com/ +codewhisperer/. +[2] +Codebert. https://github.com/microsoft/CodeBERT. +[3] +Codebert-nt. https://github.com/Ahmedfir/CodeBERT-nt. +[4] +Github copilot. https://github.com/features/copilot. +[5] +mberta. https://github.com/Ahmedfir/mBERTa. +[6] +Pitest. http://pitest.org/. +[7] +Pitest-rv-plugin. https://github.com/pitest/pitest-rv-plugin. +[8] +Paul Ammann and Jeff Offutt. +Introduction to Software Testing. +Cambridge University Press, 2008. +[9] +James H. 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Association for Computing Machinery. + diff --git a/-NE1T4oBgHgl3EQf8QVm/content/tmp_files/load_file.txt b/-NE1T4oBgHgl3EQf8QVm/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..66e011440a74a476dbb80b515442dfa7062e3e53 --- /dev/null +++ b/-NE1T4oBgHgl3EQf8QVm/content/tmp_files/load_file.txt @@ -0,0 +1,942 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf,len=941 +page_content='1 Efficient Mutation Testing via Pre-Trained Language Models Ahmed Khanfir , Renzo Degiovanni , Mike Papadakis and Yves Le Traon SnT, University of Luxembourg, Luxembourg Abstract—Mutation testing is an established fault-based testing technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' It operates by seeding faults into the programs under test and asking developers to write tests that reveal these faults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' These tests have the potential to reveal a large number of faults – those that couple with the seeded ones – and thus are deemed important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' To this end, mutation testing should seed faults that are both “natural” in a sense easily understood by developers and strong (have high chances to reveal faults).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' To achieve this we propose using pre-trained generative language models (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' CodeBERT) that have the ability to produce developer-like code that operates similarly, but not exactly, as the target code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' This means that the models have the ability to seed natural faults, thereby offering opportunities to perform mutation testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' We realise this idea by implementing µBERT, a mutation testing technique that performs mutation testing using CodeBert and empirically evaluated it using 689 faulty program versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Our results show that the fault revelation ability of µBERT is higher than that of a state-of-the-art mutation testing (PiTest), yielding tests that have up to 17% higher fault detection potential than that of PiTest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Moreover, we observe that µBERT can complement PiTest, being able to detect 47 bugs missed by PiTest, while at the same time, PiTest can find 13 bugs missed by µBERT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Index Terms—Fault Injection, Mutation Testing, Pre-Trained Language Models !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' 1 INTRODUCTION Mutation testing aims at seeding faults using simple syntac- tic transformations [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' These transformations, also known as mutation operators are typically constructed based on syntactic rules crafted based on the grammar of the target programming language [8], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' replacing an arithmetic op- erator with another such as a + by a -.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Unfortunately, such techniques generate mutants (seeded faults), many of which are “unatural”, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=', non-conforming to the way developers code, thereby perceived as unrealistic by developers [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' At the same time, the syntactic-based fault seeding fails to capture the semantics of the code snippets that they apply, leading to numerous trivial or low utility faults [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' To deal with the above issue we propose forming natural mutations by using big code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Thus, we aim at introducing modifications that follow the implicit rules, norms and cod- ing conventions followed by programmers, by leveraging the capabilities of pre-trained language models to capture the underlying distribution of code and its writing, as learned by the pre-training process on big code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' To this end, we rely on CodeBERT [22], an NL-PL bimodal language model that has been trained on over 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='4 million programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' More precisely, we use its Masking Modelling Language (MLM) functionality, which given a code sequence with a masked token, predicts alternative replacements to that token, that is best matching the se- quence context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' This is important, since the predictions do not follow fixed predefined patterns as is the case of conventional mutation testing, but are instead adapted to fit best the target code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' For instance, given a sequence A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Khanfir, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Degiovanni, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Papadakis, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Le Traon are with the University of Luxembourg, Luxembourg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' int a = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=', we pass a masked version of it as int a = ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=', then CodeBERT by default proposes 5 predictions sorted by likelihood score: 0, 1, b, 2, and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Being the most likely fitting tokens to the code context, our intuition is that replacing the masked token with these predictions would induce “natural” mutants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Precisely, we introduce µBERT, a mutation testing ap- proach that uses a pre-trained language model (CodeBERT) to generate mutants by masking and replacing tokens with the aim of forming natural mutants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' µBERT iterates through the program statements and modifies their token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' In par- ticular, µBERT proceeds as follows: (1) it selects and masks one token at a time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' (2) feeds CodeBERT with the masked sequence and obtains the predictions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' (3) creates mutants by replacing the masked token with the predicted ones;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' and (4) discards non-compilable, duplicate and equivalent mutants (mutants syntactically equal to original code).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Recent research [32] has shown that some real faults are only captured by using complex patterns, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' patterns that require more than one token mutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' To account for such cases, µBERT is equipped with additive mutations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=', mutations that add code (instead of deleting or altering).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' For example, consider a boolean expression e1 (typically present in if, do, while and return statements), which is mutated by µBERT by adding a new condition e2, thereby generating a new condition e1||e2 (or e1&&e2), which is then masked and completed by CodeBERT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' For instance, given a condi- tion if(a == b), µBERT produces a new condition if(a == b || a > 0) that is masked and produces if(a == b || b > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' We implement µBERT, and evaluate its ability to serve the main purposes of mutation testing, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' guiding the testing towards finding faults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' We thus, evaluate it using 689 faults from Defects4J and asses µBERT effectiveness and arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='03543v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='SE] 9 Jan 2023 2 cost-efficiency to reveal1 them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Our results show that µBERT is very effective in terms of fault revelation, finding on average 84% of the faults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' This implies that µBERT mutants cover efficiently faulty behaviours caused by real bugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' More importantly, the approach is noticeably more effec- tive and cost-efficient than a traditional mutation testing technique, namely PiTest [17], that we use as a baseline in our evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Precisely, we consider three different configurations for PiTest that uses different sets of mutation operators (DEFAULT, ALL and RV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' In fact, test suites that kill all mutants of µBERT find on average between 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='5% to 33% more faults than those generated to kill all mutants introduced by PiTest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Moreover, even when analysing the same number of mutants, µBERT induces test suites that find on average 6% to 16% more faults than PiTest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' These results are promising and endorse the usage of µBERT over the considered mutation testing technique, as a test generation and assessment criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' We also study the impact of the condition-seeding-based mutations in the fault detection capability of µBERT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' We observe that test-suites designed to kill both kinds of µBERT mutants – induced by 1) direct CodeBERT predictions and 2) a combination of conditions-seeding with CodeBERT predictions – find on average over 9% more bugs than the ones designed to kill direct CodeBERT prediction mutants only (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Overall, our main contributions are: We introduce µBERT, the first mutation testing ap- proach that uses pre-trained language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' It lever- ages the model’s code knowledge captured during its pretraining on large code corpora and its ability to capture the program context, to produce “natural” mu- tants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' We propose new additive mutations which operate by seeding new conditions in the existing conditional expressions of the target code, then masking and re- placing their tokens with the model predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' We provide empirical evidence that µBERT mutants can guide testing towards higher fault detection capabili- ties, outperforming those achieved by SOA techniques (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' PiTest), in terms of effectiveness and cost-efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' In our empirical study, we validate also the advantage of employing the new additive mutation patterns, w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='t improving the effectiveness and cost-efficiency in writ- ing test suites with higher fault revelation capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' 2 BACKGROUND 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='1 Mutation Testing Mutation analysis [47] is a test adequacy criterion repre- senting test requirements by the mean of mutants, which are obtained by performing slight syntactic modifications to the original program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' For instance, an expression like x > 0 can be mutated to x < 0 by replacing the relational operator > with <.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' These mutants are then used to assess the effectiveness and thoroughness of a test suite in detecting their corresponding code modification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Tests are written/generated to kill (reveal) the mutants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' A bug is revealed by a mutation testing approach, if the written tests to kill its mutants also reveal the bug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' A test case detects a mutant if it is capable of producing distinguishable observable outputs between the mutant and the original program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' A mutant is said to be killed if it is detected by a test case or a test suite;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' otherwise, it is called live or survived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Some mutants cannot be killed as they are functionally equivalent to the original program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' The mutation score measures the test suite adequacy and is computed as the ratio of killed mutants over the total number of generated mutants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='2 Generative Language Models Advances in deep learning approaches gave birth to new language models for code generation [1], [4], [15], [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' These models are trained on large corpora counting multiple projects, thereby acquiring a decent knowledge of code, enabling them to predict accurately source code to devel- opers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Among these pre-trained models, CodeBERT [22], a language model that has been recently introduced and made openly accessible for researchers by Microsoft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' CodeBERT is an NL-PL bimodal pre-trained language model (Natural Language Programming Language) that supports multiple applications such as code search, code documentation generation, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Same as most large pre- trained models, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' BERT [20], CodeBERT’s developing adopts a Multilayer Transformer [55] architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' It has been trained on a large corpus collected from over 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='4 million projects available on GitHub, counting 6 different programming languages, including Java.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' The model was trained in a cross-modal fashion, through bimodal NL-PL data, where the input data is formed by pairs of source code and its related documentation, as well-as unimodal data, including either natural language or programming language sequences per input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' This way, it enables the model to offer both – PL and NL-PL – functionalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' The training targets a hybrid objective function, that is based on replaced token detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' µBERT incorporates the Masked Language Modeling (MLM) functionality [2] of CodeBERT in its workflow, to generate “natural” mutants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' The CodeBERT MLM pipeline takes as input a code sequence of maximum 512 tokens, including among them one masked as , whose value will be predicted by the model based on the context captured from the remaining tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' CodeBERT provides by default 5 predictions per token, among which we use the inaccurate and compilable predicted codes as mutants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' 3 APPROACH We propose µBERT, a generative language-model-based mu- tation testing approach, which is described step by step in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Given an input source code, µBERT leverages CodeBERT’s knowledge of code and its capability in captur- ing the program’s context to produce “natural” mutations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' that are similar to eventual developer mistakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='To do so, µBERT proceeds as follows in six steps: 1) First, it extracts relevant locations (AST 2 nodes) where to mutate 2) Second, it masks the identified node-tokens, creating one masked version per selected token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' AST: Abstract Syntax Tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' 3 if (a !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='= b) return a !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='= d || b>0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Source-code AST nodes locations 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' AST Nodes selection Y Y 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Masked code prediction Model Predictions Predict° 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Conditions seeding 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Injection & Compilation check Injected faults if (a !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='= b) return a !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='= d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' if (a !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='= b && b>i) return a !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='= d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' a = b + c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' return a d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' AST Nodes Masking a = b + c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' return a == d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' a = + c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' return a == d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' 1: µBERT Workflow: (1) it parses the Java code given as input, and extracts the expressions to mutate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' (2) it creates simple-replacement mutants by masking the tokens of interest and invoking CodeBERT;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' (3) it generates the mutants by replacing the masked token with CodeBERT predictions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' (4) it generates complex mutants via a) conditions-seeding, b) tokens masking then c) replacing by CodeBERT predictions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' and finally, (5) it discards not compiling and syntactically identical mutants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' 3) Then, it invokes CodeBERT to predict replacements for these masked tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' 4) In addition to the mutants produced in Step (3), µBERT also implements some condition-seeding additive mu- tations that modify more than one token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Precisely, it modifies the conditional expressions in the control flow (typically present in if, do, while and return state- ments) by extending the original condition with a new one, combined with the logical operator && or ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Then, the new conditional expression is mutated by following the same steps (2) and (3) – masking and replacing the masked tokens by the CodeBERT predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' 5) Finally, the approach discards duplicate predictions or those inducing similar code to the original one, or not compiling, and outputs the remaining ones as mutants, from diverse locations of the target code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' More precisely, it iterates through the statements in random order and outputs in every iteration one mutant per line, until achieving the desired number of mutants or all mutants are outputted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='1 AST Nodes Selection µBERT parses the AST of the input source code and selects the lines that are more likely to carry the program’s specifi- cation implementation, excluding the import statements and the declaration ones, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' the statements declaring a class, a method, an attribute, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' This way, the approach focuses the mutation on the business-logic portion of the program and excludes the lines that are probably of lower impact on the program behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' It proceeds then, by selecting from each of these statements, the relevant nodes to mutate, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' the operators, the operands, the method calls and variables, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=', and excluding the language-specific ones, like the separators and the flow controls, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' semicolons, brackets, if, else, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Table 1 summarises the type of targeted AST nodes by µBERT, with corresponding example expressions and induced mutants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' We refer to these as the conventional mutations provided by µBERT, denoted by µBERTconv in our evaluation, previously introduced in the preliminary version of the approach [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='2 Token Masking In this step, we mask the selected nodes one by one, pro- ducing a masked version from the original source code for each node of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' This means that every masked version contains the original code with one missing node, replaced by the placeholder .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' This way, µBERT can generate several mutants in the same program location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' For instance, for an assignment ex- pression like res = a + b, µBERT will create (potentially 25) mutants from the following masked sequences: = a + b res = a + b res = + b res = a b res = a + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='3 CodeBERT-MLM prediction µBERT invokes CodeBERT to predict replacements for the masked nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' To do so, it tokenizes every masked version into a tokens vector then crops it to a subset one that fits the maximum size allowed by the model (512) and counts the masked token with the surrounding code-tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Next, our approach feeds these vectors to CodeBERT MLM to predict the most probable replacements of the masked token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Our intuition is that the larger the code portion accompanying the mask placeholder, the better CodeBERT would be able to capture the code context, and consequently, the more meaningful its predictions would be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' This step ends with the generation of five predictions per masked token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='4 Condition seeding µBERT generates second-order mutants by combining con- dition seeding with CodeBERT prediction capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' To do so, our approach modifies the conditions in control flow and ava4 TABLE 1: Example of µBERT conventional mutations, available in the preliminary version of the approach [18], denoted by µBERTconv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Ast node Expression Masked Expression Mutant Example literals res + 10 res + res + 0 identifiers res + 10 + 10 a + 10 binary expressions a && b a b a || b unary expressions --a a ++a assignments sum += current sum = current sum -= current object fields node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='next node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='prev method calls list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='add(node) list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='(node) list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='push(node) array access arr[index + 1] arr[] arr[index] static type references Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='random() * 10 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='random() * 10 Random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='random() * 10 return statements, including if, do, while and return conditional expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' For every one of these statements, it starts by extending the original condition by a new one, separated with the logical operator && or ||, in both orders (original condition first or the other way around) and with or without negation (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Next, all substitute conditions are put one by one in place in the original code, forming multiple condition-seeded code versions, that we pass as input to Step (2), in which their tokens are masked and then (3) passed each to Code- BERT to predict the best substitute of their corresponding masked tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' The seeded conditions are created in two ways: 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='1 Using existing conditions in the same class To mutate a given condition – if, do, while and return conditional expressions –, we collect all other conditions existing in the same class, then combine each one of them with the target condition, using logical operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Precisely, let Expt a conditional expression to mutate and SE = {Exp0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=', Expn} the set of other conditional expressions appearing in the same class, excluding the null- check ones (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' var == null).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' The alternative replacement conditions generated for Expt are the combinations of: Expt op neg Expi and Expi op neg Expt, where op is a binary logical operator taking the values in {&&,||}, neg is either the negation operator !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' or nothing and Expi is a condition from SE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='2 Using existing variables in the same class When the target if conditional expression to mutate con- tains variables (including fields), we create new additional conditions by combining these variables with others of the same type from the same class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Then we combine each one of the newly created conditions with the original one, using logical operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Precisely, let Expt be a conditional expression to mutate containing a set of variables Svt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' For every variable vart in Svt, we load Sv = {var0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=', varn} the set of other variables appearing in the same class and of the same type T as vart, then we generate the following new conditions: Expt op (vart relop vari) and (vart relop vari) op Expt, where op is a binary logical operator taking the values in {&&,||}, relop is a relational operator applicable on the type T and vari is a variable from Sv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='5 Mutant filtering In this step, our approach starts by discarding accurate and duplicate predictions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' the redundant predictions and the ones that are exactly the same as the original code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Then, it iterates through the statements and selects in every iteration one compilable prediction by line, while discarding not compilable ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Once all first-order mutants are selected (issued by one single token replacement), our approach proceeds by selecting second-order ones (issued by the com- bination of condition seeding and one token replacement) in the same iterative manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' µBERT continues iterating until achieving the desired number of mutants or all mutants are outputted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' 4 RESEARCH QUESTIONS We start our analysis by investigating the advantage brought by the additive mutations (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' conditions seeding ones) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' the fault detection capabilities of test suites designed to kill µBERT’s mutants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Thus, we ask: RQ1 (µBERT Additive mutations) What is the added value of the additive mutations on the fault detection capabili- ties of test suites designed to kill µBERT’s mutants?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' To answer this question, we generate two sets of mutants using µBERT: 1) the first set using all possible mutations that we denote as µBERT and 2) a second one using only the conventional µBERT’ mutations – part of our preliminary implementation [18], excluding the additive ones – that we denote as µBERTconv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Then we evaluate the fault detection ability of test suites selected to kill the mutants from each set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' The answer of this question provides evidence that the additive mutations increase the fault detection capability of µBERT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Yet, to assess its general performance we compare it to state-of-the-art (SOA) mutation testing, particularly PiTest [17], and thus, we ask: RQ2 (Fault detection) How does µBERT compare with state- of-the-art mutation testing, in terms of fault detection?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' To answer this question we generate mutants using the latest version of PiTest [17], on the same target projects as for RQ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' As we are interested in comparing the approaches and not the implementations of the tools, we exclude the subjects on which PiTest did not run correctly or did not generate any mutant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' This way we ensure having a fair base of comparison by counting exactly the same study subjects for both approaches (further details are given in Section 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Then, we compare the fault detection capability of test suites 5 selected to kill the same number of mutants produced by each approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Finally, we qualitatively analyse some of the mutants generated with µBERT and ask: RQ3 (Qualitative analysis) Does µBERT generate different mutants than traditional mutation testing operators?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' To answer this question, we showcase the mutants gen- erated by µBERT that help in detecting faults not found by PiTest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Additionally, we discuss the program-context- capturing importance in µBERT’s functioning, by rerunning it with a reduced size of the masked codes passed to the model, and comparing examples of yielded mutants with those obtained in our original setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' 5 EXPERIMENTAL SETUP 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='1 Dataset & Benchmark To evaluate µBERT’s fault detection, we use real bugs from a popular dataset in the software engineering research area – Defects4J [29] v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' In this benchmark, every subject bug is provided with a buggy version of the source code, its corresponding fixed version, and equipped with a test suite that passes on the fixed version and fails with at least one test on the buggy one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' The dataset includes over 800 bugs from which, we exclude the ones presenting issues, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' with wrong revision ids, not compiling or with execution issues, or having failing tests on the fixed version, at the reporting time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Next, we run µBERT and PiTest on the corresponding classes impacted by the bug from the fixed versions of the remaining bugs and exclude the ones where no tool generated any mutant, ending up with 689 bugs covered by µBERT and 457 covered by PitTest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' As we’re interested in comparing the approaches and not the tools’ implementations, and to exclude eventual threats related to the environment (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' supported java and juint versions by each technique, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=') or the limitations and shortages of the dataset, we establish every comparison study on a dataset counting only bugs covered by all considered approaches: 689 bugs to answer RQ1 and 457 to answer RQ2 and RQ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='2 Experimental Procedure To assess the complementary and added value in terms of fault revelation of the condition-seeding-based mutations (answer to RQ1), we run our approach with and without those additional mutations – that we name respectively µBERT and µBERTconv–, and thus, generating all possible mutants on our dataset programs’ fixed versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Next, we compare the average effectiveness of the test suites generated to kill the mutants of each set;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' induced by µBERT and µBERTconv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Once the added value of the proposed condition- seeding-based mutations is validated, we compare its per- formance to S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' mutation testing (answer to RQ2 and RQ3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' We use PiTest [17], a stable and mature Java mutation testing tool, because it has been more effective at finding faults than other tools [33] and it is among the most com- monly used by researchers and practitioners [47], [52], as of today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' The tool proposes different configurations to adapt the produced mutations and their general cost to the target users, by excluding or including mutators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Among these configurations we used the three following: Pit-all (ALL) which counts all available mutation oper- ators available in the current version3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Pit-default (DEFAULTS) whose mutators are selected to form a stable and cost-efficient subset of operators by producing less but more relevant mutants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Pit-rv-all (ALL) which is a version4 that includes the mutators of Pit-all and extra experimental [7] ones that are made available for research studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' To compare the different approaches, we evaluate their effectiveness and cost-efficiency in achieving one of the main purposes of mutation testing, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=', to guide the testing towards higher fault detection capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' For this reason, we simulate a mutation testing use-case scenario, where a developer/tester selects mutants and writes tests to kill them [13], [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' We run every approach on the fixed versions and test suites provided by Defects4J, then collect the mutants and their test execution results;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' whether the mutant is killed (breaks at least one test of the test suite) and if yes by which tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Next, we suppose that the not killed mutants are equivalent or irrelevant, explaining why no tests have been written to kill them by the developers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Then, we simulate the scenario of a developer testing the fixed version, in a state where 1) it did not have any test 2) thus all mutants did not have killing tests and 3) the developer had no knowledge of which mutants are equivalent or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' This way, we can reproduce the developer flow of 1) selecting and analysing one mutant, 2) to either (a) discard it from the mutant set if it is equivalent (not killed in the actual test suite) or (b) write a test to kill it (by selecting one of the actual killing tests of the mutant), 3) then discarding all killed mutants by that test and 4) iterating similarly over the remaining mutants until all of them are analysed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' We say that a bug is found by a mutation testing technique if the resulting test suite – formed by the written (selected) tests by the developer – contains at least one test that reveals it;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' a test that breaks when executed on the buggy version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' We express the testing cost in terms of mutants analysed, and hence, we consider the effort required to find a bug as the number of mutants analysed until the first bug-revealing test is written.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' To set a common basis of comparison be- tween the approaches, accounting for the different number of generated mutants, we run the simulations until the same maximum effort is reached (maximum number of mutants to analyse), which we set to the least cost required to kill all the mutants by one of the compared approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' During our evaluation study, we use the same mutation selection strategy for all compared approaches, iterating through the lines in random order and selecting 1 arbitrary mutant per line per iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' To reduce the process randomness impact 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Version 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='4 available in PitTest’s [6] GitHub repository (branch=master, repo=https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='com/hcoles/pitest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='git, rev- id=17e1eecf) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Version 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='4 available in PitTest’s [6] GitHub repository (branch=master, repo=https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='com/hcoles/pitest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='git, rev- id=2ec1178a) 6 on our results (in the selection of mutants and tests), we run every simulation 100 times, then average their results for every target-bug and considered approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Finally, we aggregate these averages computed on all target bugs and normalise them as global percentages of achieved fault detection by spent effort, in terms of mutants analysed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Finally, to answer RQ3, we select example mutants that enabled µBERT to find bugs exclusively (not found by any of PiTest versions), from the results of RQ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Then we discuss the added value of µBERT mutations through the analysis of the mutants’ behavioural difference from the fixed version and similarity with the buggy one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='3 Implementation We implemented µBERT’s approach as described in Sec- tion 3: we have used Spoon [51] and Jdt [21] libraries to parse and extract the business logic related AST nodes and apply condition-seeding mutators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' To predict the masked tokens we have used the implementation proposed by CodeBERT-nt [3], [31], using CodeBERT Masked Language Modeling (MLM) functionality [2], [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' We provide the implementation of our approach and the reproduction package of its evaluation at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' com/Ahmedfir/mBERTa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' 6 EXPERIMENTAL RESULTS 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='1 RQ1: µBERT Additive mutations To answer this question we compare the fault detection effectiveness of test suites written to kill mutants generated by µBERT with and without additive mutations, noted re- spectively µBERT and µBERTconv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Figure 2 depicts the fault detection improvement when extending µBERT mutations by the additive ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' In fact, µBERT fault detection increased on average by over 9% compared to the one achieved by µBERTconv, achieving 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='64% on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' We can also see that besides outliers, the majority of bugs are found in 100% of the times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Moreover, when examining the bugs separately, we find that µBERT finds 20 more bugs than µBERTconv (with fault detection > 0%), and 70 more when considering bugs found with fault detection percentages above 90%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' This confirms that the additive patterns induce relevant mutants ensuring the detection of some bugs always or in most of the cases, as well as representing better new types of faults, which were not detectable otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' To check the significance of the fault detection advantage brought by the additive patterns, we performed a statistical test (Wilcoxon paired test) on the data of Figure 2a to vali- date the hypothesis ”the fault detection yielded by µBERT is greater than the one by µBERTconv ”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' The very small obtained p-values of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='92e-21 (≪ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='05) showed that the dif- ferences are significant, indicating the low probability of this fault detection amelioration to be happening by chance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' The difference size confirms also the same advantage, with ˆA12 values of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='5827 (> 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='5), indicating that µBERT induces test- suites with higher fault detection capability in the majority of the cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Next, we compare the fault detection performance of µBERT and µBERTconv when analysing the same number of mutants, and illustrate in Figure 3 their average fault BERT BERTconv tool 0 20 40 60 80 100 Fault detection % 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='64% 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='30% (a) Effectiveness: mean fault-detection per subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' 0 20 40 60 80 100 Effort % (number of analysed mutants) 0 20 40 60 80 Fault detection % tool BERT BERTconv (b) Cost-efficiency: fault detection by the number of mutants analysed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' 2: Fault-detection performance improvement when us- ing additive patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Comparison between µBERT and µBERTconv, w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' the fault-detection of test suites written to kill all generated mutants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' detection effectiveness and cost-efficiency in terms of anal- ysed mutants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' The box-plots of the Subfigure 3a show that even when spending the same effort as µBERTconv, µBERT keeps a similar advantage of on average 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='05% higher fault detection, achieving a maximum of 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='35%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' From the line- plots of the Subfigure 3b, we can see that both approaches achieve a comparable fault detection (≈ 70%) at (≤≈ 40%) of the maximum costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' At higher costs, µBERTconv’s curve increases slowly until achieving a plateau at ≈ 60% of the effort, whereas µBERT’s curve keeps increasing to- wards higher fault detection ratios even when achieving the ≈ 100% of the fixed maximum effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' To validate these findings we re-conducted the same statistical tests on the data of Subfigure 3a and found that µBERT outperforms significantly µBERTconv with negligible p-values of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='15e-19 and ˆA12 values of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='5711.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' 7 BERT BERTconv tool 0 20 40 60 80 100 Fault detection % 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='35% 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='30% (a) Effectiveness: mean fault-detection per subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' 0 20 40 60 80 100 Effort % (number of analysed mutants) 0 10 20 30 40 50 60 70 80 Fault detection % tool BERT BERTconv (b) Cost-efficiency: fault detection by the number of mutants analysed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' 3: Fault-detection comparison between µBERT and µBERTconv, with the same effort: where the maximum effort is limited to the minimum effort required to analyse all mutants of any of them, which is µBERTconv in most of the cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='2 RQ2: Fault Detection comparison with PiTest To answer this research question we reduce our dataset to the bugs covered by µBERT and the 3 considered versions of PitTest approaches: ”Pit-default” which contains the default mutation operators of PiTest, ”Pit-all” containing all PiTest operators including the default ones and ”Pit-rv-all” which contains experimental operators [7] in addition to the ”Pit- all” ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Then, we perform the same study as in RQ1, where we compare the considered approaches’ effectiveness and cost-efficiency based on the fault detection capability of test suites written to kill their generated mutants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' To have a fair base of comparison, we compare the approaches under the same effort in analysing mutants, which is equal to the least average effort required to kill all mutants of one of the approaches (which is the one of Pit-default in the majority of the cases).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' As we are interested in comparing the mutation testing approaches and not mutant selection strategies, we run the simulation with the same one-mutant- BERT Pit-all Pit-default Pit-rv-all tool 0 20 40 60 80 100 Fault detection % 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='43% 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='87% 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='90% 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='33% (a) Effectiveness: mean fault-detection per subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' 0 20 40 60 80 100 Effort % (number of analysed mutants) 0 10 20 30 40 50 60 Fault detection % tool BERT Pit-all Pit-default Pit-rv-all (b) Cost-efficiency: fault detection by the number of mutants analysed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' 4: Fault-detection comparison between µBERT and PiTest, with the same effort: where the maximum effort is limited to the minimum effort required to analyse all mutants of any of them, which is Pit-default in most of the cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' per-line random sampling of mutants for all techniques (see Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Figure 4b shows that with small effort (≤≈ 5%) all approaches yield comparable fault detection scores (≈ 10%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' However, the difference becomes more noticeable when spending more effort, with µBERT outperforming all ver- sions of PiTest;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' achieving on average 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='53% higher fault detection scores than Pit-default, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='10% higher than Pit-rv- all and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='56% higher than Pit-all (see Figure 4a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' To validate these results, we performed the same statis- tical tests as in RQ1, checking the hypothesis that ”µBERT yields better fault detection capabilities than the other ap- proaches”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' We illustrate in the first row of Tables 2a and 2b the corresponding computed Wilcoxon paired test p-values and Vargha and Delaney ˆA12 values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Our results show that µBERT has a significant advantage over the considered SOA approaches with p-values under 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Additionally, µBERT scores ˆA12 values above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='5 which confirms that guiding 8 TABLE 2: Paired (per subject bug) statistical tests of the average fault detection of test suites written to kill the same number of mutants generated by each approach (data of Figure 4a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' (a) Wilcoxon paired test p-values computed on every dataset subject, comparing each approach (A1) from the first column to the other approaches (A2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' p-values smaller than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='05 in- dicate that (A1) yields an average fault detection significantly higher than that of (A2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' p-values Pit-rv-all Pit-default Pit-all µBERT 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='78e-11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='18e-12 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='32e-02 Pit-all 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='54e-22 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='87e-06 – Pit-default 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='55e-01 – – (b) Vargha and Delaney ˆA12 values computed on every dataset subject, comparing each approach (A1) from the first column to the other approaches (A2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' ˆA12 values higher than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='5 indicate that (A1) yields an average fault detection higher than that of (A2) in the majority of the cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' ˆA12 Pit-rv-all Pit-default Pit-all µBERT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='6488 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='5514 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='5066 Pit-all 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='7210 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='4956 – Pit-default 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='5449 – – the testing by µBERT mutants instead of those generated by SOA techniques yields comparable or higher fault detection ratios, in the majority of the cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Indeed, the ˆA12 differ- ence between Pit-all and µBERT is small (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='5066), indicating that both approaches perform similarly or better on some studied subjects and worst on others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' We notice also from the sub-figure 4b that Pit-default achieves a plateau at around 60% of the effort while the other tools keep increasing, showing that they are able to achieve higher fault detection capabilities, at a higher cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' This is very noticeable when we compare the sub-figures (a) and (b) of Figure 4 with the figure 2, where the average fault detection of µBERT is way lower than what it achieves in RQ1 – around 66% instead of 84%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' This is a direct consequence of the fact that Pit default produces fewer mutants than the other approaches, limiting considerably the maximum effort of the mutation campaigns and thus the fault detection ratios, in the majority of the cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Indeed, as illustrated in Figure 5, all approaches score higher fault detection percentages when spending more effort, achieving on average ≈65% for Pit-all, ≈66% for Pit-rv-all and ≈83% for µBERT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' We explain the small decrease of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='72% in the mean fault detection achieved by µBERT in comparison with RQ1 (82,92% in RQ2 instead of 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='64% in RQ1) by the difference in the considered dataset for each RQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' In Table 3, we illustrate the ˆA12 and p-values computed on data of the boxplots in Sub-figure 5a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' The results confirm that µBERT outperforms significantly SOA mutation testing w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='t the fault detection capability of test suites written to all kill mutants generated by each approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Next, we turned our interest to the set of particular bugs that every approach can and cannot reveal when spending the same effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Hence, we map each bug with its revealing tool, from the simulation results of Figure 4a and illustrate their corresponding Venn diagrams in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' BERT Pit-all Pit-default Pit-rv-all tool 0 20 40 60 80 100 Fault detection % 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='92% 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='49% 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='90% 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='35% (a) Effectiveness: mean fault-detection per subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' 0 20 40 60 80 100 Effort % (number of analysed mutants) 0 20 40 60 80 Fault detection % tool BERT Pit-all Pit-default Pit-rv-all (b) Cost-efficiency: fault detection by the number of mutants analysed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' 5: Comparison between µBERT and PiTest, relative to the fault-detection of test suites written to kill all generated mutants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' From the disjoint sets in Sub-figure 6a, we notice a clear advantage in using µBERT over the considered SOA baselines, as it finds most of the bugs they find in addition to finding exclusively 47 bugs when spending the same effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' More precisely, µBERT finds 52, 77 and 52 more bugs than Pit-all, Pit-default and Pit-rv-all, respectively, whereas they find each 13, 10 and 13 bugs that µBERT missed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' This endorses the fact that µBERT introduces mutants that represent more real bugs than SOA mutation tech- niques, and encourages the investigation of the eventual complementary between the approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' This observation is more noticeable when considering the overlapping be- tween bugs found by each approach in at least 90% of the simulations (Sub-figure 6b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' We notice that the approaches perform comparably, with a particular distinction of Pit-all and Pit-default results which find exclusively 19 and 21 bugs with these high fault detection percentages instead of 0, as observed in Sub-figure 6a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Nevertheless, µBERT conserves the same advantage over the considered baselines in this 9 TABLE 3: Paired (per subject bug) statistical tests of the average fault detection of test suites written to kill all the mutants generated by each approach (data of Figure 5a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' (a) Wilcoxon paired test p-values computed on every dataset subject, comparing each approach (A1) from the first column to the other approaches (A2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' p-values smaller than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='05 in- dicate that (A1) yields an average fault detection significantly higher than that of (A2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' p-values Pit-rv-all Pit-default Pit-all µBERT 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='49e-13 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='14e-33 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='47e-14 Pit-all 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='71e-01 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='76e-23 – Pit-default 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='00e+00 – – (b) Vargha and Delaney ˆA12 values computed on every dataset subject, comparing each approach (A1) from the first column to the other approaches (A2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' ˆA12 values higher than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='5 indicate that (A1) yields an average fault detection higher than that of (A2) in the majority of the cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' ˆA12 Pit-rv-all Pit-default Pit-all µBERT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='6028 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='7123 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='6061 Pit-all 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='5077 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='6400 – Pit-default 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='3676 – – regard, finding exclusively 42 bugs more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' It finds also 50, 63 and 69 more bugs than respectively Pit-all, Pit-default and Pit-rv-all, whereas they find each 59, 58 and 27 bugs that µBERT missed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='3 RQ3: Qualitative Analysis of µBERT Mutants To answer this research question we investigate the mutants generated by µBERT, which induced test suites able to find bugs that were not detected otherwise, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' by the considered SOA approaches (see Figure 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Meaning that the mutants break similar tests as the target real buggy version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' As a simple bug example (requiring only one change to fix it), we consider Lang-49 from Defects4J and we investigate mutants that have been generated by µBERT and helped in generating tests that reveal it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' This bug impacts the results of the method reduce() from the class org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='apache.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='commons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='lang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='Fraction, which returns a new reduced fraction instance, if possible, or the same instance, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' The bug is caused by a miss-implementation of a specific corner case, which con- sists of calling the method on a fraction instance that has 0 as numerator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' In Table 4, we illustrate example mutants generated by µBERT that helped in revealing this bug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Every mutant is represented by a diff between the fixed and the mutated version by µBERT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' As can be seen, µBERT can generate mutants that can be induced by applying conventional pattern-based mutations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=', Mutant 1 replaces a relational operator (==) with an- other (>) and Mutant 2 replaces an integer operand (0) with another one (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' In addition, it proposes more complex mutations that are unlikely achievable without any knowledge of either the AST or the context of the considered program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' For instance, it can generate Mutant 4 by changing a conditional return statement with (this) the current instance of Fraction, which matches the return type of the method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Similarly, to 47 0 1 0 1 0 3 0 3 3 23 0 1 10 354 Pit-all Pit-default Pit-rv-all BERT (a) Faults discovered at least once per 100 runs (Fault detection > 0%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' 42 2 3 21 3 0 2 19 10 3 8 15 14 22 114 Pit-all Pit-default Pit-rv-all BERT (b) Faults discovered in over 90% of the runs (Fault detection≥ 90%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' 6: Number of faults discovered by test-suites written to kill mutants generated by µBERT and PiTest versions when analysing the same number of mutants (same effort).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' generate Mutant 5, it replaces (this) the current instance of the class Fraction by an existent instance of the same type (ONE), making the statement returning either the object ONE or the object ZERO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' To produce more complex mutants, µBERT applies a condition seeding followed by token-masking and Code- BERT prediction, such as adding || (numerator == other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='numerator) to the original condition of a return statement, inducing Mutant 8, or adding || !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' (numerator == Integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='MIN_VALUE) to the original condition of an if statement, inducing Mutant 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' To investigate further the impact of the code context captured by the model on the generated mutants, we have rerun µBERT on 5 subjects from our dataset, with a max- imum number of surrounding tokens equal to 10 (instead of 512).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Then, we compared manually the induced mutants with those generated by our default setup, in the same locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' From our results, we observed a noticeable de- crease in the number of compilable predictions, indicating the general performance decrease of the model when it lacks information about the code context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Particularly, we notice 10 TABLE 4: Example of mutants generated by µBERT that helped find the bug Lang-49 from Defects4J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Mutant 1: replacing binary operator @@ org .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' apache .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' commons .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' lang .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' math .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Fraction : 466 @@ − i f ( numerator == 0) { + i f ( numerator > 0) { Mutant 2: replacing literal implementation @@ org .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' apache .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' commons .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' lang .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' math .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Fraction : 466 @@ − i f ( numerator == 0) { + i f ( numerator == 1) { Mutant 3: adding a condition to an if statement @@ org .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' apache .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' commons .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' lang .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' math .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Fraction : 466 @@ − i f ( numerator == 0) { + i f ( ( numerator == 0) + | | !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' ( numerator==Integer .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='MIN VALUE) ) { Mutant 4: replacing a condition @@ org .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' apache .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' commons .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' lang .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' math .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Fraction : 467 @@ − return equals (ZERO) ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' t h i s : ZERO;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' + return t h i s ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Mutant 5: replacing this access by another object @@ org .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' apache .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' commons .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' lang .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' math .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Fraction : 467 @@ − return equals (ZERO) ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' t h i s : ZERO;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' + return equals (ZERO) ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' ONE: ZERO;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Mutant 6: replacing method argument @@ org .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' apache .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' commons .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' lang .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' math .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Fraction : 469 @@ int gcd = greatestCommonDivisor ( − Math .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' abs ( numerator ) , denominator ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' + Math .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' abs ( numerator ) , 1 ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Mutant 7: replacing a variable @@ org .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' apache .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' commons .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' lang .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' math .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Fraction : 473 @@ − return Fraction .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' getFraction ( numerator / gcd , + return Fraction .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' getFraction ( numerator / 3 , denominator / gcd ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Mutant 8: adding a condition to a return statement @@ org .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' apache .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' commons .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' lang .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' math .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Fraction : 840 @@ return ( getNumerator ( ) == other .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' getNumerator ( ) − && getDenominator ( ) == other .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' getDenominator ( ) ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' + && getDenominator ( ) == other .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' getDenominator ( ) ) ) + | | ( numerator == other .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' numerator ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' that it is not able to produce program-specific mutants, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' by changing an object by another or a method call with another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' In Table 5, we illustrate some example mutants that helped find each of the studied subjects (breaking same tests as the original bug), which µBERT failed to generate when the maximum number of surrounding tokens is limited to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' 7 THREATS TO VALIDITY One external threat to validity concerns the generalisation of our findings and results in the empirical evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' To reduce this threat, we used a large number of real bugs from popular open-source projects with their associated developer test-suites, provided by an established and in- dependently built benchmark (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Defects4J [29]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Never- theless, we acknowledge that the results may be different considering projects in different domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Other threats may arise from our way of assessing the fault detection capability of mutation testing approaches, based on their capability of guiding the testing via a devel- oper/tester simulation in which we assume that the current test suites are complete and the not killed mutants are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Although we acknowledge that this may not be the case in real-world scenarios, we believe that this process is sufficient to evaluate our approach, particularly considering the fact the test suites provided by Defects4J are relatively strong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Additionally, to mitigate any com- parison threat between the considered approaches, we use consistently and similarly the same test-suites, setups and simulation assumptions in all our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' The choice of our comparison baseline may form other threats to the validity of our findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' While different fault- seeding approaches have been proposed recently, PiTest remains among the most mature and stable mutation test- ing tools for Java programs, thus, forming an appropriate comparison baseline to evaluate our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Furthermore, we compared our results with those obtained by mutants from different configurations proposed by PiTest, enlarging our study to the different audiences targeted by this latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' We acknowledge however that the results may change when considering other techniques and consider the evaluation of the effectiveness and cost-efficiency of different mutation testing techniques as out of the scope of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Other construct threats may arise from considering the number of mutants analysed as metric to measure the effort required to find a fault.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' In addition to the fact that this metric has been widely used by the literature [9], [34], [47], we believe that it is intuitive and representative of the global manual effort of the tester in analysing the mutants, dis- carding them or writing tests to kill them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' While being the standard in the literature, we acknowledge that this measure does not account for the cost difference between mutants, attributing the same cost to all mutants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' This is simply because we do not know the specific effort required to analyse every specific mutant or to write every specific test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Additionally, our cost-efficiency results may be impacted by costs that are not captured with this metric, such as the execution or the developing effort of either CodeBERT, the key component of µBERT, or the set of patterns and execution enhancements over the different releases of PiTest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Nevertheless, we tried to mitigate any major threats that can be induced by the implementation of the different tools, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' we reduce the dataset subjects to those on which every approach generated at least one mutant and consider any implementation difference between the approaches as out of the current scope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' 8 RELATED WORK Since the 1970s, mutation testing has been the main focus of multiple research works [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Their findings have proven that artificial faults can be useful in multiple software en- gineering applications, such as testing [47], debugging [37], [48], assessing fault tolerance [42], risk analysis [16], [56] and dependability evaluation [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Despite this long history of research, the generation of relevant mutants remains an open question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Most of the related research has focused on the design of fault 11 TABLE 5: Example of mutants generated by µBERT that helped in finding bugs from Defects4J and could not be generated when limiting the maximum number of surrounding tokens to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Mutant 1 (JacksonCore-4) : replacing a method call @@ com .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' fasterxml .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' jackson .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' core .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' u t i l .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' TextBuffer : 515 @@ − unshare ( 1 ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' + expand ( 1 ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Mutant 2 (Closure-26) : replacing an object @@ com .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' google .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' j a v a s c r i p t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' jscomp .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' ProcessCommonJSModules : 89 @@ − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' replaceAll ( Pattern .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' quote ( F i l e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' separator ) , MODULE NAME SEPARATOR) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' replaceAll ( Pattern .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' quote ( filename ) , MODULE NAME SEPARATOR) Mutant 3 (Closure-35) : replacing a method call @@ com .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' google .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' j a v a s c r i p t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' jscomp .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' TypeInference : 1092 @@ − scope = traverseChildren (n , scope ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' + scope = traverse (n , scope ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Mutant 4 (Lang-27) : replacing a method call @@ org .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' apache .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' commons .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' lang3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' math .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' NumberUtils : 526 @@ − i f ( !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' ( f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' i s I n f i n i t e ( ) | | ( f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' floatValue ( ) == 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='0 F && !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' allZeros ) ) ) { + i f ( !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' ( f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' i s I n f i n i t e ( ) | | ( f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' round ( ) == 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='0 F && !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' allZeros ) ) ) { / / a l s o ” f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' f l o a t V a l u e ( ) ” to ” f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' s c a l e ( ) ” Mutant 5 (Math-64) : replacing an object @@ org .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' apache .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' commons .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' lang .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' math .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Fraction : 852 @@ − for ( i nt j = k ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' j < jacobian .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' length ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' ++ j ) { + for ( i nt j = k ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' j < beta .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' length ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' ++ j ) { Mutant 6 (Lang-27) : replacing an object @@ org .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' apache .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' commons .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' lang3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' math .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' NumberUtils : 526 @@ − i f ( !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' ( f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' i s I n f i n i t e ( ) | | ( f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' floatValue ( ) == 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='0 F && !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' allZeros ) ) ) { + i f ( !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' ( f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' i s I n f i n i t e ( ) | | ( f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' round ( ) == 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='0 F && !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' zero ) ) ) { patterns (mutation operators) which are usually defined based on the target language grammar [8], [47] then refined through empirical studies [33], [40], [44] aiming at reducing the redundancy and noise among their generated mutants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' The continuous advances in this sense were followed by a constant emergence of pattern-based mutation testing tools and releases [17], [35], [39], among which some are becoming popular and widely adopted by researchers and practitioners, such as PiTest [17], from which we consider three configurations as our comparison baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Recent research has focused their interest on improving the representativeness of artificial faults aiming at reducing the mutation space to real-like faults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' For instance, instead of basing the mutation operators’ design on the programming language grammar, Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' [12] proposed inferring them from real bug fixes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Similarly, Tufano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' [54] pro- posed a neural machine translation technique that learns how to inject faults from real bug fixes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Along the same line, Patra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' [50] proposed a semantic-aware learning approach, that learns and then adapts fault patterns to the project of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Their results are promising, however, the fact that these techniques depend on the availability of numerous, diverse, comprehensive and untangled fix commits [27] of not coupled faults [43], which is often hard to fulfil in practice, may hinder their performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Acknowl- edging for the injection location [13], [42], Khanfir et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' [32] combined the usage of information retrieved from bug reports with inverted automated-program-repair patterns to replicate real faults fixable by the original fix-patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Their results showed that they can generate faults that mimic real ones, however, their approach remains dependent and lim- ited to the presence of good bug reports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Overall, designing the mutation operators based on the known faults space yields more diverse mutants that represent more fault types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' However, these extended operator sets tend to increase the number of generated mutants and consequently the general cost of the mutation campaign i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' the fault patterns pro- posed by Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' and Khanfir et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' counted also most of the conventional mutators in addition to new ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Unlike these techniques, µBERT leverages pre-trained models to introduce mutants based on code knowledge instead of the faults one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' As code is more available than faults, it offers a more flexible and complete knowledge base than faults, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' it perms to overcome the limitations and efforts required 1) to collect clean bug-fixing commits, 2) to capture the faulty behaviour and 3) design fault patterns, be it manually or via machine learning techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Aiming at reducing the number of generated mutants, researchers have proposed different strategies to generate relevant mutants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' For instance, studies that show that mu- tant strength resides in not only its inducing pattern but also where it is injected [13], [42], motivated the importance of selecting relevant locations to mutate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' In this regard, Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' [53] suggest mutating multiple places within diverse program execution paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Gong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' [26] also propose the mutation in diverse locations of the program extracted from graph analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Similarly, Mirshokraie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' [41] compute complexity metrics from program executions to extract loca- 12 tions with good observability to mutate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Other approaches restrict the fault injection on specific locations of the pro- gram, such as the code impacted by the last commits [38], [58] for better usability in continuous integration, or target- ing locations related to a given bug-report [32] to target a specific feature or behaviour, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' More recent advances have resulted in powerful techniques for cost-effectively selecting mutants, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=', by avoiding the analysis of redundant mutants (basically, equivalent and subsumed ones) [24], [25], [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' In particular, the work of Garg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' [24] utilises the knowledge of mutants’ surrounding context, embedded into the vector space, to predict whether a mutant is likely subsuming or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' In this work, we do not target any specific code part or any narrow use case, but instead, perform fault injection in a brute-force way similarly to mutation testing, by iterating every program statement and masking every involved token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Multiple studies have been focused on the relationship between artificial and real faults [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' The results of the stud- ies conducted by Ojdanic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' [45], Papadakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' [49], Just et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' [30] and Andrews et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' [9] showed that there is a correlation between tests broken by a bug and tests killing mutants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Meaning that artificial faults can be used as alternatives to real faults in controlled studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Moreover, the findings of Chekam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' [14], Frankl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' [23] and Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' [36] show that guiding testing by mutants leads to significantly higher fault revelation capability than the ones of other test adequacy criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Based on these findings, we assess our approach based on the relation between the injected and real faults, in terms of breaking tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' More precisely, we conduct a fault detection effectiveness and cost-efficiency study to evaluate our approach’s mutants in guiding testing and compare it to state-of-the-art techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Furthermore, we discuss the diversity and readability of µBERT mutants through real examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' The closest related work is a preliminary implementation of µBERT that was recently presented in the 2022 mutation workshop [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' This implementation, denoted as µBERTconv in our evaluation, includes the conventional mutations (to mask and replace tokens by the model predictiosn), but it does not include the condition-seeding additive mutations that provide major benefits for fault detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Moreover, µBERTconv was evaluated only on 40 bugs from Defects4J, and compared only to an early version of PiTest (similar to Pit-rv-all).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' In this work, we perform an extensive exper- imental evaluation including 689 bugs from Defects4J and compare µBERT effectiveness with three different configura- tions from PiTest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Moreover, we show that µBERT finds on average more bugs than µBERTconv without requiring more effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' 9 CONCLUSION We presented µBERT;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' a pre-trained language model based fault injection approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' µBERT provides researchers and practitioners with easy-to-understand “natural” mutantsto help them in writing tests of higher fault revelation capabil- ities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' Unlike state-of-the-art approaches, it does neither re- quire nor depend on any kind of faults knowledge or language grammar but instead on the actual code definition and distribution, as written by developers in numerous projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' This facilitates its developing, maintainability, inte- gration and extension to different programming languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' In fact, it reduces the overhead of learning how to mutate, be it via creating and selecting patterns or collecting good bug-fixes and learning from their patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' In a nutshell, µBERT takes as input a given program and replaces different pieces of its code base with predictions made by a pretrained generative language model, produc- ing multiple likely-to-occur mutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' The approach targets diverse business code locations and injects either simple one-token replacement mutants or more complex ones by extending the control-flow conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' This provides proba- ble developer-like faults impacting different functionalities of the program with higher relevance and lower cost to developers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' This is further endorsed by our results where µBERT induces high fault detection test suites at low effort, outperforming state-of-the-art techniques (PiTest), in this regard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' We have made our implementation and results avail- able [5] to enable reproducibility and support future re- search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' ACKNOWLEDGMENT This work was supported by the Luxembourg National Research Fund (FNR) projects C20/IS/14761415/TestFlakes and TestFast, ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' 12630949.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' REFERENCES [1] Amazon codewhisperer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content=' https://aws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='amazon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQf8QVm/content/2301.03543v1.pdf'} +page_content='com/ 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Berkeley +KENNETH LIEN, UC Berkeley +ISAAC ONG, UC Berkeley +TONY HONG, UC Berkeley +SANGBIN CHO, Anyscale +ERIC LIANG, Anyscale +ION STOICA, UC Berkeley and Anyscale +1 +INTRODUCTION +We present Exoshuffle-CloudSort, a sorting application running on top of Ray using the Exoshuffle archi- +tecture [4]. Exoshuffle-CloudSort runs on Amazon EC2, with input and output data stored on Amazon S3. +Using 40× i4i.4xlarge workers, Exoshuffle-CloudSort completes the 100 TB CloudSort Benchmark (Indy +category [6]) in 5378 seconds, with an average total cost of $97. +2 +IMPLEMENTATION +2.1 +Overview +Exoshuffle-CloudSort is a distributed futures program running on top of Ray, a task-based distributed +execution system. The program acts as the control plane to coordinate map and reduce tasks; the Ray system +acts as the data plane, responsible for executing tasks, transferring blocks, and recovering from failures. +Exoshuffle-CloudSort implements a two-stage external sort algorithm. The first stage is map and shuffle. +Each map task reads an input partition, sorts it, and partitions the result into 𝑊 output partitions, each sent +to a merger on a worker node. A merger receives 𝑊 map output partitions, merges and sorts them, and +further partition the result into 𝑅/𝑊 output partitions, all of which are spilled to local disk. +The second stage is reduce. Once the map and shuffle stage finishes, each reduce task reads 𝑊 shuffled +partitions, merges and sorts them, and writes the final output partition. +For the 100 TB CloudSort Benchmark, we set the following parameters: +• Total data size is 100 TB. +• Number of input partitions 𝑀 = 50 000. Each input partition is 2 GB. +∗Author’s address: lsf@berkeley.edu, 465 Soda Hall, Berkeley, CA, USA. +1 +arXiv:2301.03734v1 [cs.DC] 10 Jan 2023 + +2 +Luan et al. +• Number of workers 𝑊 = 40. +• Number of output partitions 𝑅 = 25 000. +2.2 +Preparation +The first step in Exoshuffle-CloudSort is to compute the partition boundary values. For a sort record with +10-byte key, we view the first 8 bytes as a 64-bit unsigned integer partition key. We partition the key space +[0, 264 − 1) into 𝑅 = 25 000 equal ranges, such that all the records within a key range should be sent to one +reducer. +Every 𝑅1 = 𝑅/𝑊 = 625 reducer ranges are combined into a worker range, and records in each worker +range will be sent to one worker node. This yields 𝑊 = 40 equally-partitioned worker ranges. +2.3 +Map and Shuffle Stage +In the map and shuffle stage, Exoshuffle-CloudSort schedules the 𝑀 = 50 000 map tasks onto all worker +nodes. In our experiments we set the map parallelism, i.e. the number of map tasks running on a single +worker node, to be 3/4 of the total number of vCPU cores. Extra tasks are queued on the driver node. +Whenever a worker node finishes a map task, the driver assigns a new task from the queue to this node. +In a map task, we first download the input partition from S3. We then sort the input data in memory, +then partition it into 𝑊 = 40 slices. Each slice is eagerly sent to a merge controller on each worker. The +map task returns when all slices are sent. +On the receiving end, the merge controller accumulates the map blocks in memory until a threshold +is reached. We set the threshold to 40 blocks, or about 2 GB of data. Once the threshold is reached, the +controller launches a merge task to merge the already-sorted map blocks, and further partitions it into +𝑅1 = 625 merged blocks, each corresponding to a reduce task on this node. These blocks are spilled to the +local SSD for use by the reducers. +The merge parallelism is set to be the same as the map parallelism. When the number of merge tasks +reaches the maximum parallelism, and the merge controller’s in-memory buffer is filled up, it will hold off +acknowledging the receipt of a map block until a merge task finishes and a new merge task can launch. This +effectively creates back pressure to the map task scheduler to ensure the map, shuffle, and merge progresses +are in sync. +In our experiments, the average map task duration is 24 seconds; 15 seconds are used for downloading +input data. The average shuffle time (i.e. time to send and receive blocks) is 7 seconds. The merge task takes +17 seconds on average. +2.4 +Reduce Stage +Once all map and merge tasks finish, Exoshuffle-CloudSort enters the reduce stage. Each reduce task loads +𝑅1 = 625 from the local SSD, merges them, and uploads the sorted output partition to S3. In our experiments, +each reduce task takes 22 seconds on average. + +Exoshuffle-CloudSort +3 +2.5 +The Execution System +A highlight of the Exoshuffle architecture is that the application program only implements the control plane +logic, and the distributed futures system, Ray, handles execution. This is reflected in Exoshuffle-CloudSort. +Here is an incomplete list of features provided by Ray that we take “for free”: +• Task scheduling: The program specifies when and where to schedule tasks; the system handles the +RPC, serialization, and other bookkeeping. +• Network transfer: The program instructs data to be transferred by passing distributed futures as task +arguments; the system implements high-performance network transfer. +• Memory management and disk spilling: The program manipulates data references in a virtual, infinite +address space; the system uses reference counting to manage distributed memory, spills objects to +local disks when memory is low, and restores objects from local disks when they are needed. +• Pipelining of network and disk I/O: The network transfer, spilling and recovery of objects are trans- +parent to the application and are performed asynchronously. For example, the system shuffles map +output blocks while other map and merge tasks are running; it spills merge task output to disk while +other merge tasks are executing, and it restores merged blocks while reduce tasks are executing. +• Fault tolerance: this is transparent to the application: the system automatically retries the operation +when it encounters network failures and worker process failures. +For more details, we refer the reader to the Ray Architecture Whitepaper [7], the ownership design for +distributed futures systems [8], and the Exoshuffle paper [4]. +2.6 +Source Code +Exoshuffle-CloudSort is implemented in about 1000 lines of Python, and about 300 lines of C++. The +C++ component implements two functionalities: sorting and partitioning records, and merging sorted +record arrays. Exoshuffle-CloudSort runs on top of Ray, which is implemented in Python and C++. All of +Exoshuffle-CloudSort’s source code is available at https://github.com/exoshuffle/cloudsort. +3 +EVALUATION +3.1 +Environment Setup +We run Exoshuffle-CloudSort on AWS on a compute cluster configured as follows: +• 1× r6i.2xlarge master node. This node runs on 8 cores of an Intel Xeon 8375C CPU at 2.9 GHz, and +64 GiB memory. +• 40× i4i.4xlarge worker nodes. Each node runs on 16 cores of an Intel Xeon 8375C CPU at 2.9 GHz, +and 128 GiB memory. Each node has a directly-attached 3.75 TB AWS Nitro NVMe SSD. +• Each node is attached with a 40 GiB Amazon EBS General Purpose SSD (gp3) volume. +The software stack is configured as follows: + +4 +Luan et al. +• Ubuntu 22.04.1 LTS, Linux kernel version 5.15.0-1022-aws. +• XFS 5.13.0 filesystem. +• Intel oneAPI DPC++/C++ Compiler 2022.2.0.20220730. +• Python 3.9.13. +• Ray 2.1.0. +We measure the raw system I/O performance on the worker nodes using standard benchmarking tools: +• Network bandwidth: 25 Gbps between nodes, benchmarked with iperf. +• SSD: 2.9 GB/s read, 2.2 GB/s write, benchmarked with fio. +For storage, we use 40 buckets on Amazon S3 and randomly distribute the input and output partitions +across the buckets. +3.2 +Benchmark Setup +Generating Input. We use gensort version 1.5 as provided by the Sort Benchmark committee [5]. We +run the command gensort -c -b{offset} {size} {path} to generate each partition. {size} is fixed at +𝑃 = 20 000 000 such that each partition is exactly 2 GB. {offset} takes the values {𝑖 · 𝑃 : 0 ≤ 𝑖 < 𝑀} where +the number of input partitions 𝑀 = 50 000. {path} is a unique path in tmpfs. -c provides data checksum for +validation. After generating an input file, we randomly choose a bucket and upload the partition to S3. We +use Ray to schedule the 50 000 input generation tasks to all 40 worker nodes. The result is aggregated as an +input manifest file, saved for use by Exoshuffle-CloudSort to locate the sort input. +Validating Output. Exoshuffle-CloudSort produces an output manifest file containing the bucket and keys +of each output partition on S3. In each validation task, we first download the output partition to tmpfs, then +run the command valsort -o {sumpath} {path} to validate the ordering of records in each partition. We +use Ray to schedule the 25 000 output validation tasks to all 40 worker nodes. We concatenate the contents +of the summary files from each validation task, then run valsort -s to validate the total ordering, and +generate the total output checksum. Finally, we compare the output checksum with the input checksum to +verify data integrity. +3.3 +Experimental Results +3.3.1 +Job Completion Time. On November 10, 2022, we ran the 100 TB CloudSort Benchmark in the AWS +US West (Oregon, us-west-2) region with the setup described above. We first generated the input data on +Amazon S3, then ran Exoshuffle-CloudSort 3 times, each followed by a validation step. All 3 runs succeeded +with the same output checksum as the input, indicating all bytes are preserved in the sort. Table 1 reports +the job completion times of each run. The average job completion time is 5378 seconds, or 1.4939 hours. +Figure 1 shows the system utilizations of all worker nodes in the compute cluster during run #1 of the +100 TB CloudSort Benchmark. + +Exoshuffle-CloudSort +5 +Run +Map & Shuffle Time +Reduce Time +Total Job Completion Time +#1 +3509 s +1852 s +5361 s +#2 +3496 s +1852 s +5348 s +#3 +3520 s +1906 s +5426 s +Average +3508 s +1870 s +5378 s +Table 1. Job completion times of Exoshuffle-CloudSort on the 100 TB CloudSort Benchmark. +Fig. 1. Cluster utilization during run #1 of the 100 TB CloudSort Benchmark. Each thick line represents the median +system utilization of all worker nodes; the highest and lowest lines represent the maximum and minimum utilization +among all worker nodes, respectively. +3.3.2 +Total Cost of Ownership. The total job cost comprises of two parts: compute cost (Amazon EC2), and +the storage cost (Amazon S3). The storage cost is further divided into data storage cost and data access cost. +Compute Cost. The compute cost is calculated as the compute cluster’s hourly cost times the job completion +time. The total hourly cost is calculated as follows: +Total Hourly Compute Cost = Master Node Hourly Cost ++ Worker Node Hourly Cost × Number of Workers ++ EBS Volume Hourly Cost × (Number of Workers + 1) +(1) +We obtain the compute instance hourly costs from the Amazon EC2 on-demand pricing information [2]. +For EBS, we use the Amazon EBS monthly price [1] divided by the average number of hours in a month +( 365×24 +12 += 730) as the hourly price. The hourly cost of a 40 GiB gp3 volume is $0.08/730 × 40 = $0.0044. Now +we plug the cost variables into Equation (1): + +CPU +Memory +Application Progress +50000 +100% +70 GB +40000 +60 GB +80% +30000 +50 GB +20000 +60% +40 GB +10000 +30 GB +40% +20 GB +02:40 +02:50 +03:00 +03:10 +03:20 +03:30 +03:40 +03:5004:00 +10 GB +- map_in_progress +20% +reduce_in_progress +reduce_in_progress + reduce_in_progress +0 B +02:40 +02:50 +03:0003:1003:20 +03:3003:40 +03:5004:00 +map_completed +map_completec + map_completed + map_completed +0% +02:4002:50 +03:10 +03:20 +03:30 +03:40 +03:50 +04:00 + median objmem +- reducer_completed +reducer_completed reducer_completed +- reducer_completed +03:00 +min objmem + max objmem +min workmem + median cpu - min cpu - max cpu + max workmem +merge_in_progress +merge_in_progress +merge_in_progress +nerge_in_progress +NVMe Disk I/0 +Network I/0 + Disk Usage +7 GB/s +3 GB/s +100% +6 GB/s +2.50 GB/s +80% +5 GB/s +2 GB/s +4 GB/s +60% +1.50 GB/s +3 GB/s +2 GB/s + 40% +1 GB/s +1 GB/s +500 MB/s +20% +0 B/s +02:40 +02:50 +03:00 +03:10 +03:20 +03:30 +03:40 +03:50 +04:00 +0 B/s +02:4002:50 +¥03:0003:1003:2003:30 +03:4003:50 +04:00 +median network in + min network in +max network in + median network out + median disk write - min disk write - max disk write - median disk read +min network out + min network total +max network out +02:40 +02:50 +03:00 +03:10 +03:20 +03:30 +03:40 +03:50 +04:00 +- min disk read + max disk read - + median disk total - max disk total +- max network total6 +Luan et al. +• Master node (r6i.2xlarge) hourly cost is $0.504. +• Worker node (i4i.4xlarge) hourly cost is $1.373. +• Number of workers is 40. +• EBS volume hourly cost is $0.0044. +Hence, the total hourly compute cost is $55.6044. We multiply this hourly cost by the job completion +time of 1.4939 hours to obtain the total compute cost of $83.0674. +Data Storage Cost. The storage cost comprises of data storage cost and data access cost. We first consider +the data storage cost. Amazon S3 employs a pay-as-you-go pricing model, i.e. the user does not need to +provision storage capacity ahead of time, and only pays for the storage cost of objects based on their sizes +and storage duration. Amazon S3 charges $0.023 per GB-month for the first 50 TB, then $0.022 per GB-month +for the next 450 TB [3]. Since the total data size is 100 TB, we take the average price between the first two +tiers, i.e. $0.0225 per GB-month, or $3.0822 per hour per 100 TB. +• Input: The storage cost of the 100 TB input data is simply the cost to store 100 TB for the duration of +the sort: $3.0822 × 1.4939 = $4.6045. +• Output: The 100 TB output data is uploaded to and stored on Amazon S3 during the reduce stage +of the sort. We use the duration of the reduce stage as the storage time of the 100 TB output data. +This is an over-estimation because the output partitions are uploaded as the reduce stage progresses, +and therefore most of the 100 TB is stored on S3 for less time than the entire reduce stage duration. +Table 1 shows the average reduce stage time is 1870 seconds, or 0.5194 hours. Hence we get the output +storage cost: $3.0822 × 0.5194 = $1.6009. +Adding up the input and output data storage cost, we get the total data storage cost: $6.2054. +Data Access Cost. We consider GET and PUT requests to Amazon S3. Exoshuffle-CloudSort downloads +the 100 TB input data in 50 000 map tasks. Each map task downloads a 2 GB input partition in 16 MiB chunks, +resulting in 120 GET requests per task, or 6 000 000 GET requests in total. Amazon S3 charges $0.0004 per +1000 GET requests [3]. Hence the total GET cost is $2.4000. +Exoshuffle-CloudSort uploads the output data in 25 000 reduce tasks. Each reduce task uploads approxi- +mately 4 GB data in 100 MB chunks, resulting in 40 PUT requests, or 1 000 000 PUT requests in total. Amazon +S3 charges $0.005 per 1000 PUT requests [3]. Hence the total PUT cost is $5.0000. +The actual number of requests could be marginally higher due to request failures and retries, but the +amount should be negligible. Hence, the total data access cost is $7.4000. +Total Cost of Ownership. Adding up the compute cost and storage cost, we get the total cost of ownership +for the 100 TB CloudSort Benchmark: $96.6728. Table 2 presents a summary of the cost analysis. + +Exoshuffle-CloudSort +7 +Service +Unit Price +Amount +Total Price +Compute VM Cluster +$55.6044 / hr +1.4939 hours +$83.0674 +Data Storage (Input) +$3.0822 / hr +1.4939 hours +$4.6045 +Data Storage (Output) +$3.0822 / hr +0.5194 hours +$1.6009 +Data Access (Input) +$0.0004 / 1000 requests +6 000 000 requests +$2.4000 +Data Access (Output) +$0.005 / 1000 requests +1 000 000 requests +$5.0000 +Total +– +– +$96.6728 +Table 2. Cost breakdown of Exoshuffle-CloudSort on the 100 TB CloudSort Benchmark. +ACKNOWLEDGMENTS +This work is done in the Sky Computing Lab at UC Berkeley, sponsored by Astronomer, Google, IBM, Intel, +Lacework, Nexla, Samsung SDS, and VMware. This work is done in collaboration with Anyscale. +REFERENCES +[1] Amazon. 2022. Amazon EBS High-Performance Block Storage Pricing. Amazon Web Services. https://aws.amazon.com/ebs/pricing/ +[2] Amazon. 2022. Amazon EC2 On-Demand Instance Pricing. Amazon Web Services. https://aws.amazon.com/ec2/pricing/on-demand/ +[3] Amazon. 2022. Amazon S3 Simple Storage Service Pricing. Amazon Web Services. https://aws.amazon.com/s3/pricing/ +[4] Frank Sifei Luan, Stephanie Wang, Samyukta Yagati, Sean Kim, Kenneth Lien, Isaac Ong, SangBin Cho, Eric Liang, and Ion Stoica. +2022. Exoshuffle: Large-Scale Shuffle at the Application Level. https://doi.org/10.48550/ARXIV.2203.05072 +[5] Chris Nyberg. 2022. Sort Benchmark Data Generator and Output Validator. Ordinal Technology Corp. http://www.ordinal.com/ +gensort.html +[6] Mehul A. Shah, Amiato, and Chris Nyberg. 2014. CloudSort: A TCO Sort Benchmark. http://sortbenchmark.org/2014_06_ +CloudSort_v_0_4.pdf. (Accessed on 11/10/2022). +[7] Ray Team. 2022. Ray v2 Architecture. Anyscale. https://docs.google.com/document/d/1tBw9A4j62ruI5omIJbMxly-la5w4q_TjyJgJL_ +jN2fI/preview +[8] Stephanie Wang, Eric Liang, Edward Oakes, Ben Hindman, Frank Sifei Luan, Audrey Cheng, and Ion Stoica. 2021. Ownership: A +Distributed Futures System for Fine-Grained Tasks. In 18th USENIX Symposium on Networked Systems Design and Implementation +(NSDI 21). USENIX Association, Virtual, 671–686. https://www.usenix.org/conference/nsdi21/presentation/cheng + diff --git a/1dE2T4oBgHgl3EQfNQZD/content/tmp_files/load_file.txt b/1dE2T4oBgHgl3EQfNQZD/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9d67d8d60ec6e5130d0e365c2e4a6cb5d96ecca3 --- /dev/null +++ b/1dE2T4oBgHgl3EQfNQZD/content/tmp_files/load_file.txt @@ -0,0 +1,420 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf,len=419 +page_content='Exoshuffle-CloudSort FRANK SIFEI LUAN∗, UC Berkeley STEPHANIE WANG, UC Berkeley and Anyscale SAMYUKTA YAGATI, UC Berkeley SEAN KIM, UC Berkeley KENNETH LIEN, UC Berkeley ISAAC ONG, UC Berkeley TONY HONG, UC Berkeley SANGBIN CHO, Anyscale ERIC LIANG, Anyscale ION STOICA, UC Berkeley and Anyscale 1 INTRODUCTION We present Exoshuffle-CloudSort, a sorting application running on top of Ray using the Exoshuffle archi- tecture [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' Exoshuffle-CloudSort runs on Amazon EC2, with input and output data stored on Amazon S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' Using 40× i4i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='4xlarge workers, Exoshuffle-CloudSort completes the 100 TB CloudSort Benchmark (Indy category [6]) in 5378 seconds, with an average total cost of $97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' 2 IMPLEMENTATION 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='1 Overview Exoshuffle-CloudSort is a distributed futures program running on top of Ray, a task-based distributed execution system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' The program acts as the control plane to coordinate map and reduce tasks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' the Ray system acts as the data plane, responsible for executing tasks, transferring blocks, and recovering from failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' Exoshuffle-CloudSort implements a two-stage external sort algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' The first stage is map and shuffle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' Each map task reads an input partition, sorts it, and partitions the result into 𝑊 output partitions, each sent to a merger on a worker node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' A merger receives 𝑊 map output partitions, merges and sorts them, and further partition the result into 𝑅/𝑊 output partitions, all of which are spilled to local disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' The second stage is reduce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' Once the map and shuffle stage finishes, each reduce task reads 𝑊 shuffled partitions, merges and sorts them, and writes the final output partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' For the 100 TB CloudSort Benchmark, we set the following parameters: Total data size is 100 TB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' Number of input partitions 𝑀 = 50 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' Each input partition is 2 GB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' ∗Author’s address: lsf@berkeley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='edu, 465 Soda Hall, Berkeley, CA, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='03734v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='DC] 10 Jan 2023 2 Luan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' Number of workers 𝑊 = 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' Number of output partitions 𝑅 = 25 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='2 Preparation The first step in Exoshuffle-CloudSort is to compute the partition boundary values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' For a sort record with 10-byte key, we view the first 8 bytes as a 64-bit unsigned integer partition key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' We partition the key space [0, 264 − 1) into 𝑅 = 25 000 equal ranges, such that all the records within a key range should be sent to one reducer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' Every 𝑅1 = 𝑅/𝑊 = 625 reducer ranges are combined into a worker range, and records in each worker range will be sent to one worker node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' This yields 𝑊 = 40 equally-partitioned worker ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='3 Map and Shuffle Stage In the map and shuffle stage, Exoshuffle-CloudSort schedules the 𝑀 = 50 000 map tasks onto all worker nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' In our experiments we set the map parallelism, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' the number of map tasks running on a single worker node, to be 3/4 of the total number of vCPU cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' Extra tasks are queued on the driver node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' Whenever a worker node finishes a map task, the driver assigns a new task from the queue to this node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' In a map task, we first download the input partition from S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' We then sort the input data in memory, then partition it into 𝑊 = 40 slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' Each slice is eagerly sent to a merge controller on each worker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' The map task returns when all slices are sent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' On the receiving end, the merge controller accumulates the map blocks in memory until a threshold is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' We set the threshold to 40 blocks, or about 2 GB of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' Once the threshold is reached, the controller launches a merge task to merge the already-sorted map blocks, and further partitions it into 𝑅1 = 625 merged blocks, each corresponding to a reduce task on this node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' These blocks are spilled to the local SSD for use by the reducers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' The merge parallelism is set to be the same as the map parallelism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' When the number of merge tasks reaches the maximum parallelism, and the merge controller’s in-memory buffer is filled up, it will hold off acknowledging the receipt of a map block until a merge task finishes and a new merge task can launch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' This effectively creates back pressure to the map task scheduler to ensure the map, shuffle, and merge progresses are in sync.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' In our experiments, the average map task duration is 24 seconds;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' 15 seconds are used for downloading input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' The average shuffle time (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' time to send and receive blocks) is 7 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' The merge task takes 17 seconds on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='4 Reduce Stage Once all map and merge tasks finish, Exoshuffle-CloudSort enters the reduce stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' Each reduce task loads 𝑅1 = 625 from the local SSD, merges them, and uploads the sorted output partition to S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' In our experiments, each reduce task takes 22 seconds on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' Exoshuffle-CloudSort 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='5 The Execution System A highlight of the Exoshuffle architecture is that the application program only implements the control plane logic, and the distributed futures system, Ray, handles execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' This is reflected in Exoshuffle-CloudSort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' Here is an incomplete list of features provided by Ray that we take “for free”: Task scheduling: The program specifies when and where to schedule tasks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' the system handles the RPC, serialization, and other bookkeeping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' Network transfer: The program instructs data to be transferred by passing distributed futures as task arguments;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' the system implements high-performance network transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' Memory management and disk spilling: The program manipulates data references in a virtual, infinite address space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' the system uses reference counting to manage distributed memory, spills objects to local disks when memory is low, and restores objects from local disks when they are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' Pipelining of network and disk I/O: The network transfer, spilling and recovery of objects are trans- parent to the application and are performed asynchronously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' For example, the system shuffles map output blocks while other map and merge tasks are running;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' it spills merge task output to disk while other merge tasks are executing, and it restores merged blocks while reduce tasks are executing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' Fault tolerance: this is transparent to the application: the system automatically retries the operation when it encounters network failures and worker process failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' For more details, we refer the reader to the Ray Architecture Whitepaper [7], the ownership design for distributed futures systems [8], and the Exoshuffle paper [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='6 Source Code Exoshuffle-CloudSort is implemented in about 1000 lines of Python, and about 300 lines of C++.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' The C++ component implements two functionalities: sorting and partitioning records, and merging sorted record arrays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' Exoshuffle-CloudSort runs on top of Ray, which is implemented in Python and C++.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' All of Exoshuffle-CloudSort’s source code is available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='com/exoshuffle/cloudsort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' 3 EVALUATION 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='1 Environment Setup We run Exoshuffle-CloudSort on AWS on a compute cluster configured as follows: 1× r6i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='2xlarge master node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' This node runs on 8 cores of an Intel Xeon 8375C CPU at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='9 GHz, and 64 GiB memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' 40× i4i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='4xlarge worker nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' Each node runs on 16 cores of an Intel Xeon 8375C CPU at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='9 GHz, and 128 GiB memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' Each node has a directly-attached 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='75 TB AWS Nitro NVMe SSD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' Each node is attached with a 40 GiB Amazon EBS General Purpose SSD (gp3) volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' The software stack is configured as follows: 4 Luan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' Ubuntu 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='1 LTS, Linux kernel version 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='0-1022-aws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' XFS 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='0 filesystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' Intel oneAPI DPC++/C++ Compiler 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='20220730.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' Python 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' Ray 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' We measure the raw system I/O performance on the worker nodes using standard benchmarking tools: Network bandwidth: 25 Gbps between nodes, benchmarked with iperf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' SSD: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='9 GB/s read, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='2 GB/s write, benchmarked with fio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' For storage, we use 40 buckets on Amazon S3 and randomly distribute the input and output partitions across the buckets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='2 Benchmark Setup Generating Input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' We use gensort version 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='5 as provided by the Sort Benchmark committee [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' We run the command gensort -c -b{offset} {size} {path} to generate each partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' {size} is fixed at 𝑃 = 20 000 000 such that each partition is exactly 2 GB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' {offset} takes the values {𝑖 · 𝑃 : 0 ≤ 𝑖 < 𝑀} where the number of input partitions 𝑀 = 50 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' {path} is a unique path in tmpfs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' -c provides data checksum for validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' After generating an input file, we randomly choose a bucket and upload the partition to S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' We use Ray to schedule the 50 000 input generation tasks to all 40 worker nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' The result is aggregated as an input manifest file, saved for use by Exoshuffle-CloudSort to locate the sort input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' Validating Output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' Exoshuffle-CloudSort produces an output manifest file containing the bucket and keys of each output partition on S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' In each validation task, we first download the output partition to tmpfs, then run the command valsort -o {sumpath} {path} to validate the ordering of records in each partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' We use Ray to schedule the 25 000 output validation tasks to all 40 worker nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' We concatenate the contents of the summary files from each validation task, then run valsort -s to validate the total ordering, and generate the total output checksum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' Finally, we compare the output checksum with the input checksum to verify data integrity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='3 Experimental Results 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='1 Job Completion Time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' On November 10, 2022, we ran the 100 TB CloudSort Benchmark in the AWS US West (Oregon, us-west-2) region with the setup described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' We first generated the input data on Amazon S3, then ran Exoshuffle-CloudSort 3 times, each followed by a validation step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' All 3 runs succeeded with the same output checksum as the input, indicating all bytes are preserved in the sort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' Table 1 reports the job completion times of each run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' The average job completion time is 5378 seconds, or 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='4939 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' Figure 1 shows the system utilizations of all worker nodes in the compute cluster during run #1 of the 100 TB CloudSort Benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' Exoshuffle-CloudSort 5 Run Map & Shuffle Time Reduce Time Total Job Completion Time #1 3509 s 1852 s 5361 s #2 3496 s 1852 s 5348 s #3 3520 s 1906 s 5426 s Average 3508 s 1870 s 5378 s Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' Job completion times of Exoshuffle-CloudSort on the 100 TB CloudSort Benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' Cluster utilization during run #1 of the 100 TB CloudSort Benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' Each thick line represents the median system utilization of all worker nodes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' the highest and lowest lines represent the maximum and minimum utilization among all worker nodes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='2 Total Cost of Ownership.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' The total job cost comprises of two parts: compute cost (Amazon EC2), and the storage cost (Amazon S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' The storage cost is further divided into data storage cost and data access cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' Compute Cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' The compute cost is calculated as the compute cluster’s hourly cost times the job completion time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' The total hourly cost is calculated as follows: Total Hourly Compute Cost = Master Node Hourly Cost + Worker Node Hourly Cost × Number of Workers + EBS Volume Hourly Cost × (Number of Workers + 1) (1) We obtain the compute instance hourly costs from the Amazon EC2 on-demand pricing information [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' For EBS, we use the Amazon EBS monthly price [1] divided by the average number of hours in a month ( 365×24 12 = 730) as the hourly price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' The hourly cost of a 40 GiB gp3 volume is $0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='08/730 × 40 = $0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='0044.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' Now ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='we plug the cost variables into Equation (1): ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='CPU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='Memory ' metadata={'source': 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+page_content='2 GB/s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='40% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='1 GB/s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='1 GB/s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='500 MB/s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='20% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='0 B/s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='02:40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='02:50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='03:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='03:10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='03:20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='03:30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='03:40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='03:50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='04:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='0 B/s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='02:4002:50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='¥03:0003:1003:2003:30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='03:4003:50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='04:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='median network in ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='min network in ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='max network in ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='median network out ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='median disk write - min disk write - max disk write - median disk read ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='min network out ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='min network total ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='max network out ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='02:40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='02:50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='03:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='03:10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='03:20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='03:30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='03:40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='03:50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='04:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='min disk read ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='max disk read - ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='median disk total - max disk total ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='max network total6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='Luan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' Master node (r6i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='2xlarge) hourly cost is $0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='504.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' Worker node (i4i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='4xlarge) hourly cost is $1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='373.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' Number of workers is 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' EBS volume hourly cost is $0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='0044.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' Hence, the total hourly compute cost is $55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='6044.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' We multiply this hourly cost by the job completion time of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='4939 hours to obtain the total compute cost of $83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='0674.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' Data Storage Cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' The storage cost comprises of data storage cost and data access cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' We first consider the data storage cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' Amazon S3 employs a pay-as-you-go pricing model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' the user does not need to provision storage capacity ahead of time, and only pays for the storage cost of objects based on their sizes and storage duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' Amazon S3 charges $0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='023 per GB-month for the first 50 TB, then $0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='022 per GB-month for the next 450 TB [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' Since the total data size is 100 TB, we take the average price between the first two tiers, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' $0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='0225 per GB-month, or $3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='0822 per hour per 100 TB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' Input: The storage cost of the 100 TB input data is simply the cost to store 100 TB for the duration of the sort: $3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='0822 × 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='4939 = $4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='6045.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' Output: The 100 TB output data is uploaded to and stored on Amazon S3 during the reduce stage of the sort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' We use the duration of the reduce stage as the storage time of the 100 TB output data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' This is an over-estimation because the output partitions are uploaded as the reduce stage progresses, and therefore most of the 100 TB is stored on S3 for less time than the entire reduce stage duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' Table 1 shows the average reduce stage time is 1870 seconds, or 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='5194 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' Hence we get the output storage cost: $3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='0822 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='5194 = $1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='6009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' Adding up the input and output data storage cost, we get the total data storage cost: $6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='2054.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' Data Access Cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' We consider GET and PUT requests to Amazon S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' Exoshuffle-CloudSort downloads the 100 TB input data in 50 000 map tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' Each map task downloads a 2 GB input partition in 16 MiB chunks, resulting in 120 GET requests per task, or 6 000 000 GET requests in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' Amazon S3 charges $0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='0004 per 1000 GET requests [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' Hence the total GET cost is $2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='4000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' Exoshuffle-CloudSort uploads the output data in 25 000 reduce tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' Each reduce task uploads approxi- mately 4 GB data in 100 MB chunks, resulting in 40 PUT requests, or 1 000 000 PUT requests in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' Amazon S3 charges $0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='005 per 1000 PUT requests [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' Hence the total PUT cost is $5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='0000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' The actual number of requests could be marginally higher due to request failures and retries, but the amount should be negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' Hence, the total data access cost is $7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='4000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' Total Cost of Ownership.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' Adding up the compute cost and storage cost, we get the total cost of ownership for the 100 TB CloudSort Benchmark: $96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='6728.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' Table 2 presents a summary of the cost analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' Exoshuffle-CloudSort 7 Service Unit Price Amount Total Price Compute VM Cluster $55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='6044 / hr 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='4939 hours $83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='0674 Data Storage (Input) $3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='0822 / hr 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='4939 hours $4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='6045 Data Storage (Output) $3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='0822 / hr 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='5194 hours $1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='6009 Data Access (Input) $0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='0004 / 1000 requests 6 000 000 requests $2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='4000 Data Access (Output) $0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='005 / 1000 requests 1 000 000 requests $5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='0000 Total – – $96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='6728 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' Cost breakdown of Exoshuffle-CloudSort on the 100 TB CloudSort Benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' ACKNOWLEDGMENTS This work is done in the Sky Computing Lab at UC Berkeley, sponsored by Astronomer, Google, IBM, Intel, Lacework, Nexla, Samsung SDS, and VMware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' This work is done in collaboration with Anyscale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' REFERENCES [1] Amazon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' Amazon EBS High-Performance Block Storage Pricing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' Amazon Web Services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' https://aws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='amazon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='com/ebs/pricing/ [2] Amazon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' Amazon EC2 On-Demand Instance Pricing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' Amazon Web Services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' https://aws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='amazon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='com/ec2/pricing/on-demand/ [3] Amazon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' Amazon S3 Simple Storage Service Pricing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' Amazon Web Services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' https://aws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='amazon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='com/s3/pricing/ [4] Frank Sifei Luan, Stephanie Wang, Samyukta Yagati, Sean Kim, Kenneth Lien, Isaac Ong, SangBin Cho, Eric Liang, and Ion Stoica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' Exoshuffle: Large-Scale Shuffle at the Application Level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='48550/ARXIV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='05072 [5] Chris Nyberg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' Sort Benchmark Data Generator and Output Validator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' Ordinal Technology Corp.' metadata={'source': 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+page_content=' CloudSort: A TCO Sort Benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' http://sortbenchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='org/2014_06_ CloudSort_v_0_4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='pdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' (Accessed on 11/10/2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' [7] Ray Team.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' 2022.' metadata={'source': 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+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' Ownership: A Distributed Futures System for Fine-Grained Tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' In 18th USENIX Symposium on Networked Systems Design and Implementation (NSDI 21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' USENIX Association, Virtual, 671–686.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'} +page_content='usenix.' metadata={'source': 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+version https://git-lfs.github.com/spec/v1 +oid sha256:c2b160dcfb7fcd866e3452ddb93eba54b176455dee995ae5ea0fe222a23f057b +size 252938 diff --git a/4NAzT4oBgHgl3EQfffyv/content/tmp_files/2301.01454v1.pdf.txt b/4NAzT4oBgHgl3EQfffyv/content/tmp_files/2301.01454v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..8a53b701bd4bb7e9682312356150f90d582670f7 --- /dev/null +++ b/4NAzT4oBgHgl3EQfffyv/content/tmp_files/2301.01454v1.pdf.txt @@ -0,0 +1,1097 @@ +Accurate, Low-latency, Efficient SAR Automatic +Target Recognition on FPGA +Bingyi Zhang∗, Rajgopal Kannan†, Viktor Prasanna∗, Carl Busart† +∗University of Southern California †DEVCOM US Army Research Lab +∗{bingyizh, prasanna}@usc.edu †{rajgopal.kannan.civ, carl.e.busart.civ}@army.mil +Abstract—Synthetic aperture radar (SAR) automatic target +recognition (ATR) is the key technique for remote-sensing image +recognition. The state-of-the-art convolutional neural networks +(CNNs) for SAR ATR suffer from high computation cost and +large memory footprint, making them unsuitable to be deployed +on resource-limited platforms, such as small/micro satellites. +In this paper, we propose a comprehensive GNN-based model- +architecture co-design on FPGA to address the above issues. +Model design: we design a novel graph neural network (GNN) for +SAR ATR. The proposed GNN model incorporates GraphSAGE +layer operators and attention mechanism, achieving comparable +accuracy as the state-of-the-art work with near 1/100 computa- +tion cost. Then, we propose a pruning approach including weight +pruning and input pruning. While weight pruning through lasso +regression reduces most parameters without accuracy drop, input +pruning eliminates most input pixels with negligible accuracy +drop. Architecture design: to fully unleash the computation +parallelism within the proposed model, we develop a novel unified +hardware architecture that can execute various computation +kernels (feature aggregation, feature transformation, graph pool- +ing). The proposed hardware design adopts the Scatter-Gather +paradigm to efficiently handle the irregular computation patterns +of various computation kernels. We deploy the proposed design +on an embedded FPGA (AMD Xilinx ZCU104) and evaluate +the performance using MSTAR dataset. Compared with the +state-of-the-art CNNs, the proposed GNN achieves comparable +accuracy with 1/3258 computation cost and 1/83 model size. +Compared with the state-of-the-art CPU/GPU, our FPGA accel- +erator achieves 14.8×/2.5× speedup (latency) and is 62×/39× +more energy efficient. +Index Terms—SAR ATR, graph neural network (GNN), hard- +ware architecture +I. INTRODUCTION +Synthetic aperture radar (SAR) can acquire remote-sensing +data in all-weather conditions to observe target on the earth +ground. SAR has been widely used in real-world applications, +such as agriculture [1], [2], civilization [3], [4], etc. SAR +automatic target recognition (ATR) is the key technique to +classify the target in a SAR image. Convolutional neural +networks (CNNs) [5]–[9] have been extensively studied for +ATR SAR since CNNs can extract discriminative features +from an image. However, the CNN-based approaches [5]– +[9] suffer from two issues: (1) high computation cost: to +achieve high accuracy, the authors [5]–[9] develop large CNN +models with high computation complexity, (2) large memory +requirement: these large CNN models have large number of +parameters, which require large memory footprint. Therefore, +it is unsuitable to deploy large CNNs on resource-limited +platforms, such as small/micro satellites [10]–[14]. +The causes of the above issues are (1) heavy convolutional +operations in CNNs, and (2) CNNs are hard to exploit data +sparsity in SAR images because CNNs need to use the whole +image as the input. In a SAR image (Figure 1), only a +small set of pixels belongs to the target (defined as pixels of +interest, POI), which can be easily extracted through applying +a constant threshold [15]. However, the extracted POI has +irregular structure that is hard to be processed by CNNs, +where Graph Neural Network (GNN) provides an opportunity. +Intuitively, we can use the POI to construct a graph and +use GNN to perform classification for the graph. Fortunately, +GNNs have been proven to be powerful models [16] to +classify graphs based on graph structural information and +vertex features. Therefore GNNs [17]–[19] have been applied +to many graph classification tasks [20]–[24]. Recently, GNNs +have been successfully applied to many image classification +tasks [25]–[27]. Motivated by that, we design a novel GNN +model for SAR ATR (Section III-A). We propose a graph +representation G(V, E) for a SAR image. The proposed GNN +model can extract the structural information of the target from +the constructed graph. To improve classification accuracy, we +leverage the attention mechanism including spatial attention +and channel attention to identify the important vertices and +features. To further reduce the computation complexity, we +perform weight pruning by training the GNN model through +lasso regression and pruning the GNN model weights that have +small absolute values. Taking advantage of the GNN model, +we perform input pruning (POI extraction). By eliminating the +vertices that have small value, the computation complexity is +reduced by 92.8% with small accuracy loss (< 0.17%). +The proposed GNN has the following advantages: (1) even +without weight/input pruning, the proposed GNN has near +1/100 computation cost as the state-of-the-art CNNs with +similar accuracy, (2) while weight pruning can potentially be +exploited by CNNs, input pruning is hard to be exploited by +CNNs because CNNs need to use the whole image as the +input. GNN is flexible to use a small set of input pixels as +the input. Therefore, despite that we can accelerate the CNNs +[5]–[9] on advanced CNN accelerators [28], their latency is +still significant (Section VI-D). +While the proposed GNN is lightweight that can be de- +ployed on the resource limited platforms, accelerating GNNs +is challenging. GNNs have irregular computation pattern and +heterogeneous computation kernels [29], making them ineffi- +cient to be deployed on the general purpose processors. The +arXiv:2301.01454v1 [cs.AR] 4 Jan 2023 + +pruned GNN model introduces additional irregularity through +weight pruning. Moreover, the proposed model has various +heterogeneous computation kernels (feature aggregation, fea- +ture transformation, graph pooling) that need to be mapped +on an accelerator. While there are many GNN accelerators +[29]–[35] proposed, none of them exploits the sparsity of the +weight matrices or deals with graph pooling, which are still +inefficient for the proposed model. While the proposed GNN +achieves high accuracy with small computation complexity, +we believe that low-latency execution of SAR ATR must be +achieved through careful model-architecture co-design. +Therefore, we develop a novel unified hardware architecture +for the proposed GNN model. We demonstrate the methods +of mapping various computation kernels onto the proposed +accelerator. In the accelerator design, we adopt Scatter-Gather +paradigm to efficient deal with the irregular computation +patterns of various kernels. To the best of our knowledge, this +is the first GNN-based model-architecture co-design for SAR +ATR. Our main contributions are: +• We propose a lightweight GNN for SAR ATR that +achieves comparable accuracy as state-of-the-art GNNs +with significant less computation complexity. +• We perform weight pruning and input pruning to dramat- +ically reduce the computation complexity and the number +of model weights. +• We design a unified hardware architecture that can exe- +cute various computation kernels in the proposed model. +We adopt Scatter-Gather paradigm to deal with the irreg- +ular computation patterns. +• Taking advantage of the proposed hardware mapping +strategy, we further optimize the load balance of various +computation kernels (Section V-A). +• We deploy our co-design on Xilinx ZCU104. We evaluate +our co-design using MSTAR dataset. Compared with +the state-of-the-art CNNs, the proposed GNN achieves +comparable accuracy with 1/3258 computation cost and +1/83 model size. Compared with the state-of-the-art +CPU/GPU, our FPGA accelerator achieves 14.8×/2.5× +speedup (latency) and is 62×/39× more energy efficient. +II. BACKGROUND AND RELATED WORK +A. Related Work +Fig. 1: The SAR images of various targets (vehicles) +SAR ATR is to automatically classify the target in a given +SAR images (Figure 1). To achieve high accuracy, deep +learning based methods have been extensively studied. David +[6] demonstrates that CNNs outperform traditional methods, +such as Support Vector Machine, etc. TAI-SARNET [9] is a +TABLE I: Notations +Notation +Description +Notation +Description +G(V, E, X0) +input graph +vi +ith vertex +V +set of vertices +eij +edge from vi to vj +E +set of edges +L +number of GNN layers +hl +i +feature vector of vi at layer l +N(i) +neighbors of vi +CNN model that incorporates atrous convolution and inception +module to achieve high accuracy for SAR ATR. The authors +[8] combine multi-view features to classify the target in SAR +images. The authors [5] propose the Convolutional Block At- +tention Module by exploiting the spatial attention and channel +attention. However, the state-of-the-art CNNs [5], [8], [9] suf- +fer from high computation cost, making them unsuitable to be +deployed on resource-limited platforms. Recently, the authors +[15] exploit GNN for SAR ATR. They construct graphs from +SAR images by connecting the pixels by the declined order of +pixel grayscale value. However, the constructed graphs lose the +structural information of targets, making it extremely sensitive +to the variations of input pixel values. +B. Graph Neural Network +The notations are defined in Table I. Graph Neural Networks +(GNN) [17]–[19] are proposed for representation learning on +graph G(V, E, X0). GNNs can learn from the structural infor- +mation and vertex/edge features of the graph, and embed these +information into low-dimension vector representation/graph +embedding (For example, hL +i is the embedding of vertex vi). +The vector representation can be used for many downstream +tasks, such as node classification [17], [18], link prediction +[36], graph classification [37], etc. GNNs follow the message- +passing paradigm that vertices recursively aggregate informa- +tion from the neighbors, for example: +GraphSAGE: GraphSAGE is proposed in [18] for inductive +representation learning on graphs. The GraphSAGE layer +follows the aggregate-update paradigm: +aggregate:zl +i = Mean +� +hl−1 +j +: j ∈ N(i) ∪ {i} +� +update:hl +i = ReLU +� +zl +iW l +neighbor + bl +neighbor||hl−1 +i +W l +self + bl +self +� (1) +III. MODEL-ARCHITECTURE CO-DESIGN +To achieve accurate and efficient SAR ATR on FPGA +platform, we perform comprehensive model-architecture co- +design. The proposed co-design consists of a novel GNN +model for SAR ATR (Section III-A), a pruning strategy to +reduce the computation complexity (Section III-B), a novel +hardware design to efficiently execute the proposed GNN +(Section III-C), and the strategy to keep load balance within +various computation kernels (Section V-A). The key novelty of +our hardware design is that it can execute various computation +kernels in the proposed model, and it can efficiently handle +the irregular computation patterns caused by the sparsity of +weight matrices. We use the widely used MSTAR dataset [38] +for performance evaluation. We target various performance + +BTR70 +BRDM2 +D7 +T62 +U +20 +20 +20 +40 +40 +D +40 +60 +60 +60 +08 +80 +80 +100 +100 +0 +100 +120 +120 +0 +120 +0 +2550 +100 +125 +25 +50 +75 +100 +0 +255075100 +125 +0 +2550 +75100 +125metrics: (1) Accuracy: the accuracy on MSTAR dataset, (2) +Computation complexity: the total computation complexity for +inferring a SAR image, (3) Number of parameters: the total +number of parameters in the model, (4) Latency: the latency for +inferring a SAR image, (5) Energy Consumption: the energy +consumption for inferring a SAR image. +A. GNN Model Design +Graph representation +GNNL +Pooling +Attention +GNNL +Pooling +Attention +… +… +… +GNNL +MLP +Classification result +SAR image +Spatial +Attention +Channel +Attention +x +x ++ +Attention module +GNNL-1 +Pooling-1 +Attention-1 +GNNL-2 +Pooling-2 +Attention-2 +GNNL-L +Pooling +within each +2 × 2 range +Fig. 2: The proposed GNN model +Graph representation: We represent a SAR image as a graph +G(V, E), with each pixel viewed as a vertex. Each pixel/vertex +is connected to its four neighbors (up, down, left, right) +with edges. The feature of a vertex is the grayscale value +of the pixel. Such graph representation maintains structural +information of the target that can be learned by GNN for +classification. It also provides the opportunity for input pruning +(Section III-B). +GNN model: As shown in Figure 2, the proposed GNN model +has a sequence of layers, including GNN layer (GNNL), graph +pooling layer (Pooling), Attention module (Attention). For +GNN layer, we use the GraphSAGE layer operators [18], +which have been proven to achieve superior accuracy in +various application domains. For graph pooling layer, since +the input graph has 2-D grid structure, we adopt the similar +pooling strategy as the CNN for 2-D image. Within each local +s × s range having s2 vertices, the pooling operator (e.g., +Max(), Min()) is performed on the s2 vertices to obtain an +output vertex. Figure 2 demonstrates the pooling operation of +size 2×2 with stride 2. Motivated by the attention mechanism +in CNN [39], the proposed Attention module consists of a +Channel Attention module and a Spatial Attention module. +Suppose the input to Attention Module is {hi : vi ∈ G}, +where hi ∈ Rc is the feature vector of vi and c is the +length of the feature vector. The Channel Attention calculates +the attention score Fch of each feature through a Multi- +layer perceptron. Then, each vertex is multiplied by Fch to +obtain {(hi)′ : (hi)′ = hi ⊗ Fch, vi ∈ G} where ⊗ is +the element-wise multiplication. The Spatial Attention module +calculates the attention score of each vertex using a GNN layer +(GraphSAGE layer operators): +{αi : vi ∈ G} = sigmoid(GNNL({hi : vi ∈ G})), +Then, each vertex feature vector is multiplied by its attention +score: {(hi)′′ : (hi)′′ = αihi, vi ∈ G}. The output of the +Attention module is calculated by: +{houtput +i +: houtput +i += hi + (hi)′ + (hi)′′, vi ∈ G} +(2) +After GNNL-L, all the feature vectors are flattened to a +vector which becomes the input to the last MLP (Multi-layer +Perceptron) for classification. +B. Network Pruning +Weight pruning: To reduce the total computation complexity, +we perform weight pruning by training the model using lasso +regression [40]. We add a L1 penalty term to the loss function: +loss = +N +� +i=1 +(yi − Model(Gi))2 + λ +W +� +w +|w| +The penalty term results in weight shrinkage. Some model +weights become zeros and are eliminated from the model. +After training, we set a threshold Iweight and the weights with +absolute values smaller than Iweight are pruned. +Input pruning: In a SAR image, most pixels outside of the +target have negligible grayscale values. Therefore, in the graph +representation G(V, E) of a SAR image, we set a threshold +Ivertex and prune the vertices that have grayscale values smaller +than Ivertex. The edges connected to the pruned vertices are also +pruned. After input pruning, the eliminated vertices maintain +the same positions in the graph pooling layer and do not +participate in the pooling operation. For example, in a local +2 × 2 range, if a vertex is pruned, the pooling operator will +operate on the remaining three vertices. For the input to last +MLP, the feature vectors of the pruned vertices are padded +using zeros. +C. Architecture design +The objective of the architecture design is to (1) support +various computation kernels in the proposed model, (2) han- +dle the irregular computation patterns caused by the feature +aggregation in the GNN layer and the sparsity of the weight +matrices. Figure 3 shows the proposed architecture design +on the embedded FPGA platform. The system consists of an +Application Processing Unit (APU) and an FPGA accelerator +in Programmable Logic Region. The FPGA accelerator exe- +cutes the inference process of the GNN model. In the FPGA +accelerator, there is a Weight/Edge Buffer (WEB) to store +the model weights and edges of input graph, an Input Buffer +(IB) to store the input vertex feature vectors, a Results Buffer +(RB) to store the output vertex feature vectors. The Matrix +Transformation Unit (MTU) performs matrix transformation +to prepare the require data layout for the next layer. Thanks +to the proposed lightweight model, the trained model is fully + +APU +DDR controller +Scatter 1 +Scatter 2 +…… +Scatter ������������ +Gather 1 +Gather 2 +…… +Gather p +Routing +Network +MTU +Bank 1 +Bank 2 +…… +Bank ������������ +Result Buffer +Input +Buffer +Weight +/Edge +Buffer +DMA +Programmable Logic +Scatter +Gather 1 +demux +mux +x +demux +mux +x +….. +…… +MTU +Matrix Transformation Unit +FPGA +APU +Application Processing Unit (e.g., ARM Cortex-A53) +mux +demux +mux +ACC +Max +mux +ReLU +Sigmoid +demux +mux +ACC +Max +mux +ReLU +Sigmoid +Fig. 3: The diagram of the system architecture +stored in the Weight Buffer, eliminating the memory traffic of +loading the model weights at runtime. +Run Time: At runtime, the APU receives an input SAR image +and transform it into the graph presentation. During the trans- +formation, the pixels that have grayscale value smaller than +Ivertex are pruned. Then, the APU sends the input graph to the +Input Buffer of the accelerator. The accelerator executes each +layer using Scatter-Gather paradigm (SGP). The accelerator +exploits the computation parallelism within each layer. After +finishing the execution of all layers, the accelerator sends the +classification result back to the APU. +IV. HARDWARE MAPPING +A. Computation kernels +We categorize the computation kernels into two classes: +Vertex aggregation kernel (VAK): VAKs include (1) feature +aggregation (in GNN layer, and in Spatial Attention module) +(2) graph pooling. In VAKs, each vertex propagates its feature +vector to the neighbors or within a local range (graph pooling). +Vertex updating kernel (VUK): VUKs include (1) feature +update (in GNN layer, and in Spatial Attention module) (2) +Channel attention of Attention module, (3) the last MLP. In +the VUKs, the feature vector of each vertex is multiplied by a +weight matrix to obtain the updated feature vector. Due to our +weight pruning, the weight matrices have high data sparsity +(1%-33% data density). +B. Kernel Mapping using Scatter-Gather Paradigm +Algorithm 1 Scatter-Gather paradigm +while not done do +Scatter Unit: +for each edge e⟨src, dst, weight⟩ do +Produce update u ←Scatter(src.vector, e.weight) +end for +Gather Unit: +for each update u⟨dst, vector⟩ do +Update vertex vdst ← Gather(u.vector) +end for +end while +The accelerator design is based on the Scatter-Gather +paradigm (Algorithm 1). There are p parallel pipelines. Each +pipeline consists of a Scatter Unit and a Gather Unit. The +Routing Network routes the intermediate results to the des- +tination based on index dst. To map the VAKs and VUKs +������������1 +������������2 +������������3 +������������4 +������������1 +������������2 +������������3 +������������4 +������������1 ������������2 ������������3������������4 +������������������������������������ +������������������������������������ +Input Feature vectors +������������1 +������������2 +������������3 +������������4 +Output Feature vectors +Vertex aggregation kernel +������������1 +������������2 +������������3 +������������4 +1 +2 +3 +4 +5 +1 2 3 4 +Adjacency matrix +������������������������������������ +������������������������������������������������ = ������������������������������������ +������������������������������������ = 5 +������������������������������������ +������������������������������������ +������������1 +������������2 +������������3 +������������4 +������������������������������������������������ = 4 +Weight matrix +1 2 3 4 5 +2 +1 +3 4 +Vertex updating +kernel +������������1 +������������������������������������������������������������ +������������1 +������������������������������������������������������������������������ +Input Feature +vectors +Output Feature vectors +Fig. 4: The diagram of mapping the two types of kernels using +Scatter-Gather paradigm +to the accelerator, we propose the following mapping strategy +(An example is shown in Figure 4): +Mapping VAK: VAK can be directly mapped to the accelera- +tor. For each edge e⟨src, dst, weight⟩, the Scatter Unit loads +the feature vector of vsrc from input buffer and produces an +update u⟨dst, vector⟩. The update u⟨dst, vector⟩ is routed to +the corresponding Gather Unit and the Gather Unit applies the +update to the destination vertex vdst. +Mapping VUK: For VUK, we group a batch of vertices batch +and the feature vector of each vertex {hinput +i +: vi ∈ batch} +is multiplied by the weight matrix W simultaneously. The +output feature vectors are {houtput +i +: hinput +i +W , vi ∈ batch}. +To apply the Scatter-Gather paradigm, we perform feature +concatenation. For example, we concatenate the first feature +of each vertex {hi(1) : vi ∈ batch} as a vector rinput +1 +. +The vector rinput +1 +has src index 1 since its contains the 1st +feature of each input feature vector. For the weight matrix +W , we represent each non-zero element in the weight matrix +as an edge e⟨src, dst, weight⟩. During execution, for each +non-zero weight e⟨src, dst, weight⟩, the Scatter Unit loads +the rinput +src +from the input buffer and produces an update +u⟨dst, vector = e.weight × rinput +src +⟩. Then, the Gather Unit +applies the update u⟨dst, vector⟩ to the destination routput +dst +. +routput +dst +contains the dstth features of each output feature +vector in the batch. +Note that VAK and VUK have different data layouts. In +VAK, the input/output feature vectors are stored in vertex- +major order. In VUK, the input/output feature vectors are +stored in feature-major order. To switch between the two data +layouts, we implement a Matrix Transformation Unit (MTU) + +to perform data layout transformation. +C. hardware modules +Scatter/Gather Unit: A Scatter Unit has an array of q +processing elements. Each processing element has a multiplier +to perform the multiplication between an edge/weight and a +vertex feature. Similar to the Scatter Unit, a Gather Unit has an +array of q processing elements. Each processing element has +an Accumulator (ACC), a Max Unit, a ReLU Unit, a sigmoid +Unit. The multiplexer (MUX) and demultiplexer (DEMUX) +select the datapath for the current layer. +Routing Network: The routing network is implemented using +a hardware-efficient butterfly network [41]. +Sigmoid Unit: We exploit the piecewise linear approximation +(PLA) [42] for Sigmoid Function. +V. LOAD BALANCE AND PERFORMANCE MODEL +A. Load Balance +Load balance in VAK: The workload balance of VAK +depends on how to partition the vertices into p memory +banks of the Result Buffer. Load imbalance is a significant +issue in GNN [43] if the graph has highly imbalanced degree +distribution. Thanks to our graph representation, the vertices +in the graph have degrees ranging from 0 to 4. We use a +greedy approach to keep the load balance of the p parallel +pipelines. For VAK, the destination vertices that have same +degree i (0 ⩽ i ⩽ 4) are evenly partitioned into p banks of +the Result Buffer. Through the proposed partitioning strategy, +each pipeline has the same amount of workload. The graph +partitioning has a small overhead O(|V|Lp) and is performed +by the APU, where Lp is the number of graph pooling layers in +the model. The proposed partitioning algorithm can be easily +parallelized using multiple threads on APU. +Load balance in VUK: To execute VUK, we need to partition +the weight matrix along the dst dimension (Figure 4). Each +Gather Unit is responsible for accumulating the partial results +of a partition. To achieve perfect load balance, each partition +should have the same number of non-zero elements. Since +the partitioning of weight matrix is an offline process, we are +able to adopt complexity algorithm to find the near optimal +data partitioning. In this work, we exploit Longest-processing- +time (LPT) first algorithm that is proved to achieve 4/3 +approximation factor [44] to the optimal partition solution. +B. Performance Model +Modeling VAK: For a VAK kernel, the length of input feature +vector cin is same as the length of output feature vector cout: +cin = cout. A Scatter Unit or a Gather Unit can process q +features in each clock cycle. The p parallel pipelines can +process p edges simultaneously. Therefore, the execution time +of a VAK kernel is: +tVAK = +�|E| +p +� +· +�cin +q +� +(3) +Modeling VUK: To execute a VUK, the accelerator groups +a batch of q vertices at a time to fully utilize the Scatter +Unit/Gather Unit. The p parallel pipelines can process p non- +zero elements in the weight matrix. Therefore, the execution +time of a VUK kernel is: +tVUK = +�|V| +q +� +· +�nnz(W ) +p +� +(4) +where nnz(W ) is the number of non-zero elements in the +weight matrix W . Since our accelerator exploits the computa- +tion parallelism within each kernel, the total execution time is +the sum of the execution time of all kernels and preprocessing +overhead. +VI. IMPLEMENTATION AND EXPERIMENTAL RESULTS +A. Implementation Details and Resource Utilizations +We implement our accelerator on an embedded FPGA plat- +form – Xilinx ZCU104. We implement 8 pipelines (8 Scatter +Units and 8 Gather Units). Each Scatter/Gather Unit has 16 +processing elements (PEs). In a Scatter Unit, a PE consumes +3 DSPs and in a Gather Unit, a PE consumes 7 DSPs. The +routing network has 8 input ports and 8 output ports. Each +port is 512-bit that can +receive/send 16 32-bit data. The +APU is a quad-core ARM-A53 processor running at 1.3 GHz. +The accelerator is developed using High-Level Synthesis. The +accelerator consumes 1280 DSPs, 96 URAMs, 221 BRAMs, +178K LUTs. The accelerator runs at 125 MHz. The resource +utilization and frequency are reported after Place&Route. +B. Benchmark and Baseline Platform +Benchmark: We conduct experiments using the widely used +MSTAR dataset. The setting of MSTAR dataset follows the +state-of-the-art work [5], [6], [8], [9]. The dataset contains the +SAR images of 10 classes of ground vehicles. The training set +has 2747 images and the testing set has 2427 images. Each +SAR image has size 128×128 and each pixel has a grayscale +value indicating the magnitude of the SAR signal. +TABLE II: Specifications of various platforms +Platforms +CPU +AMD Ryzen 3990x +GPU +Nvidia RTX3090 +FPGA +ZCU 104 +Release Year +2020 +2020 +2018 +Technology +TSMC 7 nm +TSMC 7 nm +TSMC 16 nm +Frequency +2.9 GHz +1.7 GHz +125 MHz +On-chip Memory +256 MB L3 cache +6 MB L2 cache +4.8 MB +Baseline Platform: We compare our performance with the +state-of-the-art CPU and GPU platforms as shown in Table II. +On the CPU platform and GPU platform, we run the proposed +model using Pytorch Geometry (PyG) [45] of 1.8.0 version. +For CPU platform, PyG uses the Intel MKL as the backend +and for the GPU platform, PyG uses the CUDA 11.1 as the +backend. To exploit the sparsity of the weight matrices on the +CPU and GPU platforms, we modify the GraphSAGE layer1 +of PyG by using the torch.sspaddmm() for efficient +multiplication of feature vectors and sparse weight matrices. +1https://pytorch-geometric.readthedocs.io/en/latest/ +modules/torch geometric/nn/conv/sage conv.html#SAGEConv + +C. Accuracy, Computation Complexity, Model Size +Weight/Input pruning: The magnitude of the SAR signal +ranges from 0 to 8. we set the Ivertex as 0.1 because it can filter +out most irrelevant pixels. We compare Accuracy, computation +Type +Accuracy +# of FLOPs +# of Para. +Model Size +[5] +CNN +92.3% +1 +12 × +0.5 × 106 +16 Mb +[8] +CNN +97.97% +1 +10 × +0.65 × 106 +20.8 Mb +[9] +CNN +98.52% +1 +3 × +2.1 × 106 +67.2 Mb +[6] +CNN +99.3% +1× (6.94 GFLOPs) +2.5 × 106 +80 Mb +This work +GNN +99.09% +1 +3258 × +0.03 × 106 +0.96 Mb +complexity, number of parameters with state-of-the-art work +[5], [6], [8], [9]. Compared with the state-of-the-art CNN [6], +the proposed model achieves comparable accuracy with only +1 +3258 computation complexity and +1 +83 number of parameters +on average. +D. Evaluation of Latency +Fig. 5: X-axis is the index of the SAR image (training set + +testing set). Y-axis is the inference latency of a SAR image. +To compare the latency of various platforms, we set the +batch size as 1. The measured latency on FPGA accelerator is +end-to-end from the time when APU receives the SAR image +to the time when APU gets the classification results from +the accelerator, which means the preprocessing overhead is +included in the measured latency. We measure the inference +latency on all images in training and testing sets. The com- +parison results are shown in Figure 5. On average, our FPGA +accelerator is 14.8×, 2.5× faster than the CPU and GPU +platforms in terms of latency. Since we use the input pruning, +the graph representations of the images after input pruning +have various number of vertices. Therefore, the inference +latency fluctuates with images. Compared with CPU/GPU, our +accelerator has lower latency. Because CPU/GPU has complex +cache hierarchy and large cache latency (e.g., CPU has high +cache latency: L3 cache 32ns, L2 cache 12ns). Therefore, +loading feature vectors and weight matrices leads to large +latency. In contrast, our FPGA accelerator can access data in +one-clock cycle due to our customized on-chip memory orga- +nization. Moreover, our FPGA accelerator adopts the Scatter- +Gather paradigm to efficiently deal with irregular computation +in various computation kernels. +Impact of model design: To compare the inference latency +with the state-of-the-art CNNs, we deploy AMD Xilinx DPU +[28] (2 * B4096 @ 300 MHz configuration) on the same +TABLE III: Latency comparison on ZCU 104 and GPU +Model +[5] +[8] +[9] +[6] +Proposed model +[Xilinx DPU] +[Proposed design] +ZCU104 +0.88 ms +1.23 ms +3.09 ms +12.1 ms +0.105 ms +GPU (RTX3090) +1.53 ms +2.5 ms +9.5 ms +31.2 ms +0.269 ms +FPGA platform (ZCU 104) to execute the CNN models in [5], +[6], [8], [9]. AMD Xilinx DPU is the state-of-the-art FPGA +overlay accelerator for CNNs. The average inference latency is +shown in Table III. The proposed GNN on the proposed design +(The column 6 of Table III) is 115× faster than [6] on DPU. +Note that DPU uses 8-bit data quantization for the weights and +activations. Our work uses 32-bit floating point data format. +DPU has more computation parallelism by operating on 8-bit +data. +Preprocessing Overhead: We measure the preprocessing +overhead on APU. For a SAR image, APU transforms it +into graph representation (Section III-A) with input pruning +(Section III-B), and graph partitioning (V-A). The average +preprocessing time is 11.8 us for a SAR image, which is +negligible compared with the total latency. +TABLE IV: Comparison of Energy Consumption +Platform +Inference Speed +Power +Energy (mJ/image) +Ryzen 3990X +644 (image/s) +26.5W +41.1 (mJ/image) +Nvidia RTX3090 +3717 (image/s) +97W +26.0 (mJ/image) +ZCU104 +9500 (image/s) +6.3W +0.66 (mJ/image) +E. Evaluation of Energy Consumption +Table IV shows the comparison of energy consumption +on various platforms. On the CPU platform, we measure +the power consumption of the inference program using +PowerTOP [46]. On the GPU platform, we measure power +consumption using nvidia-smi [47] command tool. For the +FPGA board (ZCU 104), we use an external power meter +to measure its power consumption. The reported numbers in +Table IV are the average power consumption during inference. +The results show that our FPGA accelerator is 62×, 39× more +energy efficient than CPU and GPU platform, respectively. +VII. CONCLUSION +In this paper, we propose a novel model-architecture co- +design for SAR ATR on FPGA. The proposed lightweight +GNN model achieves similar accuracy with state-of-the-art +models with only 1/3258 computation complexity and 1/83 +model size. The proposed accelerator on an embedded FPGA +platform has lower latency than the state-of-the-art CPU/GPU +with significant less energy consumption. +ACKNOWLEDGMENT +This work is supported by the National Science Foundation +(NSF) under grants OAC-1911229, CNS-2009057, and in +part by DEVCOM Army Research Lab (ARL) under ARL- +USC collaborative grant DIRA-ECI:DEC21-CI-037. The au- +thor Bingyi Zhang is supported by the Summer Research +Program from the Army Research Lab West (ARL West). + +Comparison of Latency +FPGA (ZCU104) +CPU (Ryzen 3990X) +GPU (RTX3090) +(second +atency +a +10 +500 +1000 +1500 +2000 +2500 +3000 +3500 +4000 +4500 +5000 +SAR image indexREFERENCES +[1] L. Landuyt, A. Van Wesemael, G. J.-P. Schumann, R. Hostache, N. E. +Verhoest, and F. M. Van Coillie, “Flood mapping based on synthetic +aperture radar: An assessment of established approaches,” IEEE Trans- +actions on Geoscience and Remote Sensing, vol. 57, no. 2, pp. 722–739, +2018. +[2] P. Zhan, W. Zhu, and N. Li, “An automated rice mapping method based +on flooding signals in synthetic aperture radar time series,” Remote +Sensing of Environment, vol. 252, p. 112112, 2021. +[3] N. Li, Z. Guo, J. Zhao, L. Wu, and Z. 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Available: https://github.com/fenrus75/powertop +[47] “nvidia-smi.” +[Online]. +Available: +https://developer.download.nvidia. +com/compute/DCGM/docs/nvidia-smi-367.38.pdf + diff --git a/4NAzT4oBgHgl3EQfffyv/content/tmp_files/load_file.txt b/4NAzT4oBgHgl3EQfffyv/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f7fd56f64fba3939c394e25a83035e22325bf827 --- /dev/null +++ b/4NAzT4oBgHgl3EQfffyv/content/tmp_files/load_file.txt @@ -0,0 +1,766 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf,len=765 +page_content='Accurate, Low-latency, Efficient SAR Automatic Target Recognition on FPGA Bingyi Zhang∗, Rajgopal Kannan†, Viktor Prasanna∗, Carl Busart† ∗University of Southern California †DEVCOM US Army Research Lab ∗{bingyizh, prasanna}@usc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='edu †{rajgopal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='kannan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='civ, carl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='busart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='civ}@army.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='mil Abstract—Synthetic aperture radar (SAR) automatic target recognition (ATR) is the key technique for remote-sensing image recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' The state-of-the-art convolutional neural networks (CNNs) for SAR ATR suffer from high computation cost and large memory footprint, making them unsuitable to be deployed on resource-limited platforms, such as small/micro satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' In this paper, we propose a comprehensive GNN-based model- architecture co-design on FPGA to address the above issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' Model design: we design a novel graph neural network (GNN) for SAR ATR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' The proposed GNN model incorporates GraphSAGE layer operators and attention mechanism, achieving comparable accuracy as the state-of-the-art work with near 1/100 computa- tion cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' Then, we propose a pruning approach including weight pruning and input pruning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' While weight pruning through lasso regression reduces most parameters without accuracy drop, input pruning eliminates most input pixels with negligible accuracy drop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' Architecture design: to fully unleash the computation parallelism within the proposed model, we develop a novel unified hardware architecture that can execute various computation kernels (feature aggregation, feature transformation, graph pool- ing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' The proposed hardware design adopts the Scatter-Gather paradigm to efficiently handle the irregular computation patterns of various computation kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' We deploy the proposed design on an embedded FPGA (AMD Xilinx ZCU104) and evaluate the performance using MSTAR dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' Compared with the state-of-the-art CNNs, the proposed GNN achieves comparable accuracy with 1/3258 computation cost and 1/83 model size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' Compared with the state-of-the-art CPU/GPU, our FPGA accel- erator achieves 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='8×/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='5× speedup (latency) and is 62×/39× more energy efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' Index Terms—SAR ATR, graph neural network (GNN), hard- ware architecture I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' INTRODUCTION Synthetic aperture radar (SAR) can acquire remote-sensing data in all-weather conditions to observe target on the earth ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' SAR has been widely used in real-world applications, such as agriculture [1], [2], civilization [3], [4], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' SAR automatic target recognition (ATR) is the key technique to classify the target in a SAR image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' Convolutional neural networks (CNNs) [5]–[9] have been extensively studied for ATR SAR since CNNs can extract discriminative features from an image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' However, the CNN-based approaches [5]– [9] suffer from two issues: (1) high computation cost: to achieve high accuracy, the authors [5]–[9] develop large CNN models with high computation complexity, (2) large memory requirement: these large CNN models have large number of parameters, which require large memory footprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' Therefore, it is unsuitable to deploy large CNNs on resource-limited platforms, such as small/micro satellites [10]–[14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' The causes of the above issues are (1) heavy convolutional operations in CNNs, and (2) CNNs are hard to exploit data sparsity in SAR images because CNNs need to use the whole image as the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' In a SAR image (Figure 1), only a small set of pixels belongs to the target (defined as pixels of interest, POI), which can be easily extracted through applying a constant threshold [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' However, the extracted POI has irregular structure that is hard to be processed by CNNs, where Graph Neural Network (GNN) provides an opportunity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' Intuitively, we can use the POI to construct a graph and use GNN to perform classification for the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' Fortunately, GNNs have been proven to be powerful models [16] to classify graphs based on graph structural information and vertex features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' Therefore GNNs [17]–[19] have been applied to many graph classification tasks [20]–[24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' Recently, GNNs have been successfully applied to many image classification tasks [25]–[27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' Motivated by that, we design a novel GNN model for SAR ATR (Section III-A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' We propose a graph representation G(V, E) for a SAR image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' The proposed GNN model can extract the structural information of the target from the constructed graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' To improve classification accuracy, we leverage the attention mechanism including spatial attention and channel attention to identify the important vertices and features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' To further reduce the computation complexity, we perform weight pruning by training the GNN model through lasso regression and pruning the GNN model weights that have small absolute values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' Taking advantage of the GNN model, we perform input pruning (POI extraction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' By eliminating the vertices that have small value, the computation complexity is reduced by 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='8% with small accuracy loss (< 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='17%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' The proposed GNN has the following advantages: (1) even without weight/input pruning, the proposed GNN has near 1/100 computation cost as the state-of-the-art CNNs with similar accuracy, (2) while weight pruning can potentially be exploited by CNNs, input pruning is hard to be exploited by CNNs because CNNs need to use the whole image as the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' GNN is flexible to use a small set of input pixels as the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' Therefore, despite that we can accelerate the CNNs [5]–[9] on advanced CNN accelerators [28], their latency is still significant (Section VI-D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' While the proposed GNN is lightweight that can be de- ployed on the resource limited platforms, accelerating GNNs is challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' GNNs have irregular computation pattern and heterogeneous computation kernels [29], making them ineffi- cient to be deployed on the general purpose processors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' The arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='01454v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='AR] 4 Jan 2023 pruned GNN model introduces additional irregularity through weight pruning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' Moreover, the proposed model has various heterogeneous computation kernels (feature aggregation, fea- ture transformation, graph pooling) that need to be mapped on an accelerator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' While there are many GNN accelerators [29]–[35] proposed, none of them exploits the sparsity of the weight matrices or deals with graph pooling, which are still inefficient for the proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' While the proposed GNN achieves high accuracy with small computation complexity, we believe that low-latency execution of SAR ATR must be achieved through careful model-architecture co-design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' Therefore, we develop a novel unified hardware architecture for the proposed GNN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' We demonstrate the methods of mapping various computation kernels onto the proposed accelerator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' In the accelerator design, we adopt Scatter-Gather paradigm to efficient deal with the irregular computation patterns of various kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' To the best of our knowledge, this is the first GNN-based model-architecture co-design for SAR ATR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' Our main contributions are: We propose a lightweight GNN for SAR ATR that achieves comparable accuracy as state-of-the-art GNNs with significant less computation complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' We perform weight pruning and input pruning to dramat- ically reduce the computation complexity and the number of model weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' We design a unified hardware architecture that can exe- cute various computation kernels in the proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' We adopt Scatter-Gather paradigm to deal with the irreg- ular computation patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' Taking advantage of the proposed hardware mapping strategy, we further optimize the load balance of various computation kernels (Section V-A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' We deploy our co-design on Xilinx ZCU104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' We evaluate our co-design using MSTAR dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' Compared with the state-of-the-art CNNs, the proposed GNN achieves comparable accuracy with 1/3258 computation cost and 1/83 model size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' Compared with the state-of-the-art CPU/GPU, our FPGA accelerator achieves 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='8×/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='5× speedup (latency) and is 62×/39× more energy efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' BACKGROUND AND RELATED WORK A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' Related Work Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' 1: The SAR images of various targets (vehicles) SAR ATR is to automatically classify the target in a given SAR images (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' To achieve high accuracy, deep learning based methods have been extensively studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' David [6] demonstrates that CNNs outperform traditional methods, such as Support Vector Machine, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' TAI-SARNET [9] is a TABLE I: Notations Notation Description Notation Description G(V, E, X0) input graph vi ith vertex V set of vertices eij edge from vi to vj E set of edges L number of GNN layers hl i feature vector of vi at layer l N(i) neighbors of vi CNN model that incorporates atrous convolution and inception module to achieve high accuracy for SAR ATR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' The authors [8] combine multi-view features to classify the target in SAR images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' The authors [5] propose the Convolutional Block At- tention Module by exploiting the spatial attention and channel attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' However, the state-of-the-art CNNs [5], [8], [9] suf- fer from high computation cost, making them unsuitable to be deployed on resource-limited platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' Recently, the authors [15] exploit GNN for SAR ATR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' They construct graphs from SAR images by connecting the pixels by the declined order of pixel grayscale value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' However, the constructed graphs lose the structural information of targets, making it extremely sensitive to the variations of input pixel values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' Graph Neural Network The notations are defined in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' Graph Neural Networks (GNN) [17]–[19] are proposed for representation learning on graph G(V, E, X0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' GNNs can learn from the structural infor- mation and vertex/edge features of the graph, and embed these information into low-dimension vector representation/graph embedding (For example, hL i is the embedding of vertex vi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' The vector representation can be used for many downstream tasks, such as node classification [17], [18], link prediction [36], graph classification [37], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' GNNs follow the message- passing paradigm that vertices recursively aggregate informa- tion from the neighbors, for example: GraphSAGE: GraphSAGE is proposed in [18] for inductive representation learning on graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' The GraphSAGE layer follows the aggregate-update paradigm: aggregate:zl i = Mean � hl−1 j : j ∈ N(i) ∪ {i} � update:hl i = ReLU � zl iW l neighbor + bl neighbor||hl−1 i W l self + bl self � (1) III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' MODEL-ARCHITECTURE CO-DESIGN To achieve accurate and efficient SAR ATR on FPGA platform, we perform comprehensive model-architecture co- design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' The proposed co-design consists of a novel GNN model for SAR ATR (Section III-A), a pruning strategy to reduce the computation complexity (Section III-B), a novel hardware design to efficiently execute the proposed GNN (Section III-C), and the strategy to keep load balance within various computation kernels (Section V-A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' The key novelty of our hardware design is that it can execute various computation kernels in the proposed model, and it can efficiently handle the irregular computation patterns caused by the sparsity of weight matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' We use the widely used MSTAR dataset [38] for performance evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' We target various performance BTR70 BRDM2 D7 T62 U 20 20 20 40 40 D 40 60 60 60 08 80 80 100 100 0 100 120 120 0 120 0 2550 100 125 25 50 75 100 0 255075100 125 0 2550 75100 125metrics: (1) Accuracy: the accuracy on MSTAR dataset,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' (2) Computation complexity: the total computation complexity for inferring a SAR image,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' (3) Number of parameters: the total number of parameters in the model,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' (4) Latency: the latency for inferring a SAR image,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' (5) Energy Consumption: the energy consumption for inferring a SAR image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' GNN Model Design Graph representation GNNL Pooling Attention GNNL Pooling Attention … … … GNNL MLP Classification result SAR image Spatial Attention Channel Attention x x + Attention module GNNL-1 Pooling-1 Attention-1 GNNL-2 Pooling-2 Attention-2 GNNL-L Pooling within each 2 × 2 range Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' 2: The proposed GNN model Graph representation: We represent a SAR image as a graph G(V, E), with each pixel viewed as a vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' Each pixel/vertex is connected to its four neighbors (up, down, left, right) with edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' The feature of a vertex is the grayscale value of the pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' Such graph representation maintains structural information of the target that can be learned by GNN for classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' It also provides the opportunity for input pruning (Section III-B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' GNN model: As shown in Figure 2, the proposed GNN model has a sequence of layers, including GNN layer (GNNL), graph pooling layer (Pooling), Attention module (Attention).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' For GNN layer, we use the GraphSAGE layer operators [18], which have been proven to achieve superior accuracy in various application domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' For graph pooling layer, since the input graph has 2-D grid structure, we adopt the similar pooling strategy as the CNN for 2-D image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' Within each local s × s range having s2 vertices, the pooling operator (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=', Max(), Min()) is performed on the s2 vertices to obtain an output vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' Figure 2 demonstrates the pooling operation of size 2×2 with stride 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' Motivated by the attention mechanism in CNN [39], the proposed Attention module consists of a Channel Attention module and a Spatial Attention module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' Suppose the input to Attention Module is {hi : vi ∈ G}, where hi ∈ Rc is the feature vector of vi and c is the length of the feature vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' The Channel Attention calculates the attention score Fch of each feature through a Multi- layer perceptron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' Then, each vertex is multiplied by Fch to obtain {(hi)′ : (hi)′ = hi ⊗ Fch, vi ∈ G} where ⊗ is the element-wise multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' The Spatial Attention module calculates the attention score of each vertex using a GNN layer (GraphSAGE layer operators): {αi : vi ∈ G} = sigmoid(GNNL({hi : vi ∈ G})), Then, each vertex feature vector is multiplied by its attention score: {(hi)′′ : (hi)′′ = αihi, vi ∈ G}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' The output of the Attention module is calculated by: {houtput i : houtput i = hi + (hi)′ + (hi)′′, vi ∈ G} (2) After GNNL-L, all the feature vectors are flattened to a vector which becomes the input to the last MLP (Multi-layer Perceptron) for classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' Network Pruning Weight pruning: To reduce the total computation complexity, we perform weight pruning by training the model using lasso regression [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' We add a L1 penalty term to the loss function: loss = N � i=1 (yi − Model(Gi))2 + λ W � w |w| The penalty term results in weight shrinkage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' Some model weights become zeros and are eliminated from the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' After training, we set a threshold Iweight and the weights with absolute values smaller than Iweight are pruned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' Input pruning: In a SAR image, most pixels outside of the target have negligible grayscale values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' Therefore, in the graph representation G(V, E) of a SAR image, we set a threshold Ivertex and prune the vertices that have grayscale values smaller than Ivertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' The edges connected to the pruned vertices are also pruned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' After input pruning, the eliminated vertices maintain the same positions in the graph pooling layer and do not participate in the pooling operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' For example, in a local 2 × 2 range, if a vertex is pruned, the pooling operator will operate on the remaining three vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' For the input to last MLP, the feature vectors of the pruned vertices are padded using zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' Architecture design The objective of the architecture design is to (1) support various computation kernels in the proposed model, (2) han- dle the irregular computation patterns caused by the feature aggregation in the GNN layer and the sparsity of the weight matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' Figure 3 shows the proposed architecture design on the embedded FPGA platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' The system consists of an Application Processing Unit (APU) and an FPGA accelerator in Programmable Logic Region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' The FPGA accelerator exe- cutes the inference process of the GNN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' In the FPGA accelerator, there is a Weight/Edge Buffer (WEB) to store the model weights and edges of input graph, an Input Buffer (IB) to store the input vertex feature vectors, a Results Buffer (RB) to store the output vertex feature vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' The Matrix Transformation Unit (MTU) performs matrix transformation to prepare the require data layout for the next layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' Thanks to the proposed lightweight model, the trained model is fully APU DDR controller Scatter 1 Scatter 2 ……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' Scatter ������������ Gather 1 Gather 2 ……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' Gather p Routing Network MTU Bank 1 Bank 2 ……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' Bank ������������ Result Buffer Input Buffer Weight /Edge Buffer DMA Programmable Logic Scatter Gather 1 demux mux x demux mux x ….' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='. ……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' MTU Matrix Transformation Unit FPGA APU Application Processing Unit (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=', ARM Cortex-A53) mux demux mux ACC Max mux ReLU Sigmoid demux mux ACC Max mux ReLU Sigmoid Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' 3: The diagram of the system architecture stored in the Weight Buffer, eliminating the memory traffic of loading the model weights at runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' Run Time: At runtime, the APU receives an input SAR image and transform it into the graph presentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' During the trans- formation, the pixels that have grayscale value smaller than Ivertex are pruned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' Then, the APU sends the input graph to the Input Buffer of the accelerator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' The accelerator executes each layer using Scatter-Gather paradigm (SGP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' The accelerator exploits the computation parallelism within each layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' After finishing the execution of all layers, the accelerator sends the classification result back to the APU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' HARDWARE MAPPING A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' Computation kernels We categorize the computation kernels into two classes: Vertex aggregation kernel (VAK): VAKs include (1) feature aggregation (in GNN layer, and in Spatial Attention module) (2) graph pooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' In VAKs, each vertex propagates its feature vector to the neighbors or within a local range (graph pooling).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' Vertex updating kernel (VUK): VUKs include (1) feature update (in GNN layer, and in Spatial Attention module) (2) Channel attention of Attention module, (3) the last MLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' In the VUKs, the feature vector of each vertex is multiplied by a weight matrix to obtain the updated feature vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' Due to our weight pruning, the weight matrices have high data sparsity (1%-33% data density).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' Kernel Mapping using Scatter-Gather Paradigm Algorithm 1 Scatter-Gather paradigm while not done do Scatter Unit: for each edge e⟨src, dst, weight⟩ do Produce update u ←Scatter(src.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='vector, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='weight) end for Gather Unit: for each update u⟨dst, vector⟩ do Update vertex vdst ← Gather(u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='vector) end for end while The accelerator design is based on the Scatter-Gather paradigm (Algorithm 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' There are p parallel pipelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' Each pipeline consists of a Scatter Unit and a Gather Unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' The Routing Network routes the intermediate results to the des- tination based on index dst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' To map the VAKs and VUKs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='������������1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='������������2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='������������3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='������������4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='������������1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='������������2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='������������3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='������������4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='������������1 ������������2 ������������3������������4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='������������������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='������������������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='Input Feature vectors ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='������������1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='������������2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='������������3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='������������4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='Output Feature vectors ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='Vertex aggregation kernel ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='������������1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='������������2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='������������3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='������������4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='1 2 3 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='Adjacency matrix ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='������������������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='������������������������������������������������ = ������������������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='������������������������������������ = 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='������������������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='������������������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='������������1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='������������2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='������������3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='������������4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='������������������������������������������������ = 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='Weight matrix ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='1 2 3 4 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='3 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='Vertex updating ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='kernel ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='������������1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='������������������������������������������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='������������1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='������������������������������������������������������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='Input Feature ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='vectors ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='Output Feature vectors ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' 4: The diagram of mapping the two types of kernels using Scatter-Gather paradigm to the accelerator, we propose the following mapping strategy (An example is shown in Figure 4): Mapping VAK: VAK can be directly mapped to the accelera- tor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' For each edge e⟨src, dst, weight⟩, the Scatter Unit loads the feature vector of vsrc from input buffer and produces an update u⟨dst, vector⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' The update u⟨dst, vector⟩ is routed to the corresponding Gather Unit and the Gather Unit applies the update to the destination vertex vdst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' Mapping VUK: For VUK, we group a batch of vertices batch and the feature vector of each vertex {hinput i : vi ∈ batch} is multiplied by the weight matrix W simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' The output feature vectors are {houtput i : hinput i W , vi ∈ batch}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' To apply the Scatter-Gather paradigm, we perform feature concatenation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' For example, we concatenate the first feature of each vertex {hi(1) : vi ∈ batch} as a vector rinput 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' The vector rinput 1 has src index 1 since its contains the 1st feature of each input feature vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' For the weight matrix W , we represent each non-zero element in the weight matrix as an edge e⟨src, dst, weight⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' During execution, for each non-zero weight e⟨src, dst, weight⟩, the Scatter Unit loads the rinput src from the input buffer and produces an update u⟨dst, vector = e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='weight × rinput src ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' Then, the Gather Unit applies the update u⟨dst, vector⟩ to the destination routput dst .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' routput dst contains the dstth features of each output feature vector in the batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' Note that VAK and VUK have different data layouts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' In VAK, the input/output feature vectors are stored in vertex- major order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' In VUK, the input/output feature vectors are stored in feature-major order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' To switch between the two data layouts, we implement a Matrix Transformation Unit (MTU) to perform data layout transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' hardware modules Scatter/Gather Unit: A Scatter Unit has an array of q processing elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' Each processing element has a multiplier to perform the multiplication between an edge/weight and a vertex feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' Similar to the Scatter Unit, a Gather Unit has an array of q processing elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' Each processing element has an Accumulator (ACC), a Max Unit, a ReLU Unit, a sigmoid Unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' The multiplexer (MUX) and demultiplexer (DEMUX) select the datapath for the current layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' Routing Network: The routing network is implemented using a hardware-efficient butterfly network [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' Sigmoid Unit: We exploit the piecewise linear approximation (PLA) [42] for Sigmoid Function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' LOAD BALANCE AND PERFORMANCE MODEL A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' Load Balance Load balance in VAK: The workload balance of VAK depends on how to partition the vertices into p memory banks of the Result Buffer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' Load imbalance is a significant issue in GNN [43] if the graph has highly imbalanced degree distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' Thanks to our graph representation, the vertices in the graph have degrees ranging from 0 to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' We use a greedy approach to keep the load balance of the p parallel pipelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' For VAK, the destination vertices that have same degree i (0 ⩽ i ⩽ 4) are evenly partitioned into p banks of the Result Buffer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' Through the proposed partitioning strategy, each pipeline has the same amount of workload.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' The graph partitioning has a small overhead O(|V|Lp) and is performed by the APU, where Lp is the number of graph pooling layers in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' The proposed partitioning algorithm can be easily parallelized using multiple threads on APU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' Load balance in VUK: To execute VUK, we need to partition the weight matrix along the dst dimension (Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' Each Gather Unit is responsible for accumulating the partial results of a partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' To achieve perfect load balance, each partition should have the same number of non-zero elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' Since the partitioning of weight matrix is an offline process, we are able to adopt complexity algorithm to find the near optimal data partitioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' In this work, we exploit Longest-processing- time (LPT) first algorithm that is proved to achieve 4/3 approximation factor [44] to the optimal partition solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' Performance Model Modeling VAK: For a VAK kernel, the length of input feature vector cin is same as the length of output feature vector cout: cin = cout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' A Scatter Unit or a Gather Unit can process q features in each clock cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' The p parallel pipelines can process p edges simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' Therefore, the execution time of a VAK kernel is: tVAK = �|E| p � �cin q � (3) Modeling VUK: To execute a VUK, the accelerator groups a batch of q vertices at a time to fully utilize the Scatter Unit/Gather Unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' The p parallel pipelines can process p non- zero elements in the weight matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' Therefore, the execution time of a VUK kernel is: tVUK = �|V| q � �nnz(W ) p � (4) where nnz(W ) is the number of non-zero elements in the weight matrix W .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' Since our accelerator exploits the computa- tion parallelism within each kernel, the total execution time is the sum of the execution time of all kernels and preprocessing overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' IMPLEMENTATION AND EXPERIMENTAL RESULTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' Implementation Details and Resource Utilizations We implement our accelerator on an embedded FPGA plat- form – Xilinx ZCU104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' We implement 8 pipelines (8 Scatter Units and 8 Gather Units).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' Each Scatter/Gather Unit has 16 processing elements (PEs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' In a Scatter Unit, a PE consumes 3 DSPs and in a Gather Unit, a PE consumes 7 DSPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' The routing network has 8 input ports and 8 output ports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' Each port is 512-bit that can receive/send 16 32-bit data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' The APU is a quad-core ARM-A53 processor running at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='3 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' The accelerator is developed using High-Level Synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' The accelerator consumes 1280 DSPs, 96 URAMs, 221 BRAMs, 178K LUTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' The accelerator runs at 125 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' The resource utilization and frequency are reported after Place&Route.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' Benchmark and Baseline Platform Benchmark: We conduct experiments using the widely used MSTAR dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' The setting of MSTAR dataset follows the state-of-the-art work [5], [6], [8], [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' The dataset contains the SAR images of 10 classes of ground vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' The training set has 2747 images and the testing set has 2427 images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' Each SAR image has size 128×128 and each pixel has a grayscale value indicating the magnitude of the SAR signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' TABLE II: Specifications of various platforms Platforms CPU AMD Ryzen 3990x GPU Nvidia RTX3090 FPGA ZCU 104 Release Year 2020 2020 2018 Technology TSMC 7 nm TSMC 7 nm TSMC 16 nm Frequency 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='9 GHz 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='7 GHz 125 MHz On-chip Memory 256 MB L3 cache 6 MB L2 cache 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='8 MB Baseline Platform: We compare our performance with the state-of-the-art CPU and GPU platforms as shown in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' On the CPU platform and GPU platform, we run the proposed model using Pytorch Geometry (PyG) [45] of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='0 version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' For CPU platform, PyG uses the Intel MKL as the backend and for the GPU platform, PyG uses the CUDA 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='1 as the backend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' To exploit the sparsity of the weight matrices on the CPU and GPU platforms, we modify the GraphSAGE layer1 of PyG by using the torch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='sspaddmm() for efficient multiplication of feature vectors and sparse weight matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' 1https://pytorch-geometric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='readthedocs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='io/en/latest/ modules/torch geometric/nn/conv/sage conv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='html#SAGEConv C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' Accuracy, Computation Complexity, Model Size Weight/Input pruning: The magnitude of the SAR signal ranges from 0 to 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' we set the Ivertex as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='1 because it can filter out most irrelevant pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' We compare Accuracy, computation Type Accuracy # of FLOPs # of Para.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' Model Size [5] CNN 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='3% 1 12 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='5 × 106 16 Mb [8] CNN 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='97% 1 10 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='65 × 106 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='8 Mb [9] CNN 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='52% 1 3 × 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='1 × 106 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='2 Mb [6] CNN 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='3% 1× (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='94 GFLOPs) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='5 × 106 80 Mb This work GNN 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='09% 1 3258 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='03 × 106 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='96 Mb complexity, number of parameters with state-of-the-art work [5], [6], [8], [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' Compared with the state-of-the-art CNN [6], the proposed model achieves comparable accuracy with only 1 3258 computation complexity and 1 83 number of parameters on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' Evaluation of Latency Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' 5: X-axis is the index of the SAR image (training set + testing set).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' Y-axis is the inference latency of a SAR image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' To compare the latency of various platforms, we set the batch size as 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' The measured latency on FPGA accelerator is end-to-end from the time when APU receives the SAR image to the time when APU gets the classification results from the accelerator, which means the preprocessing overhead is included in the measured latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' We measure the inference latency on all images in training and testing sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' The com- parison results are shown in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' On average, our FPGA accelerator is 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='8×, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='5× faster than the CPU and GPU platforms in terms of latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' Since we use the input pruning, the graph representations of the images after input pruning have various number of vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' Therefore, the inference latency fluctuates with images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' Compared with CPU/GPU, our accelerator has lower latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' Because CPU/GPU has complex cache hierarchy and large cache latency (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=', CPU has high cache latency: L3 cache 32ns, L2 cache 12ns).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' Therefore, loading feature vectors and weight matrices leads to large latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' In contrast, our FPGA accelerator can access data in one-clock cycle due to our customized on-chip memory orga- nization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' Moreover, our FPGA accelerator adopts the Scatter- Gather paradigm to efficiently deal with irregular computation in various computation kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' Impact of model design: To compare the inference latency with the state-of-the-art CNNs, we deploy AMD Xilinx DPU [28] (2 * B4096 @ 300 MHz configuration) on the same TABLE III: Latency comparison on ZCU 104 and GPU Model [5] [8] [9] [6] Proposed model [Xilinx DPU] [Proposed design] ZCU104 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='88 ms 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='23 ms 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='09 ms 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='1 ms 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='105 ms GPU (RTX3090) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='53 ms 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='5 ms 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='5 ms 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='2 ms 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='269 ms FPGA platform (ZCU 104) to execute the CNN models in [5], [6], [8], [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' AMD Xilinx DPU is the state-of-the-art FPGA overlay accelerator for CNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' The average inference latency is shown in Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' The proposed GNN on the proposed design (The column 6 of Table III) is 115× faster than [6] on DPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' Note that DPU uses 8-bit data quantization for the weights and activations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' Our work uses 32-bit floating point data format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' DPU has more computation parallelism by operating on 8-bit data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' Preprocessing Overhead: We measure the preprocessing overhead on APU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' For a SAR image, APU transforms it into graph representation (Section III-A) with input pruning (Section III-B), and graph partitioning (V-A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' The average preprocessing time is 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='8 us for a SAR image, which is negligible compared with the total latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' TABLE IV: Comparison of Energy Consumption Platform Inference Speed Power Energy (mJ/image) Ryzen 3990X 644 (image/s) 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='5W 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='1 (mJ/image) Nvidia RTX3090 3717 (image/s) 97W 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='0 (mJ/image) ZCU104 9500 (image/s) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='3W 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content='66 (mJ/image) E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' Evaluation of Energy Consumption Table IV shows the comparison of energy consumption on various platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' On the CPU platform, we measure the power consumption of the inference program using PowerTOP [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' On the GPU platform, we measure power consumption using nvidia-smi [47] command tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' For the FPGA board (ZCU 104), we use an external power meter to measure its power consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' The reported numbers in Table IV are the average power consumption during inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' The results show that our FPGA accelerator is 62×, 39× more energy efficient than CPU and GPU platform, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' CONCLUSION In this paper, we propose a novel model-architecture co- design for SAR ATR on FPGA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' The proposed lightweight GNN model achieves similar accuracy with state-of-the-art models with only 1/3258 computation complexity and 1/83 model size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' The proposed accelerator on an embedded FPGA platform has lower latency than the state-of-the-art CPU/GPU with significant less energy consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' ACKNOWLEDGMENT This work is supported by the National Science Foundation (NSF) under grants OAC-1911229, CNS-2009057, and in part by DEVCOM Army Research Lab (ARL) under ARL- USC collaborative grant DIRA-ECI:DEC21-CI-037.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' The au- thor Bingyi Zhang is supported by the Summer Research Program from the Army Research Lab West (ARL West).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfffyv/content/2301.01454v1.pdf'} +page_content=' 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+Abstract +In this paper we introduce some simple models, based on rolling dice, to explore mechanisms +proposed to explain planetary habitability. The idea is to study these selection mechanisms +in an analytically tractable setting, isolating their consequences from other details which can +confound or obscure their effect in more realistic models. We find that while the observable +of interest, the face value shown on the die, ‘improves’ over time in all models, for two of the +more popular ideas: Selection by Survival and Sequential Selection, this is down to sampling +effects. A modified version of Sequential Selection, Sequential Selection with Memory, implies +a statistical tendency for systems to improve over time. We discuss the implications of this and +its relationship to the ideas of the ‘inhabitance paradox’ and the ‘Gaian bottleneck’. +1 +Introduction +Relatively recent discussion about the persistence of life over long periods has brought to the fore +various selection principles [1, 2, 3, 4]. With the recent launch of the James Webb Space Telescope +[5, 6] these questions not only have important implications for our understanding of Earth history, +but also for the search for other inhabited planets. In the future, with a large enough catalogue of +inhabited planets, it may be possible to experimentally investigate alternative trajectories for life +[7]. Until then, understanding general principles behind planetary habitability and inhabitance is a +way to provide working hypotheses that can explain both how ‘lucky’ the Earth is to be inhabited +and what we might expect on planets orbiting other stars. +The most discussed of these selection principles is called ‘Selection by Survival’ (SBS) in [3] +though the idea has many names, see [8]. The essential idea is that a population where the entities +have different rates of survival will be ‘purified’ so that, in the long run, surviving entities have +must have properties conducive to survival. Several works e.g. [1, 2, 9], attempt to disentangle +Darwinian selection from this ‘differential persistence’ (essentially a synonym for SBS). [2] and [9] +emphasise the importance of this selection principle acting on higher order phenomena like whole +ecosystems, planetary scale biogeochemical cycles and the entire life-Earth coupled system i.e. Gaia +[10]. They argue that in systems of hereditary replicators Darwinian selection is more powerful, +however for entities like populations, ecosystems or bio-geochemical cycles [11], which do not have +∗E-mail: R.Arthur@exeter.ac.uk +†E-mail: A.E.Nicholson@exeter.ac.uk +1 +arXiv:2301.02623v1 [q-bio.PE] 6 Jan 2023 + +strict heredity and reproduction, SBS will operate to favour certain macroscopic features. Specific +examples suggested by e.g. [8] are sexual reproduction and macroevolutionary freezing. +[3] defines another, related, selection principle called Sequential Selection (SS) [12]. This is a +similar idea to SBS, but motivated by the frequent upheavals in the history of life on Earth and +meant to account for life’s apparent stabilising effect on Earth’s habitability. [3] propose a simple +algorithm - evolutionary innovations have a stabilizing or destabilizing effect on the environment. +If they have a destabilizing effect, habitability is reduced, eventually eliminating the destabilizing +innovation. In this way destabilizing effects are eliminated by ‘near fatal resets’ while stabilizing +innovations persist and accumulate. +In [13, 4] we argue for a refinement of this algorithm, emphasising that the resets are‘near fatal +i.e. the evolutionary innovations developed during the previous stable period are not completely lost. +The algorithm of [3] applies: destabilizing innovations lead to resets which greatly reduce species +abundance but have a lesser effect on species diversity. The life-earth system which arises after +the reset is selected from a larger ‘pool’, which has the potential to generate better, more stable +ecosystems. Higher species and functional diversity give Gaia more tools to generate stability. The +process is completely blind, so unstable states can also be selected, however, by definition, these +are short lived and eventually a long-lived stable state will arise. During this stable period species +diversity can increase again leading to a kind of ratcheting effect. To emphasise this cumulative +process, in contrast to the sequential selection algorithm of [3], we call this ‘Sequential Selection +with Memory’ (SSM). +A variety of abstract models of varying complexity have been proposed to explore these selection +principles e.g. +[1, 14, 15, 4, 16]. +Though these models have great value, it can sometimes be +unclear which of their features are programmed in (as alleged by [11] of the famous Daisyworld +[17] model) and which are emergent. It must also be said that the mathematical or computational +complexity of these models can give them an air of mystery - especially to biologists not well versed +in these methods. Indeed, the very fact that the key model features are often emergent means that +understanding how they emerge requires a detailed understanding of each model’s dynamics. +Here we propose an extremely simple probability model as a setting to study selection princi- +ples. The aim is to strip out as much complexity as possible to understand the core meaning of +these principles and their consequences. A very loose analogy would be trying to understand the +approximately Gaussian distribution of, say, human height. This has some genetic and environ- +mental causes which, with great difficultly, could be experimentally isolated and formulated into a +mechanistic model of height, simulated and shown to result in a Gaussian distribution. However, a +much simpler, and in many ways more satisfactory, explanation is that a Gaussian distribution is +the expected outcome for an observable which is a sum of independent effects. +Continuing the height analogy, by simulating sums of random variables and showing this results +in a Gaussian distribution we might start to suspect that a more general principle is operating, +one which isn’t affected by the particular details of our model, in this case, the Central Limit +Theorem. This paper doesn’t propose anything as general as a statistical convergence theorem, +what we do propose are models simple enough to be analytically solved but complex enough to +see selection principles operating. Theses models will be shown to exhibit interesting behaviour +which is also observed in more complex models. The aim is to provide some clarity on exactly what +non-Darwinian selection principles can do in a clear and tractable setting. +2 + +2 +Introducing the Model +Consider an M sided die with the rule that, once rolled, whatever number is showing on the top face +gives the number of steps to wait before rolling again or finishing the game. For r ≥ 1 dice we roll +each one independently to get x1, x2, . . . , xr, and take the highest face value: max(x1, x2, . . . , xr). +Based on this consider the following dice games: +1. Selection By Survival(SBS): Roll N (where N is a very large number) independent dice +once each. +2. Sequential Selection (SS): Roll one die repeatedly for T time steps. +3. Sequential Selection with Memory (SSM): +(a) Starting with r = 1, roll r dice repeatedly for T time steps. Add a new die every time +the top face shows the maximum value, M. +(b) Starting with M = 1, roll an M sided die repeatedly for T times steps. Every time the +top face shows the maximum value M, increase M by 1. +The SSM games are reminiscent of the Polya Urn model, though have not been studied before to our +knowledge. The quantity of interest will be the expected face value at time t. The names chosen are +based on the discussion in the Introduction and follow the conventions of [3] and [4]. Our version of +Selection By Survival is much simpler than the (mostly verbal) models proposed by others e.g. [9] +and most closely follows the graphical model from [2]. +As a rough mapping to reality - a die represents an inhabited ‘planet’. Each roll is a period +of stability for the planet’s biosphere. The face value represents something akin to the ‘fitness’ of +the biosphere on that planet, i.e. how long it will persist. If we observe an inhabited planet at +some random point in its history we may see a biosphere with properties conducive to long term +stability (high face value) or only short term stability (low face value). The question of interest for +astrobiology is, if we were to survey a large catalogue of inhabited planets, what would be the average +‘fitness’? For Earth history (or for the history of any inhabited planet) the equivalent question is, +if we were to observe a planet at a random point in its history, what should we expect about the +habitability properties of that planet? +3 + +3 +Selection by Survival +t = 1 +t = 2 +t = 3 +t = 4 +t = 5 +t = 6 +Figure 1: One possible unfolding of the SBS game with N = 25 and M = 6. At t = 1 we have our +initial ensemble, at t = 2 we have removed all the 1s, at t = 3 we remove all the 2s etc. +Figure 1 shows one realisation of the SBS game. At t = 1 all of the dice are in play and the average +face value (over very large N or many different realisations of the same game) is +(1 + 2 + . . . + M)/M +at t = 2 all of the dice showing 1 on the top face are removed. Restricting our survey to inhabited +planets, the average face value is now +(2 + 3 + . . . + M)/(M − 1) +At time t ≤ M the average face value is +(t + (t + 1) + . . . + M) +(M − t) += M + t +2 +(1) +So that average face value increases linearly with time. +4 + +: +:围 +880 +围Figure 2: Average face value in the SBS game as a function of t for an M = 10 sided dice over +N = 1000 dice rolls. +Figure 2 shows the result of simulations of the game compared to equation 1. In terms of ‘planets’ +this model is simply stating the (obvious) fact that planets which survive have properties (high face +value) which allow them to survive! Looking at the catalogue of inhabited planets will necessarily +yield planets with properties conducive to maintaining life, without the need for any additional +mechanism. +This realisation has all planets are seeded with life at the same time. +More complex games +could be devised (say a constant rate of habitable planet generation) to study how the generateion +rate interacts with this simple selection mechanism. For this paper, SBS represents a basic null +model - older inhabited planets must have features which have enabled them to remain inhabited. +The growth in fitness of the ‘surviving’ planets is simply a sampling artefact, the average fitness +of an inhabited planet increases because we throw away more and more of the unfit planets from +our average. Considering our solar system according to SBS, the single inhabited planet we see is +habitable because if it wasn’t, we wouldn’t be looking at it, or living on it. Thus in this context, +SBS is nothing more than an observer effect or anthropic principle. +5 + +10 +Exact +Average of 1000 runs +8 +Face Value +6 +4 +2 +0 +2 +6 +8 +10 +4 +t4 +Sequential Selection +The Earth has experienced numerous mass extinction events, had very different planetary regulation +mechanisms, atmospheric composition, levels of volcanic activity and life has persisted the entire +time [18]. We seek to model these sequential resets with another simple game: repeatedly rolling a +single die. +Figure 3: One possible unfolding of the SS game with T = 20 and M = 6. At t = 1 we roll 2 which +shows for 2 steps, we roll 1 which shows for 1 step, then 3 for 3 steps etc. The game is played a +large number, N, of times as in figure 1 so we are interested in average behaviour. +When observing the die at a random time t, what should we expect the face of the die to show, +on average? The chance of the die showing k is proportional to the probability of rolling a k, p(k), +times the number of ‘slots’ where the observation could occur e.g. if the die is showing 3 this could +be an observation of the die on the first, second or third step where it is face up. We normalise this +probability and compute the expected value for the top face as +M +� +k=1 +k +kp(k) +�m +k=1 kp(k) +(2) +For one die p(k) = 1/M and this simplifies to +2M + 1 +3 +(3) +Note this is larger than the expected value of a single dice roll, M+1 +2 +for M > 1. +6 + +·Figure 4: Average face value in the SS game as a function of t for an M = 10 sided dice, averaged +over N = 1000 independent instances. The expected value of a single 10 sided dice roll is 5.5 which +is less than the expected face value in the sequential sampling game, 7. Note the logged x-axis. +Figure 4 shows the results of simulations of the game compared to equation 3. The results here +imply that when observing a ‘planet’ at a random time, it is more likely to be in a state with stability +enhancing properties (high face value). Again this reflects the obvious fact that if we depict Earth’s +history as a time line and pick a random point on the line we are more likely to pick a point in a +long stable period. In particular, our present time is most likely to be a stable period, without the +need for any additional mechanism. Like with SBS, Earth’s current stability is simply an observer +effect. +One thing missing from this game (and from the algorithm of [3]) is the possibility of total extinc- +tion, that is, finishing the game early. We could implement an additional rule, say when we roll a 1, +stop the game. This would give a model where SBS and SS are both operating simultaneously. Here, +for simplicity and clarity, we don’t account for total extinction, so as not to mix the mechanisms. +All of our SS games persist for the same amount of time. More complex models e.g. [4, 19] do have +the possibility of early stopping and come to very similar conclusions. +7 + +7.4 +Average of 1000 runs +Exact +7.2 +7.0 +6.8 +Face Value +6.6 +6.4 +6.2 +6.0 +100 +101 +102 +103 +t5 +Sequential Selection with Memory +The continued inhabitance of Earth and the fact that biodiversity has increased over time motivates +the final games. Each reset does not start from scratch, but builds on evolutionary and ecological +innovations that came before. +We propose 2 models with an extremely simple ‘memory’. +This +memory is implemented in two ways, first by adding extra dice at fixed M, second by increasing M. +5.1 +Game A: Adding dice +The face value in this game is determined by rolling multiple dice and choosing the one with the +maximum face value. The idea is that stable biospheres outlast unstable ones. One could imagine +independent ecosystems co-existing with the final ‘reset’ only occurring when the most stable sub- +system collapses. If this seems contrived, in more complex models e.g. [4], a similar feature emerges +as a consequence of model dynamics rather than being enforced. +Figure 5: One possible unfolding of SSM game A with T = 20 and M = 6. At t = 3 we roll 6 +which adds an extra die. At t = 13 we roll six again which adds a third die. The bottom row is the +observable, the other rows show dice rolls which are not observed. +The analysis is a little more complex that the previous two games. The first thing we need is the +probability to get the face value f when rolling r dice and applying the rule f = max(x1, x2, . . . , xr). +This is +pr(f) = +r +� +i=1 +�r +i +� +p(f)ip(x < f)r−i +(4) +where p(f) = 1/M and p(x < f) = f−1 +M . This is just the probability to get at least one f and +nothing higher, multiplied by a combinatoric factor. To simplify this, consider arranging all the +possible outcomes of r rolls in an r-dimensional hypercube. The number of ways to obtain f is given +by the difference in volumes between an f and f − 1 sided hypercube so +pr(f) = f r − (f − 1)r +M r +(5) +It is shown in Appendix A that the expected face value for large M is +E[f|r, M ≫ 1] = M +r +r + 1 + 1 +2. +(6) +In a game with r dice, the expected number of dice rolls before hitting the value M, where we +add an extra die, is 1/pr(M). Therefore, the expected time spent playing with exactly r dice is +T(r) = +�M +i=1 ipr(i) +pr(M) +. +(7) +8 + +JFor large M (using the summation result from Appendix A) this is +T(r) ≃ M 2 +r + 1 +(8) +To calculate the expected face value at t, we first compute the expected number of dice at time t by +solving +r +� +k +T(k) = t +(9) +for r. Using the large M approximation +r +� +k +T(k) ≃ M 2 +r +� +k +1 +i + 1 = M 2(Hr − 1) +(10) +where Hr is the rth harmonic number. This has a standard approximation, valid for large r but quite +accurate even at r = 1: Hr ≃ γ +ln(r), where γ is the Euler-Mascheroni constant. Substituting and +solving for r gives +r(t) = exp +� t +M 2 + 1 − γ +� += A exp +� t +M 2 +� +(11) +where A = exp(1 − γ). The number of dice grows exponentially, with growth rate 1/M 2. The more +faces a die has, the longer we have to wait to land on a specific one, for example the time to go from +r dice to 2r dice is ≃ M 2 ln 2. +The expected face value at time t is the expected face value with r(t) dice. Still working in the +large M limit this is +E[f : M ≫ 1](t) ≃ M +r +r + 1 = M +A exp +� +t +M 2 +� +1 + A exp +� +t +M 2 +� +(12) +This is a sigmoid function in the variable t/M 2. At large values of t the value is M, as expected, +we have so many dice we are virtually guaranteed to roll at least one M. The small t behaviour +is interesting, the function is roughly linear which means, despite the exponential growth in the +number of dice suggested by equation 11, the expected face value grows much more slowly. +9 + +Figure 6: Average face value in the SSM game as a function of t for an M = 10 sided dice, averaged +over N = 1000 independent instances. +Figure 6 shows the results of 1000 simulations of the game with M = 10 compared to the ‘exact’ +answer, equation 12. We observe convergence to the upper bound M at a rate that is roughly linear +in log time. Such slow convergence is seen in more complex evolutionary models, especially the +Tangled Nature Model [20, 21] and its variants [13, 4]. There, it arises from the simulated ecosystem +successively crossing ‘entropic barriers’ [22]. Each time a barrier is crossed the system is likely to be +in a more stable configuration with a higher barrier. This behaviour has been discussed before in +the language of record statistics and is also observed in physical systems like spin glasses, colloids +and high temperature superconductors [23]. +This model is simple enough for an approximate analytic solution. This shows that there is +competition between the growth in the number of dice over time against the growth in the time taken +between trials. There is also a trade-off between large values of M, leading to higher average rolls, +versus time taken to add a new die. What this model suggests is that selection plus accumulation +leads to slow growth in stability. This model implies that older inhabited planets should be more +habitable, so our presence on Earth is not just an observer effect but a statistically more likely +outcome. +10 + +10.0 +Average of 1000 runs +Exact +9.5 +9.0 +8.5 +Value +8.0 +Face +7.5 +7.0 +6.5 +6.0 +100 +101 +102 +103 +t5.2 +Game B: Increasing M +This game similar to the previous one, except instead of adding extra dice we have just one die +and add a extra faces to it, which makes this harder to play with real dice! A rough analogy to +a real ecosystem is to assume that species diversity is not lost after each collapse (dice roll) and +that species persist at low abundance, in dormant states or isolated refugia. Reaching a fitness peak +(hitting the max value of M) generates more latent diversity and allows ecosystems to explore more +of the so-called fitness landscape [24]. Thus each reset has the potential to find a more stable state +because the space of possibilities is wider. +Figure 7: One possible unfolding of SSM game B with T = 20. The bottom row shows the actual +face values and the top row shows the number of sides of the die. For example at t = 11 we roll a 4 +on a 4-sided die, increasing the number of sides to 5 for the next roll. +If the die has M sides, the expected number of rolls required to hit the M face is just 1/M. +Each roll is expected to last M+1 +2 +steps so the expected waiting time before increasing the number +of faces is +T(M) = M M + 1 +2 +(13) +Summing up the wait times from each M gives the total duration of the experiment +t = +M +� +i=1 +i(i + 1) +2 += 1 +2 +�M(M + 1)(2M + 1) +6 ++ M(M + 1) +2 +� +(14) +Keeping only the terms of highest order in M and solving for t gives +M = +3√ +6t +(15) +Substituting into equation 3 gives +E[f](t) = 2 +3√ +6t + 1 +3 +(16) +11 + +M=1 +2 2 +33 +4 +4 +5 +5 +....r.Figure 8: Average face value in SSM game B as a function of t, averaged over N = 1000 independent +instances. +Figure 8 shows the results of simulations of the game compared to the exact answer, equation 16. +Unlike game A there is no convergence and the expected face value grows without bound, though +fairly slowly. Again there is a trade off between increasing M by performing a large number of trials +and the time it takes to complete those trials. This is again reminiscent of Tangled Nature Model +dynamics [22, 13, 4] and other physical systems which cross energetic or entropic barriers [23]. +6 +Discussion +These three mechanisms give three reasonable ideas about what to expect when surveying large +catalogues of inhabited planets, or looking at an inhabited one at a random point in its history. +The first two (SBS, SS) have no role for life. +The stability properties of inhabited planets are +down to observer effects - unless they had these properties we wouldn’t be looking at them. The +third mechanism is more interesting. Once a planet is inhabited life can have a positive effect on +habitability. In particular - inhabited planets have properties conducive to stability because of their +history of inhabitance. +This idea has appeared previously as ‘The inhabitance paradox’ in [25] and is closely related +12 + +16 +Average of 1000 runs +Exact +14 +12 +Face Value +10 +8 +6 +4 +2 +100 +101 +102 +103 +tto the idea of the Gaian bottleneck [26]. +This paradox says that for a planet to be habitable, +it must be inhabited. This means life must seize the reins and exert a stabilising effect early in a +planet’s history or go extinct due to deteriorating geophysical conditions - an effect dubbed the Gaian +bottleneck. The SSM game shows a very simple mechanism by which this could occur, combining +the sequential selection algorithm of [3] with some method of making cumulative improvements will +tend to generate more stable systems. We have argued previously [4] that such cumulative processes +are widespread on Earth, for example: microbial seed banks, dormancy [27] and lateral gene transfer +[28] all contribute to the maintenance of microbial diversity and therefore the stabilising effect of +functional redundancy. +We hope that this model and its analysis provides some clarity on selection principles as well as +providing a sandbox for studying selection effects. In particular, we believe that Sequential Selection +with Memory provides a plausible way for a complex system, like an inhabited planet, to become +more stable over time. We propose that Gaia - the stabilising and symbiotic feedback of life and the +environment - can be born through this kind of natural, but non-Darwinian, selection. +A +Expected value for the max of r, M-sided dice. +Rolling r, M sided dice gives the face value f with probability +p(f) = kr − (k − 1)r +M r +as discussed in the text. The expectation for the face value is therefore +E[f] = +M +� +k=1 +k kr − (k − 1)r +M r +Writing out the sum explicitly +1.1r + 2.2r + 3.3r + . . . + M.M r +−(1.0r + 2.1r + 3.2r + . . . + M.(M − 1)r) +Shows that we can regroup and rewrite as +E[f] = +1 +M r +� +M r+1 − +M−1 +� +k=1 +kr +� +The sum can be simplified using Faulhaber’s formula [29] +M−1 +� +k=1 +kr = (M − 1)r+1 +r + 1 ++ (M − 1)r +2 ++ O(M r−1) +where the lower order terms are fairly complex coefficients involving the Bernoulli numbers. Substi- +tuting and taking the limit of large M we get +E[f] = M +r +r + 1 + 1 +2 +as stated in the text. +13 + +References +[1] Pierrick Bourrat. From survivors to replicators: evolution by natural selection revisited. Biology +& Philosophy, 29(4):517–538, 2014. +[2] W Ford Doolittle. Natural selection through survival alone, and the possibility of gaia. 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Met Office, 2008. +14 + +[13] Rudy Arthur and Arwen Nicholson. An entropic model of gaia. Journal of theoretical biology, +430:177–184, 2017. +[14] Arwen E Nicholson, David M Wilkinson, Hywel TP Williams, and Timothy M Lenton. Alter- +native mechanisms for gaia. Journal of theoretical biology, 457:249–257, 2018. +[15] Timothy M Lenton, Timothy A Kohler, Pablo A Marquet, Richard A Boyle, Michel Crucifix, +David M Wilkinson, and Marten Scheffer. +Survival of the systems. +Trends in Ecology & +Evolution, 36(4):333–344, 2021. +[16] Richard A Boyle and Timothy M Lenton. +The evolution of biogeochemical recycling by +persistence-based selection. Communications Earth & Environment, 3(1):1–14, 2022. +[17] Andrew J Watson and James E Lovelock. Biological homeostasis of the global environment: +the parable of daisyworld. Tellus B: Chemical and Physical Meteorology, 35(4):284–289, 1983. +[18] Tim Lenton and Andrew Watson. Revolutions that made the Earth. OUP Oxford, 2013. +[19] Rudy Arthur and Arwen Nicholson. A gaian habitable zone, 2023. +[20] Kim Christensen, Simone A Di Collobiano, Matt Hall, and Henrik J Jensen. Tangled nature: +a model of evolutionary ecology. Journal of theoretical Biology, 216(1):73–84, 2002. +[21] Rudy Arthur, Arwen Nicholson, Paolo Sibani, and Michael Christensen. +The tangled na- +ture model for organizational ecology. Computational and Mathematical Organization Theory, +23(1):1–31, 2017. +[22] Nikolaj Becker and Paolo Sibani. Evolution and non-equilibrium physics: A study of the tangled +nature model. EPL (Europhysics Letters), 105(1):18005, 2014. +[23] Paolo Sibani, Stefan Boettcher, and Henrik Jeldtoft Jensen. +Record dynamics of evolving +metastable systems: theory and applications. The European Physical Journal B, 94(1):1–23, +2021. +[24] Rudy Arthur and Paolo Sibani. Decision making on fitness landscapes. Physica A: Statistical +Mechanics and its Applications, 471:696–704, 2017. +[25] Colin Goldblatt. The inhabitance paradox: How habitability and inhabitancy are inseparable. +arXiv preprint arXiv:1603.00950, 2016. +[26] Aditya Chopra and Charles H Lineweaver. The case for a gaian bottleneck: the biology of +habitability. Astrobiology, 16(1):7–22, 2016. +[27] Jay T Lennon and Stuart E Jones. +Microbial seed banks: the ecological and evolutionary +implications of dormancy. Nature reviews microbiology, 9(2):119–130, 2011. +[28] Nigel Goldenfeld and Carl Woese. Biology’s next revolution. Nature, 445(7126):369–369, 2007. +[29] Eric W. Weisstein. Faulhaber’s formula. From MathWorld—A Wolfram Web Resource. Last +visited on 1/1/2023. +15 + diff --git a/6NE0T4oBgHgl3EQfvwGn/content/tmp_files/load_file.txt b/6NE0T4oBgHgl3EQfvwGn/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..628abf7d10a549c6cbfd601327f3d50c31366fac --- /dev/null +++ b/6NE0T4oBgHgl3EQfvwGn/content/tmp_files/load_file.txt @@ -0,0 +1,426 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf,len=425 +page_content='Does Gaia Play Dice?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' : Simple Models of non-Darwinian Selection Rudy Arthur,1∗Arwen Nicholson,2† 1University of Exeter, Department of Computer Science 2University of Exeter, Department of Physics and Astronomy January 9, 2023 Abstract In this paper we introduce some simple models, based on rolling dice, to explore mechanisms proposed to explain planetary habitability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' The idea is to study these selection mechanisms in an analytically tractable setting, isolating their consequences from other details which can confound or obscure their effect in more realistic models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' We find that while the observable of interest, the face value shown on the die, ‘improves’ over time in all models, for two of the more popular ideas: Selection by Survival and Sequential Selection, this is down to sampling effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' A modified version of Sequential Selection, Sequential Selection with Memory, implies a statistical tendency for systems to improve over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' We discuss the implications of this and its relationship to the ideas of the ‘inhabitance paradox’ and the ‘Gaian bottleneck’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' 1 Introduction Relatively recent discussion about the persistence of life over long periods has brought to the fore various selection principles [1, 2, 3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' With the recent launch of the James Webb Space Telescope [5, 6] these questions not only have important implications for our understanding of Earth history, but also for the search for other inhabited planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' In the future, with a large enough catalogue of inhabited planets, it may be possible to experimentally investigate alternative trajectories for life [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Until then, understanding general principles behind planetary habitability and inhabitance is a way to provide working hypotheses that can explain both how ‘lucky’ the Earth is to be inhabited and what we might expect on planets orbiting other stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' The most discussed of these selection principles is called ‘Selection by Survival’ (SBS) in [3] though the idea has many names, see [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' The essential idea is that a population where the entities have different rates of survival will be ‘purified’ so that, in the long run, surviving entities have must have properties conducive to survival.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Several works e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' [1, 2, 9], attempt to disentangle Darwinian selection from this ‘differential persistence’ (essentially a synonym for SBS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' [2] and [9] emphasise the importance of this selection principle acting on higher order phenomena like whole ecosystems, planetary scale biogeochemical cycles and the entire life-Earth coupled system i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Gaia [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' They argue that in systems of hereditary replicators Darwinian selection is more powerful, however for entities like populations, ecosystems or bio-geochemical cycles [11], which do not have ∗E-mail: R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content='Arthur@exeter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content='uk †E-mail: A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content='Nicholson@exeter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content='uk 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content='02623v1 [q-bio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content='PE] 6 Jan 2023 strict heredity and reproduction, SBS will operate to favour certain macroscopic features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Specific examples suggested by e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' [8] are sexual reproduction and macroevolutionary freezing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' [3] defines another, related, selection principle called Sequential Selection (SS) [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' This is a similar idea to SBS, but motivated by the frequent upheavals in the history of life on Earth and meant to account for life’s apparent stabilising effect on Earth’s habitability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' [3] propose a simple algorithm - evolutionary innovations have a stabilizing or destabilizing effect on the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' If they have a destabilizing effect, habitability is reduced, eventually eliminating the destabilizing innovation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' In this way destabilizing effects are eliminated by ‘near fatal resets’ while stabilizing innovations persist and accumulate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' In [13, 4] we argue for a refinement of this algorithm, emphasising that the resets are‘near fatal i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' the evolutionary innovations developed during the previous stable period are not completely lost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' The algorithm of [3] applies: destabilizing innovations lead to resets which greatly reduce species abundance but have a lesser effect on species diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' The life-earth system which arises after the reset is selected from a larger ‘pool’, which has the potential to generate better, more stable ecosystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Higher species and functional diversity give Gaia more tools to generate stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' The process is completely blind, so unstable states can also be selected, however, by definition, these are short lived and eventually a long-lived stable state will arise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' During this stable period species diversity can increase again leading to a kind of ratcheting effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' To emphasise this cumulative process, in contrast to the sequential selection algorithm of [3], we call this ‘Sequential Selection with Memory’ (SSM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' A variety of abstract models of varying complexity have been proposed to explore these selection principles e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' [1, 14, 15, 4, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Though these models have great value, it can sometimes be unclear which of their features are programmed in (as alleged by [11] of the famous Daisyworld [17] model) and which are emergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' It must also be said that the mathematical or computational complexity of these models can give them an air of mystery - especially to biologists not well versed in these methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Indeed, the very fact that the key model features are often emergent means that understanding how they emerge requires a detailed understanding of each model’s dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Here we propose an extremely simple probability model as a setting to study selection princi- ples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' The aim is to strip out as much complexity as possible to understand the core meaning of these principles and their consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' A very loose analogy would be trying to understand the approximately Gaussian distribution of, say, human height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' This has some genetic and environ- mental causes which, with great difficultly, could be experimentally isolated and formulated into a mechanistic model of height, simulated and shown to result in a Gaussian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' However, a much simpler, and in many ways more satisfactory, explanation is that a Gaussian distribution is the expected outcome for an observable which is a sum of independent effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Continuing the height analogy, by simulating sums of random variables and showing this results in a Gaussian distribution we might start to suspect that a more general principle is operating, one which isn’t affected by the particular details of our model, in this case, the Central Limit Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' This paper doesn’t propose anything as general as a statistical convergence theorem, what we do propose are models simple enough to be analytically solved but complex enough to see selection principles operating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Theses models will be shown to exhibit interesting behaviour which is also observed in more complex models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' The aim is to provide some clarity on exactly what non-Darwinian selection principles can do in a clear and tractable setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' 2 2 Introducing the Model Consider an M sided die with the rule that, once rolled, whatever number is showing on the top face gives the number of steps to wait before rolling again or finishing the game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' For r ≥ 1 dice we roll each one independently to get x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' , xr, and take the highest face value: max(x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' , xr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Based on this consider the following dice games: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Selection By Survival(SBS): Roll N (where N is a very large number) independent dice once each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Sequential Selection (SS): Roll one die repeatedly for T time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Sequential Selection with Memory (SSM): (a) Starting with r = 1, roll r dice repeatedly for T time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Add a new die every time the top face shows the maximum value, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' (b) Starting with M = 1, roll an M sided die repeatedly for T times steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Every time the top face shows the maximum value M, increase M by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' The SSM games are reminiscent of the Polya Urn model, though have not been studied before to our knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' The quantity of interest will be the expected face value at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' The names chosen are based on the discussion in the Introduction and follow the conventions of [3] and [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Our version of Selection By Survival is much simpler than the (mostly verbal) models proposed by others e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' [9] and most closely follows the graphical model from [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' As a rough mapping to reality - a die represents an inhabited ‘planet’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Each roll is a period of stability for the planet’s biosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' The face value represents something akin to the ‘fitness’ of the biosphere on that planet, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' how long it will persist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' If we observe an inhabited planet at some random point in its history we may see a biosphere with properties conducive to long term stability (high face value) or only short term stability (low face value).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' The question of interest for astrobiology is, if we were to survey a large catalogue of inhabited planets, what would be the average ‘fitness’?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' For Earth history (or for the history of any inhabited planet) the equivalent question is, if we were to observe a planet at a random point in its history, what should we expect about the habitability properties of that planet?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' 3 3 Selection by Survival t = 1 t = 2 t = 3 t = 4 t = 5 t = 6 Figure 1: One possible unfolding of the SBS game with N = 25 and M = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' At t = 1 we have our initial ensemble, at t = 2 we have removed all the 1s, at t = 3 we remove all the 2s etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Figure 1 shows one realisation of the SBS game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' At t = 1 all of the dice are in play and the average face value (over very large N or many different realisations of the same game) is (1 + 2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' + M)/M at t = 2 all of the dice showing 1 on the top face are removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Restricting our survey to inhabited planets, the average face value is now (2 + 3 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' + M)/(M − 1) At time t ≤ M the average face value is (t + (t + 1) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' + M) (M − t) = M + t 2 (1) So that average face value increases linearly with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' 4 : :围 880 围Figure 2: Average face value in the SBS game as a function of t for an M = 10 sided dice over N = 1000 dice rolls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Figure 2 shows the result of simulations of the game compared to equation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' In terms of ‘planets’ this model is simply stating the (obvious) fact that planets which survive have properties (high face value) which allow them to survive!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Looking at the catalogue of inhabited planets will necessarily yield planets with properties conducive to maintaining life, without the need for any additional mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' This realisation has all planets are seeded with life at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' More complex games could be devised (say a constant rate of habitable planet generation) to study how the generateion rate interacts with this simple selection mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' For this paper, SBS represents a basic null model - older inhabited planets must have features which have enabled them to remain inhabited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' The growth in fitness of the ‘surviving’ planets is simply a sampling artefact, the average fitness of an inhabited planet increases because we throw away more and more of the unfit planets from our average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Considering our solar system according to SBS, the single inhabited planet we see is habitable because if it wasn’t, we wouldn’t be looking at it, or living on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Thus in this context, SBS is nothing more than an observer effect or anthropic principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' 5 10 Exact Average of 1000 runs 8 Face Value 6 4 2 0 2 6 8 10 4 t4 Sequential Selection The Earth has experienced numerous mass extinction events, had very different planetary regulation mechanisms, atmospheric composition, levels of volcanic activity and life has persisted the entire time [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' We seek to model these sequential resets with another simple game: repeatedly rolling a single die.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Figure 3: One possible unfolding of the SS game with T = 20 and M = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' At t = 1 we roll 2 which shows for 2 steps, we roll 1 which shows for 1 step, then 3 for 3 steps etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' The game is played a large number, N, of times as in figure 1 so we are interested in average behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' When observing the die at a random time t, what should we expect the face of the die to show, on average?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' The chance of the die showing k is proportional to the probability of rolling a k, p(k), times the number of ‘slots’ where the observation could occur e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' if the die is showing 3 this could be an observation of the die on the first, second or third step where it is face up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' We normalise this probability and compute the expected value for the top face as M � k=1 k kp(k) �m k=1 kp(k) (2) For one die p(k) = 1/M and this simplifies to 2M + 1 3 (3) Note this is larger than the expected value of a single dice roll, M+1 2 for M > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' 6 Figure 4: Average face value in the SS game as a function of t for an M = 10 sided dice, averaged over N = 1000 independent instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' The expected value of a single 10 sided dice roll is 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content='5 which is less than the expected face value in the sequential sampling game, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Note the logged x-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Figure 4 shows the results of simulations of the game compared to equation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' The results here imply that when observing a ‘planet’ at a random time, it is more likely to be in a state with stability enhancing properties (high face value).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Again this reflects the obvious fact that if we depict Earth’s history as a time line and pick a random point on the line we are more likely to pick a point in a long stable period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' In particular, our present time is most likely to be a stable period, without the need for any additional mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Like with SBS, Earth’s current stability is simply an observer effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' One thing missing from this game (and from the algorithm of [3]) is the possibility of total extinc- tion, that is, finishing the game early.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' We could implement an additional rule, say when we roll a 1, stop the game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' This would give a model where SBS and SS are both operating simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Here, for simplicity and clarity, we don’t account for total extinction, so as not to mix the mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' All of our SS games persist for the same amount of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' More complex models e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' [4, 19] do have the possibility of early stopping and come to very similar conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' 7 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content='4 Average of 1000 runs Exact 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content='2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content='8 Face Value 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content='6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content='4 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content='2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content='0 100 101 102 103 t5 Sequential Selection with Memory The continued inhabitance of Earth and the fact that biodiversity has increased over time motivates the final games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Each reset does not start from scratch, but builds on evolutionary and ecological innovations that came before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' We propose 2 models with an extremely simple ‘memory’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' This memory is implemented in two ways, first by adding extra dice at fixed M, second by increasing M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content='1 Game A: Adding dice The face value in this game is determined by rolling multiple dice and choosing the one with the maximum face value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' The idea is that stable biospheres outlast unstable ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' One could imagine independent ecosystems co-existing with the final ‘reset’ only occurring when the most stable sub- system collapses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' If this seems contrived, in more complex models e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' [4], a similar feature emerges as a consequence of model dynamics rather than being enforced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Figure 5: One possible unfolding of SSM game A with T = 20 and M = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' At t = 3 we roll 6 which adds an extra die.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' At t = 13 we roll six again which adds a third die.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' The bottom row is the observable, the other rows show dice rolls which are not observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' The analysis is a little more complex that the previous two games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' The first thing we need is the probability to get the face value f when rolling r dice and applying the rule f = max(x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' , xr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' This is pr(f) = r � i=1 �r i � p(f)ip(x < f)r−i (4) where p(f) = 1/M and p(x < f) = f−1 M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' This is just the probability to get at least one f and nothing higher, multiplied by a combinatoric factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' To simplify this, consider arranging all the possible outcomes of r rolls in an r-dimensional hypercube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' The number of ways to obtain f is given by the difference in volumes between an f and f − 1 sided hypercube so pr(f) = f r − (f − 1)r M r (5) It is shown in Appendix A that the expected face value for large M is E[f|r, M ≫ 1] = M r r + 1 + 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' (6) In a game with r dice, the expected number of dice rolls before hitting the value M, where we add an extra die, is 1/pr(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Therefore, the expected time spent playing with exactly r dice is T(r) = �M i=1 ipr(i) pr(M) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' (7) 8 JFor large M (using the summation result from Appendix A) this is T(r) ≃ M 2 r + 1 (8) To calculate the expected face value at t, we first compute the expected number of dice at time t by solving r � k T(k) = t (9) for r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Using the large M approximation r � k T(k) ≃ M 2 r � k 1 i + 1 = M 2(Hr − 1) (10) where Hr is the rth harmonic number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' This has a standard approximation, valid for large r but quite accurate even at r = 1: Hr ≃ γ +ln(r), where γ is the Euler-Mascheroni constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Substituting and solving for r gives r(t) = exp � t M 2 + 1 − γ � = A exp � t M 2 � (11) where A = exp(1 − γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' The number of dice grows exponentially, with growth rate 1/M 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' The more faces a die has, the longer we have to wait to land on a specific one, for example the time to go from r dice to 2r dice is ≃ M 2 ln 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' The expected face value at time t is the expected face value with r(t) dice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Still working in the large M limit this is E[f : M ≫ 1](t) ≃ M r r + 1 = M A exp � t M 2 � 1 + A exp � t M 2 � (12) This is a sigmoid function in the variable t/M 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' At large values of t the value is M, as expected, we have so many dice we are virtually guaranteed to roll at least one M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' The small t behaviour is interesting, the function is roughly linear which means, despite the exponential growth in the number of dice suggested by equation 11, the expected face value grows much more slowly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' 9 Figure 6: Average face value in the SSM game as a function of t for an M = 10 sided dice, averaged over N = 1000 independent instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Figure 6 shows the results of 1000 simulations of the game with M = 10 compared to the ‘exact’ answer, equation 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' We observe convergence to the upper bound M at a rate that is roughly linear in log time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Such slow convergence is seen in more complex evolutionary models, especially the Tangled Nature Model [20, 21] and its variants [13, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' There, it arises from the simulated ecosystem successively crossing ‘entropic barriers’ [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Each time a barrier is crossed the system is likely to be in a more stable configuration with a higher barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' This behaviour has been discussed before in the language of record statistics and is also observed in physical systems like spin glasses, colloids and high temperature superconductors [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' This model is simple enough for an approximate analytic solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' This shows that there is competition between the growth in the number of dice over time against the growth in the time taken between trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' There is also a trade-off between large values of M, leading to higher average rolls, versus time taken to add a new die.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' What this model suggests is that selection plus accumulation leads to slow growth in stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' This model implies that older inhabited planets should be more habitable, so our presence on Earth is not just an observer effect but a statistically more likely outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' 10 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content='0 Average of 1000 runs Exact 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content='5 Value 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content='0 Face 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content='0 100 101 102 103 t5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content='2 Game B: Increasing M This game similar to the previous one, except instead of adding extra dice we have just one die and add a extra faces to it, which makes this harder to play with real dice!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' A rough analogy to a real ecosystem is to assume that species diversity is not lost after each collapse (dice roll) and that species persist at low abundance, in dormant states or isolated refugia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Reaching a fitness peak (hitting the max value of M) generates more latent diversity and allows ecosystems to explore more of the so-called fitness landscape [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Thus each reset has the potential to find a more stable state because the space of possibilities is wider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Figure 7: One possible unfolding of SSM game B with T = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' The bottom row shows the actual face values and the top row shows the number of sides of the die.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' For example at t = 11 we roll a 4 on a 4-sided die, increasing the number of sides to 5 for the next roll.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' If the die has M sides, the expected number of rolls required to hit the M face is just 1/M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Each roll is expected to last M+1 2 steps so the expected waiting time before increasing the number of faces is T(M) = M M + 1 2 (13) Summing up the wait times from each M gives the total duration of the experiment t = M � i=1 i(i + 1) 2 = 1 2 �M(M + 1)(2M + 1) 6 + M(M + 1) 2 � (14) Keeping only the terms of highest order in M and solving for t gives M = 3√ 6t (15) Substituting into equation 3 gives E[f](t) = 2 3√ 6t + 1 3 (16) 11 M=1 2 2 33 4 4 5 5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content='.r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content='Figure 8: Average face value in SSM game B as a function of t, averaged over N = 1000 independent instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Figure 8 shows the results of simulations of the game compared to the exact answer, equation 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Unlike game A there is no convergence and the expected face value grows without bound, though fairly slowly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Again there is a trade off between increasing M by performing a large number of trials and the time it takes to complete those trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' This is again reminiscent of Tangled Nature Model dynamics [22, 13, 4] and other physical systems which cross energetic or entropic barriers [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' 6 Discussion These three mechanisms give three reasonable ideas about what to expect when surveying large catalogues of inhabited planets, or looking at an inhabited one at a random point in its history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' The first two (SBS, SS) have no role for life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' The stability properties of inhabited planets are down to observer effects - unless they had these properties we wouldn’t be looking at them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' The third mechanism is more interesting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Once a planet is inhabited life can have a positive effect on habitability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' In particular - inhabited planets have properties conducive to stability because of their history of inhabitance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' This idea has appeared previously as ‘The inhabitance paradox’ in [25] and is closely related 12 16 Average of 1000 runs Exact 14 12 Face Value 10 8 6 4 2 100 101 102 103 tto the idea of the Gaian bottleneck [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' This paradox says that for a planet to be habitable, it must be inhabited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' This means life must seize the reins and exert a stabilising effect early in a planet’s history or go extinct due to deteriorating geophysical conditions - an effect dubbed the Gaian bottleneck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' The SSM game shows a very simple mechanism by which this could occur, combining the sequential selection algorithm of [3] with some method of making cumulative improvements will tend to generate more stable systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' We have argued previously [4] that such cumulative processes are widespread on Earth, for example: microbial seed banks, dormancy [27] and lateral gene transfer [28] all contribute to the maintenance of microbial diversity and therefore the stabilising effect of functional redundancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' We hope that this model and its analysis provides some clarity on selection principles as well as providing a sandbox for studying selection effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' In particular, we believe that Sequential Selection with Memory provides a plausible way for a complex system, like an inhabited planet, to become more stable over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' We propose that Gaia - the stabilising and symbiotic feedback of life and the environment - can be born through this kind of natural, but non-Darwinian, selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' A Expected value for the max of r, M-sided dice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Rolling r, M sided dice gives the face value f with probability p(f) = kr − (k − 1)r M r as discussed in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' The expectation for the face value is therefore E[f] = M � k=1 k kr − (k − 1)r M r Writing out the sum explicitly 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content='1r + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content='2r + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content='3r + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' + M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content='M r −(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content='0r + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content='1r + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content='2r + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' + M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content='(M − 1)r) Shows that we can regroup and rewrite as E[f] = 1 M r � M r+1 − M−1 � k=1 kr � The sum can be simplified using Faulhaber’s formula [29] M−1 � k=1 kr = (M − 1)r+1 r + 1 + (M − 1)r 2 + O(M r−1) where the lower order terms are fairly complex coefficients involving the Bernoulli numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Substi- tuting and taking the limit of large M we get E[f] = M r r + 1 + 1 2 as stated in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' 13 References [1] Pierrick Bourrat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' From survivors to replicators: evolution by natural selection revisited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Biology & Philosophy, 29(4):517–538, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' [2] W Ford Doolittle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Natural selection through survival alone, and the possibility of gaia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Biology & Philosophy, 29(3):415–423, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' [3] Timothy M Lenton, Stuart J Daines, James G Dyke, Arwen E Nicholson, David M Wilkinson, and Hywel TP Williams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Selection for gaia across multiple scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Trends in Ecology & Evolution, 33(8):633–645, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' [4] Rudy Arthur and Arwen Nicholson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Selection principles for gaia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Journal of Theoretical Biology, 533:110940, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' [5] Ignas A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Snellen, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Snik, M.' metadata={'source': 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L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Beuzit, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Biller, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Birkby, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Boccaletti, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' van Boekel, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' de Boer, Matteo Brogi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Buchhave, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Carone, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Claire, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Claudi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Demory, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' D´esert, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Desidera, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Gaudi, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Gratton, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Gillon, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Grenfell, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Guyon, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Henning, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Hink- ley, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Huby, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Janson, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Helling, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Heng, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Kasper, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Keller, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Krause, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Kreidberg, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Madhusudhan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Lagrange, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Launhardt, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Lenton, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Lopez-Puertas, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Maire, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Mayne, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Meadows, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Mennesson, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Micela, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Miguel, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Milli, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Min, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' de Mooij, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Mouillet, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' N’Diaye, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' D’Orazi, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Palle, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Pagano, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Piotto, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Queloz, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Rauer, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'} +page_content=' Ribas, G.' metadata={'source': 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Huang +Dongwon Lee +The Pennsylvania State University, State College, Pennsylvania, USA +{aza6352, suh972, dul13}@psu.edu +Abstract +Advancements in Text-to-Image synthesis over recent +years have focused more on improving the quality of gener- +ated samples on datasets with descriptive captions. How- +ever, real-world image-caption pairs present in domains +such as news data do not use simple and directly descrip- +tive captions. With captions containing information on both +the image content and underlying contextual cues, they be- +come abstractive in nature. In this paper, we launch ANNA, +an Abstractive News captioNs dAtaset extracted from on- +line news articles in a variety of different contexts. +We +explore the capabilities of current Text-to-Image synthesis +models to generate news domain-specific images using ab- +stractive captions by benchmarking them on ANNA, in both +standard training and transfer learning settings. The gen- +erated images are judged on the basis of contextual rele- +vance, visual quality, and perceptual similarity to ground- +truth image-caption pairs. Through our experiments, we +show that techniques such as transfer learning achieve lim- +ited success in understanding abstractive captions but still +fail to consistently learn the relationships between content +and context features. The ANNA Dataset is available at +https://github.com/aashish2000/ANNA. +1. Introduction +Image Generation has been improving by leaps and +bounds over the last few years thanks to advancements in +Generative Modelling approaches and availability of higher +compute capacities [20]. Areas such as text-to-image syn- +thesis have grown in prominence due to the development +of model pre-training paradigms on vast image-text pairs +mined from the internet [17]. This has promoted the use of +these generators for a variety of applications such as online +content creation, art synthesis [14] and even more malicious +use-cases such as DeepFake generation [31]. With Internet +news media and social networking websites becoming the +more preferred forms of news distribution, the impact that +generative modelling, especially semantically-relevant im- +age generation can have on the news media industry is sig- +Figure 1. Example of descriptive captions from the COCO Cap- +tions dataset [2] (Above) and abstractive captions from the ANNA +(Below). In this case, the abstractive captions contain high-level +visual content information relevant to the type of room depicted +and contextual information explaining who are its inhabitants, who +sponsored it, etc. +nificant. Images accompanying news articles are primarily +used as supporting media to convey the key message of the +article along with complementary information to aid reader +understanding. +Commonly, text-to-image synthesis has made use of de- +scriptive captions, where only visual objects present within +each image are described in detail. However, news captions +also relay contextual information correlating the image’s +contents to the article. The captions are thus abstractive +(beyond being descriptive), containing both higher-level de- +scriptive information and contextual cues. Here, we define +context of a text caption as an attribute that does not have a +direct visual translation, but contributes towards modifying +an image’s appearance in relation to the situation in which +the image is referenced. Fig. 1 provides an example of this +where both images depict rooms within living spaces, but +there is a noticeable difference in the appearance of a room +within a house and that of a shelter. The study of pragmatic +reasoning in linguistics [5] typically deals with how the in- +1 +arXiv:2301.02160v1 [cs.CV] 5 Jan 2023 + +Caption: This room has a +bed with blue sheets and a +large bookcase +Caption: A room in a shelter +for victims of domestic +violence that was able to +reopen recently because of +a contribution from a donorformativeness of text is influenced by its relevance to con- +text. Past research has established the importance of prag- +matic factors in ascertaining the true meaning of context- +driven text information and how it affects accurate caption- +ing of images [22], [21] [11]. Directly descriptive captions +lack this contextual grounding, limiting their usefulness for +describing news images. To replicate the same types of im- +ages with contextual relevance using descriptive captions, +we require intensive caption engineering efforts. This com- +bination of image content information along with contex- +tual cues make abstractive captions much more challenging +to understand, directly impacting the relevance of generated +results. +Current datasets for Text-to-Image synthesis are either +focused on narrow domains with simple, descriptive cap- +tions or contain minimally filtered image-text pairs from a +multitude of online sources. There are not many domain- +specific datasets with image caption pairs containing con- +textual information in addition to image descriptions. Addi- +tionally, while most models use improved visual quality of +output images to be indicators of superior performance, not +much focus is placed on evaluating the correlation between +the output image and input text captions. This becomes +more important when dealing with captions whose features +are only partially aligned with the ground truth images due +to its non-descriptive nature. The task of abstractive text-to- +image synthesis aims to generate images from abstractive +captions with contextual cues. To evaluate this task, we de- +sign ANNA, a dataset containing abstractive captions per- +taining to news image-caption pairs. Abstractive captions +can motivate text-to-image synthesis models to effectively +identify these different feature types along with their rel- +ative importance and represent them appropriately in gen- +erated images. +With current Text-to-Image architectures +implicitly delineating content and context features, we pro- +vide detailed visualizations of both their success and failure +cases on ANNA and the need for better understanding of +sentence structures for generating image features. +Our contributions in this paper can be summarized as the +following: +• We introduce ANNA, a dataset containing approxi- +mately 30K abstractive image-caption pairs from pop- +ular media outlet The New York Times +• We show how current Text-to-Image architectures are +able to understand abstractive captions and transfer- +learned concepts from descriptive captions for abstrac- +tive text-to-image synthesis +• Using an exhaustive set of evaluation metrics, we +benchmark popular Text-to-Image architectures on the +basis of generated image quality, image similarity to +ground truth images and contextual relevance with ref- +erence captions +2. Related Work +Text-to-Image Synthesis +Text-to-Image synthesis is a +multi-modal generation task that produces relevant images +conditioned on features described in a text caption. Ini- +tial approaches such as [16] found success by leveraging +Generative Adversarial Networks (GANS) [4] for this task. +Motivated by the success of GAN’s, StackGAN [30] uses +a stacked generator to simplify the generation pipeline into +stages: semantically relevant low-resolution image synthe- +sis followed by progressive up-scaling and defect correc- +tion. AttnGAN [26] integrates an attention mechanism to +capture sentence and word level features for increasing the +correlation between generated images and input text. At- +tnGAN proved to be a strong baseline based on which mul- +tiple advancements were developed. One such approach, +DMGAN [33] integrates a dynamic memory based refine- +ment module for improving image quality and key-word se- +lection from reference captions. [29] and [27] build on the +same model architecture by introducing Contrastive learn- +ing approaches to improve consistency between learned text +and image representations. In recent years, the success of +Vision-Language Pre-training (VLP) has prompted the de- +velopment of newer and more robust Text-to-Image synthe- +sis architectures. Contrastive Language-Image Pre-training +(CLIP) [13], is one of the largest open-source, pre-trained +models that uses raw text for supervising the learning pro- +cess of visual concepts. Using pre-trained encoders such +as CLIP for input text captions, [32], [15], [14] use differ- +ent generator architectures such as GANs, Auto-regressive +Transformers and Diffusion models respectively. +Datasets +Traditional datasets used as benchmarks for +measuring Text-to-Image synthesis include domain-specific +datasets Oxford-102 Flowers [12] and CUB-200 [23]. The +Oxford-102 Flowers contains images of 102 classes of flow- +ers along with 5 human-annotated descriptions per im- +age. Similarly the CUB-200 dataset contains 11,788 im- +ages of 200 subcategories belonging to different categories +of birds along with 5 captions per image. +The captions +for each of the images in CUB-200 and Oxford-102 were +collected and released by [16] as a part of their evalua- +tion. COCO Captions [2] is another popular dataset devel- +oped using images from the MS-COCO [9], a large-scale +object detection dataset. It contains over one and a half +million captions describing over 330,000 images contain- +ing 80 different classes of everyday objects. Some of the +other datasets used for this task include the Multi-Modal- +CelebA-HQ Dataset [25] which provides text-descriptions +of facial features for images sourced from the CelebA-HQ +dataset [6]. Conceptual Captions [18] consists of over 3 +million image-caption pairs mined from the internet. In this +dataset, all the captions are hypernymized, i.e. all proper +2 + +nouns and named-entities are replaced with their respective +hypernynms to make the captions simpler to learn and more +descriptive. [1] expands this dataset by increasing the num- +ber of image-caption pairs from 3 million to 12 million. All +the datasets discussed above focus on descriptive captions +for each image, where minimal or no contextual informa- +tion regarding the image is present. Our dataset is one of +the first to investigate the previously unexplored interaction +between content and context features for text-to-image syn- +thesis. +3. Constructing Abstractive News Captions +Dataset: ANNA +The ANNA (Abstractive News captioNs dAtaset) has +been constructed to perform news image generation us- +ing abstractive captions. We source images from the NY- +Times800K dataset [19] which contains news articles and +associated image-caption pairs scraped from the news or- +ganization The New York Times (NYT). This dataset was +originally developed for News Image Captioning. Using +news image-caption pairs from a reputable media outlet +such as NYT helps ensure the dataset’s quality. +As we +aim to observe the relationship between content and con- +text features and how it translates to generated images, we +focus on selecting generalizable entities within our dataset. +News data contains a multitude of named-entities, often +with very low repetition and distinct physical appearances, +such as faces and geographic landmarks. +The inclusion +of named-entities from news images would drastically in- +crease the complexity of the generative task. The inabil- +ity to accurately generate named-entity attributes would fur- +ther hamper context feature representation due to their inter- +dependent nature. In order to combat the mentioned issues, +we carefully curate our dataset to include image-caption +pairs containing adequate contextual and content related in- +formation. We select Image-caption pairs with lesser de- +pendence on named-entities and more general visual com- +ponents to make the task feasible. The specific preprocess- +ing and filtering approaches utilized are detailed below. +3.1. Preprocessing and Filtering Approaches +The original NYTimes800K dataset contains 445K news +articles accompanied by 793K image-caption pairs. It spans +14 years of articles published on The NYT website. The +dataset has been provided as a MongoDB dump for public +access. The first step of preprocessing focuses on removing +image-caption pairs with explicit entities described both in +images and text. We use the provided NER tags for each +caption for filtering. We exclude all captions containing the +NER tags ’PERSON’, ’GPE’, ’LOC’, ’WORK OF ART’, +’ORG’. This ensures any visually significant named-entity +without adequate description isn’t present in the dataset. +Subsequently, we also set bounds on the caption length be- +tween 4 to 70 words. Any captions lesser than 4 words +would not be informative enough for extracting usable fea- +tures and captions larger than 70 words cannot be handled +by the CLIP-based Text encoder [13] that we employ in our +experiments. +Following caption-based filtering, we also remove all +images where human faces are clearly visible in the fore- +ground. We accomplish this by using a RetinaFace-based +face detector [3], removing around 1000 additional im- +ages. Through these filtering techniques, we extract rel- +evant image-caption pairs and corresponding article head- +lines from the NYTimes800K Dataset. Data pre-processing +steps include uniformly scaling our news images to our tar- +get input resolution 256x256. To accomplish this, we rela- +tively scale the smaller dimension (height or width) of the +image to our target resolution and take its center crop. This +makes sure that we have minimal information loss and also +helps center the foreground objects in each image. Dis- +claimer: The dataset samples may use words or language +that is considered profane, vulgar, or offensive by some +readers as they are extracted from real-world news articles. +3.2. Dataset Insights +The filtered and pre-processed version of the ANNA con- +tains 29625 image-text pairs. We split the dataset into Train, +Validation and Testing sets in the ratio of 80%:10%:10% re- +spectively. All metric scores reported have been calculated +on the Test set. To better understand the composition of the +dataset, we analyze various attributes of the image-text pairs +and the articles they have been selected from. +Dataset +Unique Tokens +Caption Length +Mean +StdDev +ANNA Train +17897 +14.1 +7.75 +ANNA Validation +1622 +13.8 +7.60 +ANNA Test +1649 +14.1 +7.71 +COCO Captions Train +11046 +10.4 +1.75 +COCO Captions Validation +4758 +10.4 +1.74 +Table 1. Dataset Statistics of ANNA and COCO-Captions +3.2.1 +Caption Statistics +In this section, we evaluate different statistical measures for +quantifying the distribution of captions across the dataset. +Table 1 shows the average caption length of captions present +in the dataset and across the train, validation and test sub- +sets. +We see that the average caption lengths are simi- +lar across the different data splits with the average caption +length being slightly greater than that of the COCO Cap- +tions dataset. We also show the standard deviation in cap- +tions sizes across different image-caption pairs. We also ex- +amine the words appearing in these captions by identifying +3 + +Figure 2. Object Frequency Analysis using Treemaps +unique tokens present. To calculate the unique tokens, we +use the spaCy library for tokenizing and lemmatizing our +captions along with the removal of all stop words. Subse- +quently, we tag the different Parts of Speech (POS) present +and select tokens that belong to the classes [Common Noun, +Proper Noun, Adjectives and Verbs]. This provides a mini- +mum guarantee that the abstractive captions present are long +enough to contain adequate content and contextual features. +This analysis also ensures that the composition captions +present in the train, validation and test splits are consistent +with each other. +3.2.2 +News Image Analysis +Along with the captions, we also estimate image proper- +ties such as the number of recognizable objects present in +each image and average number of detected objects per +image. We use a YOLO-R based object detector [24] for +identifying the objects present in each image of our dataset. +The YOLO-R detector has been trained on the MS-COCO +dataset, containing 80 unique object classes of common- +place objects [2]. We test the pre-trained model with 0.4 as +the confidence threshold. We find that there are an average +of 2.57 objects per image in the ANNA. Fig. 2 shows the +most frequently appearing classes of objects in our dataset +using a treemap for visualization. +3.2.3 +Categories of News Articles Selected +In this section, we identify the different types of news ar- +ticles from which image-caption pairs were sourced for +dataset construction. In total, there exist 123 unique article +topics within our dataset. Only 13 of image-caption pairs do +not have accompanying article type information so we dis- +regard those pairs from our article topic analysis. From Fig. +3, we see that there exists a good distribution across topics +such as Dining, Business, Real Estate, etc. This shows that +the news image-caption pairs are diverse and not limited to +only a particular type of news article. +4. Experiments +In order to understand how different architectures learn +abstractive captions on the ANNA, we consider various text- +to-image synthesis models previously proposed in litera- +ture. The three model architectures we test as a part of our +evaluation are: Lafite [32], AttnGAN+CL [26] and DM- +GAN+CL [27]. These models are selected for comparison +as they are among the top-10 on the COCO Captions Text- +to-Image synthesis leaderboard and take significantly dif- +ferent approaches for tackling the same task. As all these +models have achieved State-of-the-Art scores on descrip- +tive caption datasets, we evaluate how they perform with +news domain-specific, abstractive captions in our experi- +ments and visualize our results. +Text-to-Image Synthesis Models +The Lafite model uti- +lizes a pre-trained CLIP encoder for translating text em- +beddings into the image feature space. +It adapts an un- +conditional StyleGAN2 generator [7] by injecting text- +conditional information through affine transformations. +Two Fully Connected Layers are utilized to transform the +input text features to be more semantically similar with +StyleGAN’s image Stylespace. In our experiments, we train +Lafite on ANNA in a fully-supervised setting. We train 2 +variants of Lafite, with and without Transfer Learning. In +the non-transfer learning variant, we train it on the ANNA +4 + +Object Frequency Analysis +person +chair +book +cake +spoon +couch +boat +17,069 +5,252 +2,073 +1,179 +1,087 +1,064 +1,001 +potted plant +carrot +bed +wine +fork +1,964 +890 +702 +627 +dining table +bench +4,090 +cup +868 +6op +1,717 +clock +knife +835 +bird +bowl +1,669 +3,779 +cell +phone +cat +bus +car +orange +truck +tv +6,155 +1,332 +780 +airplane +bottle +2,416 +vase +donut +train +1,183 +762 +cowFigure 3. Visualizing Article Categories of image-caption pairs present in ANNA +Model +IS (↑) +FIDCLIP (↓) +LPIPS(↓) +CLIPScore (↑) +Lafite (Transfer Learning) +16.49 +13.93 +0.7470 +0.7575 +Lafite (Base) +12.59 +20.48 +0.7432 +0.7277 +DMGAN+CL (512 dim) +14.07 +29.30 +0.7568 +0.5913 +DMGAN+CL (256 dim) +13.37 +29.87 +0.7581 +0.5861 +AttnGAN+CL (512 dim) +12.56 +41.00 +0.7623 +0.5695 +AttnGAN+CL (256 dim) +13.06 +37.41 +0.7616 +0.5748 +Table 2. Results of Abstractive Text-to-Image synthesis on ANNA +until convergence for 4000 epochs. To perform Transfer +Learning, we initialize the model with pre-trained weights +from the Conceptual Captions (CC3M) dataset [18] and +continue training on the ANNA until convergence for 2000 +epochs. +The AttnGAN+CL and DMGAN+CL models share sim- +ilar architectures, with both utilizing a Deep Attentional +Multimodal Similarity Model (DAMSM) for computing the +similarity between extracted images and text. These archi- +tectures have been supplemented with a Constrastive Learn- +ing Loss function along with their DAMSM loss to improve +pre-training performance. We first train the DAMSM mod- +ule on the Train and Validation sets of our dataset to con- +struct the mapping between image and text features. We +compare 2 different embedding sizes of the DAMSM mod- +ule for both models: 256 and 512. The default AttnGAN +and DMGAM models have 256 embedding feature vectors +by default, but the CLIP based model Lafite uses 512 em- +bedding feature vectors instead. Thus, we train the models +with both embedding sizes to ensure a fair comparison. +Evaluation Metrics +To evaluate the performance of these +architectures, we report 4 different metrics: Inception Score +(IS), Fr´echet Inception Distance (FID), Learned Perceptual +Image Patch Similarity (LPIPS) and CLIPScore. IS and FID +evaluate the quality and diversity of generated images. They +estimate probability distribution properties of the generated +images and how far it diverges from that of the reference im- +ages. For FID, we adapt the proposed FIDCLIP from [8] +due to its closer correspondence with human judgement on +real-world, diverse datasets. LPIPS judges the perceptual +similarity between the reference and generated images us- +ing deep features extracted across image patches instead of +measuring pixel-level similarity. We use LPIPS version 0.1 +for our testing. Since LPIPS is an image-wise similarity +metric, we report the average of scores obtained by the gen- +erated test set images. CLIPScore is a reference-free metric +that can be employed to evaluate the relevance of input text +captions to the content of generated images. We selected +these 4 metrics as they provide a holistic evaluation of the +different key aspects involved in measuring text-to-image +model performance. We report our scores in Table 2. +5 + +Article Categories +Metro +3,646 +Dining +3,010 +Business +RealEstate +2,573 +Science +12,224 +National +Foreign +2,023 +Travel +1,432 +Culture +1,120 +Styles +Sports +959 +Weekend +805 +Home +601 +Magazine +TStyle +412 +Automobiles +SundayBusiness +398 +Metropolitan +Arts&Leisure +1342 +OpEd +NYTNOW +1247 +BookReview +Escapes +1199 +SpecialSections +Learning +158 +CityWeekly +Express +140 +Washington +Upshot +109 +Regionals +0 +200 +400 +600 +800 +1000 +1200 1400 1600 +1800 +2000 +2200 2400 2600 2800 +3000 +32003400 +3600 3800 +Count =(a) Original Image +(b) Lafite (Transfer +Learning) +(c) Lafite (Base) +(d) +DMGAN +(512 +dim) +(e) +DMGAN +(256 +dim) +(f) +AttnGAN +(512 +dim) +(g) +AttnGAN +(256 +dim) +Figure 4. Result Visualization for Caption: The castle, draped with vines and adorned with bougainvillea, is set on 10 acres, with +gardens, a swimming pool and a private chapel. +(a) Original Image +(b) Lafite (Transfer +Learning) +(c) Lafite (Base) +(d) +DMGAN +(512 +dim) +(e) +DMGAN +(256 +dim) +(f) +AttnGAN +(512 +dim) +(g) +AttnGAN +(256 +dim) +Figure 5. Result Visualization for Caption: Pollutants in the Gowanus Canal include pesticides, heavy metals and carcinogens like +PCBs. +4.1. Evaluation of Generated Samples +Image Quality +From the reported IS and FID scores, we +can clearly identify that Lafit with Transfer Learning out- +performs all other models. Although the IS score of the +baseline model is lower than that of DMGAN+CL, this +trend is reversed in FID scores. This result can be attributed +to the fact that the Inception model feature space is aligned +to the classes present in ImageNet, hence penalizing other +datasets that diverge from this distribution [8]. The updated +CLIP feature space used for computing FIDCLIP helps +mitigate this issue and makes the metric more resistant to +fluctuations caused by image preprocessing and distortions. +These results also correlate with observed image quality on +other benchmark datasets, such as COCO Captions. We +provide visualizations of generated outputs from the test set +for all the trained models in Figures 4, 5, 6, 7, 8, 9. +Delineation between Content and Context features +The Lafite (Transfer Learning) model benefits from learned +associations between visual concepts and text represen- +tations in the absence of extremely descriptive captions, +which corroborates its high CLIPScore. +Similarly, for +the other models trained without transfer learning on our +dataset, we observe that the LPIPS score and CLIP- +Score follow the same trajectory as FIDCLIP with the +Lafite (Base) model exhibiting the best correlation between +ground truth image similarity and relevance with reference +captions. These results show that the top performing models +do have an implicit understanding of what constitutes image +content and context information. But limitations still exist +for implicit delineation of captions features, as shown in +Fig. 9. With the reference image and descriptive section of +the caption dealing with the image of an animal tracking de- +vice, the Text-to-Image models incorrectly generate an an- +6 + +(a) Original Image +(b) Lafite (Transfer +Learning) +(c) Lafite (Base) +(d) +DMGAN +(512 +dim) +(e) +DMGAN +(256 +dim) +(f) +AttnGAN +(512 +dim) +(g) +AttnGAN +(256 +dim) +Figure 6. Result Visualization for Caption: Left, the New Museum and the original adjacent building it purchased 12 years ago on +the Bowery, at right. +(a) Original Image +(b) Lafite (Transfer +Learning) +(c) Lafite (Base) +(d) +DMGAN +(512 +dim) +(e) +DMGAN +(256 +dim) +(f) +AttnGAN +(512 +dim) +(g) +AttnGAN +(256 +dim) +Figure 7. Result Visualization for Caption: The rooms at the Ace Hotel have high ceilings and oversized windows. Some of the larger +rooms and suites includes details like guitars, turntables and vinyl records. +imal as the image foreground rather than the tracker. Thus, +comprehension of caption structures and explicit feature de- +lineation must be improved. +These experiments demon- +strate the need for non-descriptive image-captions datasets, +such as ANNA for bridging the performance gap between +descriptive and abstractive captions. +5. Discussion and Conclusion +Our experiments demonstrate how existing text-to-image +architectures understand abstractive captions present in +domain-specific data such as news media. We show that +implicit delineation between content and context features +have limitations, prompting the need for explicit feature de- +lineation and modified objective functions to better suit this +task. One major impact of understanding abstractive cap- +tions such as those present in ANNA is the reduction in re- +quirements for directly descriptive captioning. As the size +of datasets keep increasing, scaling up human annotation of +images to match demand adds a huge overhead. As descrip- +tive captions need to be tightly-coupled with the reference +image’s contents, there needs to be multiple rounds of eval- +uation and filtering, making it a manually tedious task. The +use of abstractive captions for images can greatly simplify +the human annotation process for datasets. Additionally, +ANNA motivates the development of journalism assistance +solutions. The use of keywords and descriptive prompts +with current image generators involves a lot of prompt en- +gineering to get relevant images for a specific topic [10]. +High quality images are generated only when a particu- +larly restrictive sentence structure and vocabulary is used +in prompts. As models are trained to understand abstractive +captions, the requirements for intensive prompt engineering +would be significantly reduced. Similarly, achieving better +delineation between different feature types present in non- +7 + +(a) Original Image +(b) Lafite (Transfer +Learning) +(c) Lafite (Base) +(d) +DMGAN +(512 +dim) +(e) +DMGAN +(256 +dim) +(f) +AttnGAN +(512 +dim) +(g) +AttnGAN +(256 +dim) +Figure 8. Result Visualization for Caption: The Full Orange: two all-beef patties, special sauce, lettuce. +(a) Original Image +(b) Lafite (Transfer +Learning) +(c) Lafite (Base) +(d) +DMGAN +(512 +dim) +(e) +DMGAN +(256 +dim) +(f) +AttnGAN +(512 +dim) +(g) +AttnGAN +(256 +dim) +Figure 9. Result Visualization for Caption: With the RoamEO base unit, left (which includes a collar), a dog owner can get radio +signals tracking the animal’s location, up to 1.5 miles away. +descriptive captions can also benefit related tasks such as +image retrieval. The addition of context can play a major +role in influencing the quality of retrievals. +Limitations +This paper aims at introducing the potential +of abstractive captions to motivate the development of more +contextually-grounded text-to-image synthesis models, par- +ticularly when synthesizing news-domain specific images. +Although news articles contain a lot of named-entities, we +choose to filter them out and instead focus on context fea- +tures that can be inferred from text captions and depicted by +general visual concepts. Developing text-to-image synthe- +sis architectures that can take advantage of named-entities +using external knowledge bases as reference would help +overcome this limitation. Large-scale human evaluation of +images generated by text-to-image architectures on abstrac- +tive captions is another important step towards measuring +their relative performance, which we aim to perform as a +part of our future research. +Potential negative societal impacts +Image generation ar- +chitectures have the potential to be misused for nefarious +use-cases such as spreading disinformation [31] and gen- +erating neural fake news [28]. Our current preprocessing +pipeline removes most images containing named-entities, +i.e. public figures and locations of national importance, con- +tributing towards risk mitigation. However, we recognize +the threat posed by contextually-relevant Deepfake images +when dealing with news media images. Future research di- +rections include understanding the extent up to which text- +to-image models can be used for neural fake news genera- +tion and identifying appropriate detection strategies. +6. Acknowledgements +This research has been partially supported by NSF +Awards #1820609 and #2114824. +8 + +References +[1] Soravit Changpinyo, Piyush Sharma, Nan Ding, and Radu +Soricut. +Conceptual 12m: Pushing web-scale image-text +pre-training to recognize long-tail visual concepts. In 2021 +IEEE/CVF Conference on Computer Vision and Pattern +Recognition (CVPR), pages 3558–3568. IEEE, June 2021. +3 +[2] Xinlei Chen, Hao Fang, Tsung-Yi Lin, Ramakrishna Vedan- +tam, Saurabh Gupta, Piotr Dollar, and C Lawrence Zitnick. +Microsoft COCO captions: Data collection and evaluation +server. arXiv preprint arXiv:1504.00325, Apr. 2015. 1, 2, 4 +[3] Jiankang Deng, Jia Guo, Evangelos Ververas, Irene Kot- +sia, and Stefanos Zafeiriou. Retinaface: Single-shot multi- +level face localisation in the wild. +In Proceedings of +the IEEE/CVF conference on computer vision and pattern +recognition, pages 5203–5212. 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In Proceedings of the IEEE/CVF Con- +ference on Computer Vision and Pattern Recognition, pages +17907–17917. openaccess.thecvf.com, 2022. 2, 4 +[33] Minfeng Zhu, Pingbo Pan, Wei Chen, and Yi Yang. Dm-gan: +Dynamic memory generative adversarial networks for text- +to-image synthesis. In Proceedings of the IEEE/CVF con- +ference on computer vision and pattern recognition, pages +5802–5810. openaccess.thecvf.com, 2019. 2 +10 + diff --git a/7dA0T4oBgHgl3EQfOf8M/content/tmp_files/load_file.txt b/7dA0T4oBgHgl3EQfOf8M/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b39c7c6608b57dcaf019b364909bf2496f22949c --- /dev/null +++ b/7dA0T4oBgHgl3EQfOf8M/content/tmp_files/load_file.txt @@ -0,0 +1,482 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf,len=481 +page_content='ANNA: Abstractive Text-to-Image Synthesis with Filtered News Captions Aashish Anantha Ramakrishnan Sharon X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' Huang Dongwon Lee The Pennsylvania State University, State College, Pennsylvania, USA {aza6352, suh972, dul13}@psu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content='edu Abstract Advancements in Text-to-Image synthesis over recent years have focused more on improving the quality of gener- ated samples on datasets with descriptive captions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' How- ever, real-world image-caption pairs present in domains such as news data do not use simple and directly descrip- tive captions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' With captions containing information on both the image content and underlying contextual cues, they be- come abstractive in nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' In this paper, we launch ANNA, an Abstractive News captioNs dAtaset extracted from on- line news articles in a variety of different contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' We explore the capabilities of current Text-to-Image synthesis models to generate news domain-specific images using ab- stractive captions by benchmarking them on ANNA, in both standard training and transfer learning settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' The gen- erated images are judged on the basis of contextual rele- vance, visual quality, and perceptual similarity to ground- truth image-caption pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' Through our experiments, we show that techniques such as transfer learning achieve lim- ited success in understanding abstractive captions but still fail to consistently learn the relationships between content and context features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' The ANNA Dataset is available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content='com/aashish2000/ANNA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' Introduction Image Generation has been improving by leaps and bounds over the last few years thanks to advancements in Generative Modelling approaches and availability of higher compute capacities [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' Areas such as text-to-image syn- thesis have grown in prominence due to the development of model pre-training paradigms on vast image-text pairs mined from the internet [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' This has promoted the use of these generators for a variety of applications such as online content creation, art synthesis [14] and even more malicious use-cases such as DeepFake generation [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' With Internet news media and social networking websites becoming the more preferred forms of news distribution, the impact that generative modelling, especially semantically-relevant im- age generation can have on the news media industry is sig- Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' Example of descriptive captions from the COCO Cap- tions dataset [2] (Above) and abstractive captions from the ANNA (Below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' In this case, the abstractive captions contain high-level visual content information relevant to the type of room depicted and contextual information explaining who are its inhabitants, who sponsored it, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' nificant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' Images accompanying news articles are primarily used as supporting media to convey the key message of the article along with complementary information to aid reader understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' Commonly, text-to-image synthesis has made use of de- scriptive captions, where only visual objects present within each image are described in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' However, news captions also relay contextual information correlating the image’s contents to the article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' The captions are thus abstractive (beyond being descriptive), containing both higher-level de- scriptive information and contextual cues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' Here, we define context of a text caption as an attribute that does not have a direct visual translation, but contributes towards modifying an image’s appearance in relation to the situation in which the image is referenced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' 1 provides an example of this where both images depict rooms within living spaces, but there is a noticeable difference in the appearance of a room within a house and that of a shelter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' The study of pragmatic reasoning in linguistics [5] typically deals with how the in- 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content='02160v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content='CV] 5 Jan 2023 Caption: This room has a bed with blue sheets and a large bookcase Caption: A room in a shelter for victims of domestic violence that was able to reopen recently because of a contribution from a donorformativeness of text is influenced by its relevance to con- text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' Past research has established the importance of prag- matic factors in ascertaining the true meaning of context- driven text information and how it affects accurate caption- ing of images [22], [21] [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' Directly descriptive captions lack this contextual grounding, limiting their usefulness for describing news images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' To replicate the same types of im- ages with contextual relevance using descriptive captions, we require intensive caption engineering efforts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' This com- bination of image content information along with contex- tual cues make abstractive captions much more challenging to understand, directly impacting the relevance of generated results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' Current datasets for Text-to-Image synthesis are either focused on narrow domains with simple, descriptive cap- tions or contain minimally filtered image-text pairs from a multitude of online sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' There are not many domain- specific datasets with image caption pairs containing con- textual information in addition to image descriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' Addi- tionally, while most models use improved visual quality of output images to be indicators of superior performance, not much focus is placed on evaluating the correlation between the output image and input text captions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' This becomes more important when dealing with captions whose features are only partially aligned with the ground truth images due to its non-descriptive nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' The task of abstractive text-to- image synthesis aims to generate images from abstractive captions with contextual cues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' To evaluate this task, we de- sign ANNA, a dataset containing abstractive captions per- taining to news image-caption pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' Abstractive captions can motivate text-to-image synthesis models to effectively identify these different feature types along with their rel- ative importance and represent them appropriately in gen- erated images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' With current Text-to-Image architectures implicitly delineating content and context features, we pro- vide detailed visualizations of both their success and failure cases on ANNA and the need for better understanding of sentence structures for generating image features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' Our contributions in this paper can be summarized as the following: We introduce ANNA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' a dataset containing approxi- mately 30K abstractive image-caption pairs from pop- ular media outlet The New York Times We show how current Text-to-Image architectures are able to understand abstractive captions and transfer- learned concepts from descriptive captions for abstrac- tive text-to-image synthesis Using an exhaustive set of evaluation metrics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' we benchmark popular Text-to-Image architectures on the basis of generated image quality,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' image similarity to ground truth images and contextual relevance with ref- erence captions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' Related Work Text-to-Image Synthesis Text-to-Image synthesis is a multi-modal generation task that produces relevant images conditioned on features described in a text caption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' Ini- tial approaches such as [16] found success by leveraging Generative Adversarial Networks (GANS) [4] for this task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' Motivated by the success of GAN’s, StackGAN [30] uses a stacked generator to simplify the generation pipeline into stages: semantically relevant low-resolution image synthe- sis followed by progressive up-scaling and defect correc- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' AttnGAN [26] integrates an attention mechanism to capture sentence and word level features for increasing the correlation between generated images and input text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' At- tnGAN proved to be a strong baseline based on which mul- tiple advancements were developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' One such approach, DMGAN [33] integrates a dynamic memory based refine- ment module for improving image quality and key-word se- lection from reference captions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' [29] and [27] build on the same model architecture by introducing Contrastive learn- ing approaches to improve consistency between learned text and image representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' In recent years, the success of Vision-Language Pre-training (VLP) has prompted the de- velopment of newer and more robust Text-to-Image synthe- sis architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' Contrastive Language-Image Pre-training (CLIP) [13], is one of the largest open-source, pre-trained models that uses raw text for supervising the learning pro- cess of visual concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' Using pre-trained encoders such as CLIP for input text captions, [32], [15], [14] use differ- ent generator architectures such as GANs, Auto-regressive Transformers and Diffusion models respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' Datasets Traditional datasets used as benchmarks for measuring Text-to-Image synthesis include domain-specific datasets Oxford-102 Flowers [12] and CUB-200 [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' The Oxford-102 Flowers contains images of 102 classes of flow- ers along with 5 human-annotated descriptions per im- age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' Similarly the CUB-200 dataset contains 11,788 im- ages of 200 subcategories belonging to different categories of birds along with 5 captions per image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' The captions for each of the images in CUB-200 and Oxford-102 were collected and released by [16] as a part of their evalua- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' COCO Captions [2] is another popular dataset devel- oped using images from the MS-COCO [9], a large-scale object detection dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' It contains over one and a half million captions describing over 330,000 images contain- ing 80 different classes of everyday objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' Some of the other datasets used for this task include the Multi-Modal- CelebA-HQ Dataset [25] which provides text-descriptions of facial features for images sourced from the CelebA-HQ dataset [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' Conceptual Captions [18] consists of over 3 million image-caption pairs mined from the internet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' In this dataset, all the captions are hypernymized, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' all proper 2 nouns and named-entities are replaced with their respective hypernynms to make the captions simpler to learn and more descriptive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' [1] expands this dataset by increasing the num- ber of image-caption pairs from 3 million to 12 million.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' All the datasets discussed above focus on descriptive captions for each image, where minimal or no contextual informa- tion regarding the image is present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' Our dataset is one of the first to investigate the previously unexplored interaction between content and context features for text-to-image syn- thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' Constructing Abstractive News Captions Dataset: ANNA The ANNA (Abstractive News captioNs dAtaset) has been constructed to perform news image generation us- ing abstractive captions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' We source images from the NY- Times800K dataset [19] which contains news articles and associated image-caption pairs scraped from the news or- ganization The New York Times (NYT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' This dataset was originally developed for News Image Captioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' Using news image-caption pairs from a reputable media outlet such as NYT helps ensure the dataset’s quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' As we aim to observe the relationship between content and con- text features and how it translates to generated images, we focus on selecting generalizable entities within our dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' News data contains a multitude of named-entities, often with very low repetition and distinct physical appearances, such as faces and geographic landmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' The inclusion of named-entities from news images would drastically in- crease the complexity of the generative task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' The inabil- ity to accurately generate named-entity attributes would fur- ther hamper context feature representation due to their inter- dependent nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' In order to combat the mentioned issues, we carefully curate our dataset to include image-caption pairs containing adequate contextual and content related in- formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' We select Image-caption pairs with lesser de- pendence on named-entities and more general visual com- ponents to make the task feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' The specific preprocess- ing and filtering approaches utilized are detailed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' Preprocessing and Filtering Approaches The original NYTimes800K dataset contains 445K news articles accompanied by 793K image-caption pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' It spans 14 years of articles published on The NYT website.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' The dataset has been provided as a MongoDB dump for public access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' The first step of preprocessing focuses on removing image-caption pairs with explicit entities described both in images and text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' We use the provided NER tags for each caption for filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' We exclude all captions containing the NER tags ’PERSON’, ’GPE’, ’LOC’, ’WORK OF ART’, ’ORG’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' This ensures any visually significant named-entity without adequate description isn’t present in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' Subsequently, we also set bounds on the caption length be- tween 4 to 70 words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' Any captions lesser than 4 words would not be informative enough for extracting usable fea- tures and captions larger than 70 words cannot be handled by the CLIP-based Text encoder [13] that we employ in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' Following caption-based filtering, we also remove all images where human faces are clearly visible in the fore- ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' We accomplish this by using a RetinaFace-based face detector [3], removing around 1000 additional im- ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' Through these filtering techniques, we extract rel- evant image-caption pairs and corresponding article head- lines from the NYTimes800K Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' Data pre-processing steps include uniformly scaling our news images to our tar- get input resolution 256x256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' To accomplish this, we rela- tively scale the smaller dimension (height or width) of the image to our target resolution and take its center crop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' This makes sure that we have minimal information loss and also helps center the foreground objects in each image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' Dis- claimer: The dataset samples may use words or language that is considered profane, vulgar, or offensive by some readers as they are extracted from real-world news articles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' Dataset Insights The filtered and pre-processed version of the ANNA con- tains 29625 image-text pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' We split the dataset into Train, Validation and Testing sets in the ratio of 80%:10%:10% re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' All metric scores reported have been calculated on the Test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' To better understand the composition of the dataset, we analyze various attributes of the image-text pairs and the articles they have been selected from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' Dataset Unique Tokens Caption Length Mean StdDev ANNA Train 17897 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content='1 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content='75 ANNA Validation 1622 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content='8 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content='60 ANNA Test 1649 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content='1 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content='71 COCO Captions Train 11046 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content='75 COCO Captions Validation 4758 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content='74 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' Dataset Statistics of ANNA and COCO-Captions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content='1 Caption Statistics In this section, we evaluate different statistical measures for quantifying the distribution of captions across the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' Table 1 shows the average caption length of captions present in the dataset and across the train, validation and test sub- sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' We see that the average caption lengths are simi- lar across the different data splits with the average caption length being slightly greater than that of the COCO Cap- tions dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' We also show the standard deviation in cap- tions sizes across different image-caption pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' We also ex- amine the words appearing in these captions by identifying 3 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' Object Frequency Analysis using Treemaps unique tokens present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' To calculate the unique tokens, we use the spaCy library for tokenizing and lemmatizing our captions along with the removal of all stop words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' Subse- quently, we tag the different Parts of Speech (POS) present and select tokens that belong to the classes [Common Noun, Proper Noun, Adjectives and Verbs].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' This provides a mini- mum guarantee that the abstractive captions present are long enough to contain adequate content and contextual features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' This analysis also ensures that the composition captions present in the train, validation and test splits are consistent with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content='2 News Image Analysis Along with the captions, we also estimate image proper- ties such as the number of recognizable objects present in each image and average number of detected objects per image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' We use a YOLO-R based object detector [24] for identifying the objects present in each image of our dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' The YOLO-R detector has been trained on the MS-COCO dataset, containing 80 unique object classes of common- place objects [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' We test the pre-trained model with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content='4 as the confidence threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' We find that there are an average of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content='57 objects per image in the ANNA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' 2 shows the most frequently appearing classes of objects in our dataset using a treemap for visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content='3 Categories of News Articles Selected In this section, we identify the different types of news ar- ticles from which image-caption pairs were sourced for dataset construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' In total, there exist 123 unique article topics within our dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' Only 13 of image-caption pairs do not have accompanying article type information so we dis- regard those pairs from our article topic analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' 3, we see that there exists a good distribution across topics such as Dining, Business, Real Estate, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' This shows that the news image-caption pairs are diverse and not limited to only a particular type of news article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' Experiments In order to understand how different architectures learn abstractive captions on the ANNA, we consider various text- to-image synthesis models previously proposed in litera- ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' The three model architectures we test as a part of our evaluation are: Lafite [32], AttnGAN+CL [26] and DM- GAN+CL [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' These models are selected for comparison as they are among the top-10 on the COCO Captions Text- to-Image synthesis leaderboard and take significantly dif- ferent approaches for tackling the same task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' As all these models have achieved State-of-the-Art scores on descrip- tive caption datasets, we evaluate how they perform with news domain-specific, abstractive captions in our experi- ments and visualize our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' Text-to-Image Synthesis Models The Lafite model uti- lizes a pre-trained CLIP encoder for translating text em- beddings into the image feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' It adapts an un- conditional StyleGAN2 generator [7] by injecting text- conditional information through affine transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' Two Fully Connected Layers are utilized to transform the input text features to be more semantically similar with StyleGAN’s image Stylespace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' In our experiments, we train Lafite on ANNA in a fully-supervised setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' We train 2 variants of Lafite, with and without Transfer Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' In the non-transfer learning variant, we train it on the ANNA 4 Object Frequency Analysis person chair book cake spoon couch boat 17,069 5,252 2,073 1,179 1,087 1,064 1,001 potted plant carrot bed wine fork 1,964 890 702 627 dining table bench 4,090 cup 868 6op 1,717 clock knife 835 bird bowl 1,669 3,779 cell phone cat bus car orange truck tv 6,155 1,332 780 airplane bottle 2,416 vase donut train 1,183 762 cowFigure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' Visualizing Article Categories of image-caption pairs present in ANNA Model IS (↑) FIDCLIP (↓) LPIPS(↓) CLIPScore (↑) Lafite (Transfer Learning) 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content='49 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content='93 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content='7470 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content='7575 Lafite (Base) 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content='59 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content='7432 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content='7277 DMGAN+CL (512 dim) 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content='07 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content='7568 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content='5913 DMGAN+CL (256 dim) 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content='37 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content='87 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content='7581 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content='5861 AttnGAN+CL (512 dim) 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content='56 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content='7623 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content='5695 AttnGAN+CL (256 dim) 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content='06 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content='7616 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content='5748 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' Results of Abstractive Text-to-Image synthesis on ANNA until convergence for 4000 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' To perform Transfer Learning, we initialize the model with pre-trained weights from the Conceptual Captions (CC3M) dataset [18] and continue training on the ANNA until convergence for 2000 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' The AttnGAN+CL and DMGAN+CL models share sim- ilar architectures, with both utilizing a Deep Attentional Multimodal Similarity Model (DAMSM) for computing the similarity between extracted images and text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' These archi- tectures have been supplemented with a Constrastive Learn- ing Loss function along with their DAMSM loss to improve pre-training performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' We first train the DAMSM mod- ule on the Train and Validation sets of our dataset to con- struct the mapping between image and text features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' We compare 2 different embedding sizes of the DAMSM mod- ule for both models: 256 and 512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' The default AttnGAN and DMGAM models have 256 embedding feature vectors by default, but the CLIP based model Lafite uses 512 em- bedding feature vectors instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' Thus, we train the models with both embedding sizes to ensure a fair comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' Evaluation Metrics To evaluate the performance of these architectures, we report 4 different metrics: Inception Score (IS), Fr´echet Inception Distance (FID), Learned Perceptual Image Patch Similarity (LPIPS) and CLIPScore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' IS and FID evaluate the quality and diversity of generated images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' They estimate probability distribution properties of the generated images and how far it diverges from that of the reference im- ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' For FID, we adapt the proposed FIDCLIP from [8] due to its closer correspondence with human judgement on real-world, diverse datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' LPIPS judges the perceptual similarity between the reference and generated images us- ing deep features extracted across image patches instead of measuring pixel-level similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' We use LPIPS version 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content='1 for our testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' Since LPIPS is an image-wise similarity metric, we report the average of scores obtained by the gen- erated test set images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' CLIPScore is a reference-free metric that can be employed to evaluate the relevance of input text captions to the content of generated images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' We selected these 4 metrics as they provide a holistic evaluation of the different key aspects involved in measuring text-to-image model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' We report our scores in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' 5 Article Categories Metro 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content='646 Dining 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content='010 Business RealEstate 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content='573 Science 12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content='224 National Foreign 2,' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content='(b) Lafite (Transfer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content='Learning) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content='(c) Lafite (Base) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content='(d) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content='DMGAN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content='(512 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content='dim) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content='(e) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content='DMGAN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content='(256 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content='dim) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content='(f) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content='AttnGAN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content='(512 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content='dim) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content='(g) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content='AttnGAN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content='(256 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content='dim) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content='Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' Result Visualization for Caption: The castle, draped with vines and adorned with bougainvillea, is set on 10 acres, with gardens, a swimming pool and a private chapel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' (a) Original Image (b) Lafite (Transfer Learning) (c) Lafite (Base) (d) DMGAN (512 dim) (e) DMGAN (256 dim) (f) AttnGAN (512 dim) (g) AttnGAN (256 dim) Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' Result Visualization for Caption: Pollutants in the Gowanus Canal include pesticides, heavy metals and carcinogens like PCBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' Evaluation of Generated Samples Image Quality From the reported IS and FID scores, we can clearly identify that Lafit with Transfer Learning out- performs all other models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' Although the IS score of the baseline model is lower than that of DMGAN+CL, this trend is reversed in FID scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' This result can be attributed to the fact that the Inception model feature space is aligned to the classes present in ImageNet, hence penalizing other datasets that diverge from this distribution [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' The updated CLIP feature space used for computing FIDCLIP helps mitigate this issue and makes the metric more resistant to fluctuations caused by image preprocessing and distortions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' These results also correlate with observed image quality on other benchmark datasets, such as COCO Captions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' We provide visualizations of generated outputs from the test set for all the trained models in Figures 4, 5, 6, 7, 8, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' Delineation between Content and Context features The Lafite (Transfer Learning) model benefits from learned associations between visual concepts and text represen- tations in the absence of extremely descriptive captions, which corroborates its high CLIPScore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' Similarly, for the other models trained without transfer learning on our dataset, we observe that the LPIPS score and CLIP- Score follow the same trajectory as FIDCLIP with the Lafite (Base) model exhibiting the best correlation between ground truth image similarity and relevance with reference captions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' These results show that the top performing models do have an implicit understanding of what constitutes image content and context information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' But limitations still exist for implicit delineation of captions features, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' With the reference image and descriptive section of the caption dealing with the image of an animal tracking de- vice, the Text-to-Image models incorrectly generate an an- 6 (a) Original Image (b) Lafite (Transfer Learning) (c) Lafite (Base) (d) DMGAN (512 dim) (e) DMGAN (256 dim) (f) AttnGAN (512 dim) (g) AttnGAN (256 dim) Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' Result Visualization for Caption: Left, the New Museum and the original adjacent building it purchased 12 years ago on the Bowery, at right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' (a) Original Image (b) Lafite (Transfer Learning) (c) Lafite (Base) (d) DMGAN (512 dim) (e) DMGAN (256 dim) (f) AttnGAN (512 dim) (g) AttnGAN (256 dim) Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' Result Visualization for Caption: The rooms at the Ace Hotel have high ceilings and oversized windows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' Some of the larger rooms and suites includes details like guitars, turntables and vinyl records.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' imal as the image foreground rather than the tracker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' Thus, comprehension of caption structures and explicit feature de- lineation must be improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' These experiments demon- strate the need for non-descriptive image-captions datasets, such as ANNA for bridging the performance gap between descriptive and abstractive captions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' Discussion and Conclusion Our experiments demonstrate how existing text-to-image architectures understand abstractive captions present in domain-specific data such as news media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' We show that implicit delineation between content and context features have limitations, prompting the need for explicit feature de- lineation and modified objective functions to better suit this task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' One major impact of understanding abstractive cap- tions such as those present in ANNA is the reduction in re- quirements for directly descriptive captioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' As the size of datasets keep increasing, scaling up human annotation of images to match demand adds a huge overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' As descrip- tive captions need to be tightly-coupled with the reference image’s contents, there needs to be multiple rounds of eval- uation and filtering, making it a manually tedious task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' The use of abstractive captions for images can greatly simplify the human annotation process for datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' Additionally, ANNA motivates the development of journalism assistance solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' The use of keywords and descriptive prompts with current image generators involves a lot of prompt en- gineering to get relevant images for a specific topic [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' High quality images are generated only when a particu- larly restrictive sentence structure and vocabulary is used in prompts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' As models are trained to understand abstractive captions, the requirements for intensive prompt engineering would be significantly reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' Similarly, achieving better delineation between different feature types present in non- 7 (a) Original Image (b) Lafite (Transfer Learning) (c) Lafite (Base) (d) DMGAN (512 dim) (e) DMGAN (256 dim) (f) AttnGAN (512 dim) (g) AttnGAN (256 dim) Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' Result Visualization for Caption: The Full Orange: two all-beef patties, special sauce, lettuce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' (a) Original Image (b) Lafite (Transfer Learning) (c) Lafite (Base) (d) DMGAN (512 dim) (e) DMGAN (256 dim) (f) AttnGAN (512 dim) (g) AttnGAN (256 dim) Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' Result Visualization for Caption: With the RoamEO base unit, left (which includes a collar), a dog owner can get radio signals tracking the animal’s location, up to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content='5 miles away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' descriptive captions can also benefit related tasks such as image retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' The addition of context can play a major role in influencing the quality of retrievals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' Limitations This paper aims at introducing the potential of abstractive captions to motivate the development of more contextually-grounded text-to-image synthesis models, par- ticularly when synthesizing news-domain specific images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' Although news articles contain a lot of named-entities, we choose to filter them out and instead focus on context fea- tures that can be inferred from text captions and depicted by general visual concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' Developing text-to-image synthe- sis architectures that can take advantage of named-entities using external knowledge bases as reference would help overcome this limitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' Large-scale human evaluation of images generated by text-to-image architectures on abstrac- tive captions is another important step towards measuring their relative performance, which we aim to perform as a part of our future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' Potential negative societal impacts Image generation ar- chitectures have the potential to be misused for nefarious use-cases such as spreading disinformation [31] and gen- erating neural fake news [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' Our current preprocessing pipeline removes most images containing named-entities, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' public figures and locations of national importance, con- tributing towards risk mitigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' However, we recognize the threat posed by contextually-relevant Deepfake images when dealing with news media images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' Future research di- rections include understanding the extent up to which text- to-image models can be used for neural fake news genera- tion and identifying appropriate detection strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' Acknowledgements This research has been partially supported by NSF Awards #1820609 and #2114824.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dA0T4oBgHgl3EQfOf8M/content/2301.02160v1.pdf'} +page_content=' 8 References [1] Soravit Changpinyo, Piyush Sharma, Nan Ding, and Radu Soricut.' metadata={'source': 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b/8tE1T4oBgHgl3EQfngT9/content/tmp_files/2301.03311v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..c51e3d1d65393e47c5e1a34983366a901480bbe8 --- /dev/null +++ b/8tE1T4oBgHgl3EQfngT9/content/tmp_files/2301.03311v1.pdf.txt @@ -0,0 +1,2053 @@ +Astronomy & Astrophysics manuscript no. main_new +©ESO 2023 +January 10, 2023 +Unbound stars hold the key to young star cluster history +Arunima Arunima1,2, Susanne Pfalzner1,2,3, and Amith Govind1,2 +1 Jülich Supercomputing Center, Forschungszentrum Jülich, 52428 Jülich, Germany +e-mail: s.pfalzner@fz-juelich.de +2 Physics Department, University of Cologne, Cologne, Germany +3 Max Planck Institute for Radio Astronomy, Auf dem Hügel 69, 53121 Bonn, Germany +Received ... +ABSTRACT +Aims. GAIA delivers the positions and velocities of stars at an unprecedented precision. Therefore, for star clusters, there exists much +higher confidence in whether a specific star is a member of a particular cluster or not. However, membership determination is still +especially challenging for young star clusters. At ages 2–10 Myr, the gas is expelled, ending the star formation process and leading to +their expansion, while at the same time, many former members become unbound. As a first step, we aim to assess the accuracy of the +methods commonly used to distinguish between bound and unbound cluster members; after identifying the most suitable technique +for this task, we wish to understand which of the two populations is more suited to provide insights into the initial configuration and +the dynamical history of a cluster starting from its currently observed properties. +Methods. Here, we perform N-body simulations of the dynamics of such young star clusters. We investigate how cluster dynamics +and observational limitations affect the recovered information about the cluster from a theoretical perspective. +Results. We find that the much-used method of distance and velocity cutoffs for membership determination often leads to false +negatives and positives alike. Often observational studies focus on the stars remaining bound. However, bound stars quickly lose the +memory of the pre-gas expulsion phase due to their ongoing interaction with their fellow cluster members. Our study shows that it +is the unbound stars that hold the key to charting a cluster’s dynamic history. Backtracking unbound stars can provide the original +cluster size and determine the time of gas expulsion – two parameters that are currently still poorly constrained. This information +is lost in the bound population. In addition, former members are often better indicators for disc lifetimes or initial binary fractions. +We apply the backtracking analysis, with varying success, to the clusters: Upper Scorpius and NGC 6530. For highly substructured +clusters such as Upper Scorpius, backtracking to the individual subcluster centres will provide better results in future. +Key words. stars: formation – open clusters and associations: general – ISM: clouds – solar neighbourhood +1. Introduction +Star clusters are the nurseries for most stars (Porras et al. 2003; +Lada & Lada 2003). As such, young star clusters play a vital role +in our understanding of how young stars form and develop. They +signify the starting point for all that happens later on, as they pro- +vide the initial stellar mass distribution (e.g. Kroupa 2002) and +the fraction of stars forming as a single-, binary-, or multiple-star +system (e.g. Duchêne et al. 2018). It is a standard procedure to +use properties of clusters of different ages to obtain information +on the dynamical development of young binary stars or the dis- +persal time of discs (e.g. Haisch et al. 2001; Ansdell et al. 2017; +Marks et al. 2014; Ribas et al. 2014; Richert et al. 2018; Michel +et al. 2021). Often the task of determining cluster membership +and deriving the temporal development of specific properties are +separate endeavours. While distinguishing members is a chal- +lenge in itself, any bias in membership determination (i.e. false +positives and false negatives) feeds through to the derived pa- +rameters used in other applications. +This study’s central aim is to utilise cluster dynamics simula- +tions to optimise the data used to determine a cluster’s past. Un- +til recently, the role of dynamics during the formation history of +young clusters was highly uncertain (e.g. Elmegreen 2000; Fujii +et al. 2012; Ward et al. 2012; Banerjee & Kroupa 2017; Dib et al. +2018), mainly because observational limitations hampered pre- +cise velocity determination. The precision of data coming from +the Gaia satellite (Gaia Collaboration et al. 2016, 2018, 2021) +helped shed light on this issue since a complete understanding +of the dynamical evolution of present-day clusters has not been +attained yet. Investigating a sample of 28 clusters and associa- +tions with ages ≈ 1–5 Myr, Kuhn et al. (2019) found that at least +75% of these systems are expanding at typical expansion veloc- +ities of the order of ≈ 0.5 km s−1. Cluster expansion was pre- +dicted by the gas expulsion scenario (Mathieu 1983; Lada et al. +1984; Adams 2000; Kroupa et al. 2001; Baumgardt & Kroupa +2007; Pelupessy & Portegies Zwart 2012; Pfalzner & Kaczmarek +2013; Brinkmann et al. 2017; Pfalzner & Govind 2021). During +the star formation phase, the stars are embedded in the gas and +dust reservoir from which they are forming. However, after ap- +proximately 1–2 Myr, the gas starts to be expelled from the clus- +ters by various mechanisms (e.g. Krumholz et al. 2019; Fujii +et al. 2021). Due to loss in gas and dust mass, the system is no +longer in equilibrium. Therefore, a considerable portion of the +stars, bound in the embedded phase, become unbound in the gas +expulsion phase. +The three-dimensional information available from the Gaia +data has been a tremendous step forward in this field. Neverthe- +less, discriminating the members of star clusters and associations +from the foreground and background population is still challeng- +ing (Gagné et al. 2018). Many new methods have been devel- +oped for determining the members of open and globular clusters +(e.g. Sollima et al. 2019; Garro et al. 2021; Vitral 2021). Cluster +Article number, page 1 of 14 +arXiv:2301.03311v1 [astro-ph.GA] 9 Jan 2023 + +A&A proofs: manuscript no. main_new +membership determination is challenging in the early expansion +phase (< 10 Myr), especially if a clear-cut distinction between +currently bound and formerly bound (i.e. unbound) members is +required. In this case, there are additional difficulties to over- +come compared to older clusters. First, the earliest stages of the +formation of star clusters are hidden from view by gas and dust. +Thus, at this young age, veiling is a severe problem. Second, the +young clusters’ expansion requires additional attention in mem- +bership determination. Third, short- and long-lived clusters co- +exist during a 10 Myr timespan (Lada & Lada 2003). They un- +dergo very different cluster dynamics (Pfalzner & Kaczmarek +2013), and it is not always straightforward whether a specific +cluster will remain bound for a long time or not. +Here, we concentrate on these dynamical aspects of young +short-lived clusters1. Any cluster observation is just a snapshot +in time of the sequence of its dynamical evolution. Based on +simulations of the cluster dynamics, we show the importance of +cluster dynamics in membership determination. We investigate +the efficiency of backtracking cluster expansion and find that dis- +tinguishing between bound and unbound stars in the expansion +phase is vital. Finally, we show that the unbound stars hold the +key to determining a cluster’s past. +2. Cluster observation techniques +Historically, star clusters have been identified visually as stel- +lar density enhancements (Dreyer 1888; Trumpler 1930; Bailey +1908; Collinder 1931). Surveys like Hipparcos (Perryman et al. +1997), 2MASS (Skrutskie et al. 2006), and Gaia have each in- +creased the samples by hundreds of candidate clusters. Due to +Gaia’s high-precision parallax measurements, the clustering of +stars can be analysed in a higher dimensional space by combin- +ing their positions in the sky, proper motions, parallaxes, and +radial velocities (when available). For studies which do auto- +mated blind searches with clustering algorithms, the youth of the +stars is used as a confirmation of membership. Such youth indi- +cators can be X-ray activity, infrared excess (Broos et al. 2013; +Feigelson et al. 2013; Getman et al. 2017), lithium abundance +(Soderblom 2010), and gravity-sensitive spectral indices such +as TiO molecular lines (Wilking et al. 2005), empirically con- +structed spectral indices (Damiani et al. 2014), or the shape of +the H-band peak (Scholz et al. 2009). +Among the clustering algorithms, one can distinguish differ- +ent classes: Density-based spatial clustering like DBSCAN (Es- +ter et al. 1996; Wilkinson et al. 2018; Zari et al. 2019; Castro- +Ginard et al. 2019, 2020, 2022; Hunt & Reffert 2021), HDB- +SCAN (Campello et al. 2013), and OPTICS (Ordering Points +To Identify the Clustering Structure; Ankerst et al. 1999), mul- +tidimensional Gaussian-based methods (Vasiliev 2019; Cantat- +Gaudin et al. 2019; Kuhn et al. 2020), k-means clustering (Mac- +Queen 1967; Hunt & Reffert 2021), and Friend of Friend algo- +rithm (FoF; Liu & Pang 2019). In addition, there exist several un- +supervised algorithms like UPMASK (Krone-Martins & Moit- +inho 2014; Cantat-Gaudin et al. 2018; Cantat-Gaudin & Anders +2020), the nearest neighbour-based method by He et al. (2021), +and STARGO (Tang et al. 2019; Zhang et al. 2020; Pang et al. +2020). +1 The nomenclature of short-lived clusters is not unequivocal. While +referred to as clusters while embedded, they are often classified as as- +sociations when the gas is expelled, and most of their stars become un- +bound. Here, we refer to short-lived clusters as clusters and point out +expressly when talking about long-lived clusters, that is, open and glob- +ular clusters. +Young star clusters pose additional challenges compared to +open or globular clusters due to their highly dynamic nature +after gas expulsion. Although space velocity is used to iden- +tify clusters, algorithms rarely consider dynamics. Observations +only provide a snapshot in the dynamic evolution of the clus- +ter. Hence, even clustering in the velocity space at the present +moment might be a chance alignment as the velocity changes +rapidly in young star cluster members. More limitations in iden- +tifying clusters come from Gaia’s poor completeness in crowded +fields and no particular regard for binarity. Moreover, young +clusters are still embedded in natal gas and dust that can not be +penetrated by optical wavelengths, which presents another diffi- +culty in identifying and analysing young clusters. +Blaauw (1964) first gave the notion of linear expansion in +associations, assuming that all members move away from their +birthplace without any forces acting on them. Then, the recipro- +cal of the expansion coefficient can provide an estimate of the as- +sociation’s kinematic age. Alternatively, the individual motions +of the stars can be traced back until they reach the smallest con- +figuration at a past time, and the kinematic age, as well as the +initial configuration of the association, can be possibly obtained +(Blaauw 1978). +Most studies apply cutoffs to remove objects with low- +quality astrometry and outliers. The sigma-clipping method aims +to reduce the chances of contaminants or uninformative stars and +improve clusters’ signal-to-noise ratio (S/N). Alternatively, out- +liers can be modelled in the fitting procedure without rejecting +points a priori (see Hogg et al. 2010). +Before Gaia, the significant errors in astrometry and the low +number of confirmed members with available radial velocities +were the main hindrances in the analysis (Fernández et al. 2008). +The higher precision of the Gaia data allows for better trace- +back analysis. For example, recent studies by Heyl et al. (2022, +2021) trace back the stars of clusters aged 40–200 Myr using +Gaia EDR3 data and determine their kinematic ages. Similarly, +Schoettler et al. (2022) trace back runaway (RW) and slower +walkaway (WW) stars within a distance of 100 pc of NGC 2264 +to the three subclusters S Mon, IRS 1 and IRS 2. The study by +Ma et al. (2022) uses Gaia DR2 data to trace back (and extrapo- +late) the trajectories of members of the Scorpius-Centaurus (Sco- +Cen) association and find evidence of past and future close stellar +flybys. +Observational challenges like distinguishing the cluster pop- +ulation from the back and foreground stars, limiting magnitudes, +imprecision of derived properties like age and mass, etc., com- +plicate backtracking. Here we apply backtracking to snapshots +in the simulations of the cluster dynamics. Under these idealised +conditions, membership is certain, the exact positions and ve- +locities of the stars are known at all times, and last, but not least, +we know what the result should be. This certainty allows us to +determine the most expedient method and suggest measures to +optimise the backtracking technique. +3. Cluster simulation method +We use a sub-set of simulations of the dynamics of clusters +containing N stars we performed recently (Pfalzner & Govind +2021), using the simulation code NBODY6++GPU (Aarseth 2003). +The simulations try to represent the situation in real clusters as +closely as possible by adopting initial conditions backed by re- +cent observations and following the observed cluster expansion +derived from the sizes of clusters in the age range of 1–10 Myr. +Here we give only a summary of the assumptions, and the nu- +merical method we applied in Pfalzner & Govind (2021), as the +Article number, page 2 of 14 + +Arunima Arunima et al.: Unbound stars hold the key to star cluster history +actual choice of simulation parameters is uncritical for the gen- +eral challenges in membership determination and backtracking +of the cluster history. +We model the dynamics of the young clusters covering all +the phases: Starting from the embedded phase, we simulate the +subsequent gas expulsion that leaves the cluster in a super-virial +state and results in the cluster expanding until it reaches a new +equilibrium. It is assumed that all stars are already formed and +that the gas expulsion occurs at temb = 2 Myr. Observations in- +dicate that the entire gas expulsion process takes ≈ 1 – 2 Myr +(Kuhn et al. 2019). Simulations investigating the dependence of +the cluster dynamics on the gas expulsion time found that the +gas expulsion can be modelled as being instantaneous (Geyer & +Burkert 2001; Portegies Zwart et al. 2010). Stellar evolution has +not been included in this work as it has little influence on the +results. +We analyse the dynamics of clusters with different numbers +of cluster members N. The corresponding clusters’ masses Mc +and sizes, illustrated by their half-mass radius rhm, are given in +Table 1. Low-mass clusters are usually smaller than high-mass +clusters of the same age (Lada & Lada 2003; Adams 2010; +Pfalzner et al. 2016). This relation between the cluster’s mass +and its half-mass radius can be approximated by a power law: +Mc = Crhm +γ. +(1) +The values of the constant C and scaling exponent γ differ in +different observational studies due to the involved observational +uncertainties. The clusters’ sizes given in Table 1 are based on +the mass-radius relation by Pfalzner et al. (2016) where C = +717.794 and γ = 1.7 ± 0.2. We assume that the star formation +efficiency in the system is 30 % (Lada & Lada 2003), which +sets the gas mass. The gas and dust component of the embedded +phase is implemented as a background potential. +In our simulations, a test particle represents a star with a +given mass, position, and velocity. The particles’ positions are +chosen so that the resulting stellar number density distribution +obeys a King profile with King parameter, W0 = 9 (King 1966a). +The King model is an empirical law that can not be defined ana- +lytically. It consists of an energy distribution function of the form +fK(E) = +�ρ1(2πσ2 +K)−3/2(eE/σ2 +K − 1) +: E > 0, +0 +: E ≤ 0, +(2) +with E = Ψ− 1 +2ν2 and Ψ = −Φ+Φ0 being the relative energy and +relative potential of a particle, respectively. Also, f(E) > 0 for +E > 0 and σK is the King velocity dispersion. The profiles are +characterised by the King parameter W0 = Ψ/σ2 +K, an increase of +which signifies decrease in the relative size of the cluster core +Table 1. Initial cluster parameters for the simulation campaign using +mass-radius dependencies. +N +Nsim +Mc +[M⊙] +rhm +[pc] +Mt +[M⊙] +temb +[Myr] +200 +1941 +117.99 +0.26 +393.31 +2.0 +1000 +497 +589.97 +0.67 +1966.57 +2.0 +4000 +127 +2359.88 +1.3 +7866.27 +2.0 +Notes. Here N denotes the number of cluster members, Nsim the number +of simulations, temb the duration of the embedded phase, Mc the stellar +mass of the cluster, rhm the half-mass radius, and Mt the total cluster +mass (stars + gas). +rc/rhm. Observationally, determining the stellar density distribu- +tion of young star clusters can be challenging but it has been +found that young clusters are best represented by King model +with W0 ≥ 7 (Hillenbrand & Hartmann 1998; Nürnberger & +Petr-Gotzens 2002). The choice of W0 mainly affects the size +of the central high-density area. Hence, the number of expelled +stars also depends on the choice of W0. Even for a relatively steep +W0 = 9-potential, the number of escapers is < 1%. Therefore, the +conclusions about membership determination methodology are +unaffected by the choice of potential. The individual test par- +ticles are assigned masses following the initial mass function +(IMF) by Kroupa (2002), with the lower mass limit set to 0.08 +M⊙ (hydrogen-burning limit) and an upper mass limit of 150 +M⊙. Potentially existing initial mass segregation in the clusters +is neglected. The cluster members are given velocities following +a Maxwellian distribution. We assume that the cluster is initially +in virial equilibrium. +We perform (Nsim) simulations for every cluster mass, where +the actual distribution of the stars depends on the seed selected in +the randomised procedure. We analyse all the simulation results +in this statistical study. However, why a specific method works +or fails, we illustrate exemplarily for just one specific randomly +chosen realisation in Figs. 1 – 3. Figures 6 – 8 also show the +method applied to randomly chosen specific clusters for visual +understanding; however, statistical results are mentioned in the +text. +For simplicity, we exclude primordial binaries, modelling all +cluster stars as initially being single stars. The absence of pri- +mordial binaries can lead to underestimating ejections from the +cluster centre (Heggie 1975). However, in most clusters, ≪1% +of the stars are affected (Olczak et al. 2006). +4. Results +Observations investigate one specific cluster at a snapshot of its +development. Mimicking this observational situation, we ran- +domly choose one of our sets of simulations and investigate it +at a specific time. However, unlike actual observations, we have +complete temporal information available. Hence, we know the +past and the future of this particular cluster down to the path of +each star. Equally, all other observational challenges, like mem- +bership uncertainty due to back and foreground populations and +limiting magnitudes, are removed. We even know each star’s ex- +act properties like its mass, position, and velocity. This informa- +tion allows us to investigate the fundamental and unavoidable +challenges in backtracking caused by the cluster dynamics that +exist even without the mentioned additional observational diffi- +culties. +4.1. Bound and unbound stars +After gas expulsion, bound and unbound stars coexist in the +same spatial area for some time. Distinguishing the two popu- +lations is vital for some applications; it does not matter or is not +even desirable for others. An example of the latter is the use of +clusters in determining disc lifetimes (Haisch et al. 2001). Here, +it is best to identify all stars that once formed together in the clus- +ter. However, if one is interested in the long-term development +of clusters (≫ 20 Myr), one would be predominantly interested +in the portion of stars that remain bound. We subsequently see +here that using backtracking to distinguish between bound and +unbound stars after gas expulsion is the key to success in ob- +taining valuable information concerning a cluster’s past. At each +Article number, page 3 of 14 + +A&A proofs: manuscript no. main_new +6 +4 +2 +0 +2 +4 +6 +x[pc] +6 +4 +2 +0 +2 +4 +6 +y[pc] +(a) +6 +4 +2 +0 +2 +4 +6 +x[pc] +6 +4 +2 +0 +2 +4 +6 +y[pc] +(b) +6 +4 +2 +0 +2 +4 +6 +x [pc] +6 +4 +2 +0 +2 +4 +6 +y [pc] +(c) +30 +20 +10 +0 +10 +20 +30 +x [pc] +30 +20 +10 +0 +10 +20 +30 +y [pc] +(d) +6 +4 +2 +0 +2 +4 +6 +x [pc] +6 +4 +2 +0 +2 +4 +6 +y [pc] +(e) +6 +4 +2 +0 +2 +4 +6 +x [pc] +6 +4 +2 +0 +2 +4 +6 +y [pc] +(f) +6 +4 +2 +0 +2 +4 +6 +x [pc] +6 +4 +2 +0 +2 +4 +6 +y [pc] +(g) +6 +4 +2 +0 +2 +4 +6 +x [pc] +6 +4 +2 +0 +2 +4 +6 +y [pc] +(h) +Fig. 1. Snapshot of the positions and velocities of example simulations +with N = 200. Velocity vectors of bound stars are highlighted in blue, +and those of unbound stars in red. Counter-intuitive examples of (a) +outward-pointing distant bound stars and (b) inward-pointing central +unbound stars. Snapshot of the temporal development at (c) t=2 Myr +and (d) t=10 Myr. Backtracking from the results at 10 Myr to 2 Myr +considering only the stars within 6 pc from the cluster centre for (e) +bound stars only and (f) unbound stars only. Same backtracking con- +sidering all the (g) bound stars and (f) unbound stars of the cluster. +A film of the cluster dynamics and the backtracking can be found at +https://doi.org/10.5281/zenodo.6041920 +snapshot of the simulations, bound and unbound stars are de- +fined as those having positive and negative total energy respec- +tively. However, in observations, distinguishing between these +two states is often not straightforward. +4.1.1. Velocity vectors +Individual stars are sometimes classified as bound or unbound +simply because their velocity vectors point towards or away from +the cluster centre. In the past, doubts about this approach were +usually anchored on the fact that only two-dimensional informa- +tion was available. However, even with three-dimensional infor- +mation becoming more accurate, this method is not advisable +even for perfectly known 3D velocities for the following reason: +The top row of Fig. 1 shows a typical snapshot of a randomly +chosen example from our sample of simulated clusters. The clus- +ter centre is marked as a green dot as a reference point. As the +many outward-pointing velocity vectors indicate, this cluster is +in the expansion phase, with many former members becoming +unbound. Nevertheless, a considerable fraction of the outward- +pointing velocity vectors belongs to stars that remain bound in +the long term. Examples of such stars are shown in blue. Equally, +stars that point inwards and are close to the cluster centre can +nevertheless be unbound (shown in red). The dynamics of these +example stars can be seen better in the corresponding video +at https://doi.org/10.5281/zenodo.6041920. Especially +among the bound stars with outward-pointing velocity vectors, +quite a few are bound despite being located at relatively large +distances from the cluster centre. We find that there is a high +failure rate in this approach, not only for this specific cluster, but +for all clusters in our extensive sample. The situation improves +for clusters aged more than 15 Myr as many of the unbound stars +are better identifiable by their larger distances to the cluster cen- +tre. +4.1.2. Advantage of using unbound stars for backtracking +The size of a cluster before expansion sets in is an essential pa- +rameter for constraining the cluster formation process. Besides +the density profile, the size of the cluster core and half-mass +radius are good indicators of the cluster density and, thus, the +importance of the environment in the star and planet formation +process. The environment’s influence includes close stellar fly- +bys and external photo-evaporation that can truncate protoplan- +etary discs or completely destroy them (Vincke et al. 2015; Win- +ter et al. 2018; Concha-Ramírez et al. 2019). These processes +influence the type and frequency of the formed planetary sys- +tems. Another example is binary capture and destruction pro- +cesses which can alter the binary fraction in clusters (Kaczmarek +et al. 2011; Marks et al. 2014; Guszejnov et al. 2022). +We find that using just the unbound stars gives the best re- +sult in determining the pre-expansion cluster size. As an exam- +ple, the second row in Fig. 1 illustrates the cluster expansion by +showing the bound and unbound stars, including their velocity +vectors, (a) shortly after gas expulsion and (b) at 10 Myr for a +cluster with N = 200. We note the different scales. Using only the +bound stars for backtracking (see Fig. 1g) results in a relatively +poor constraint on the pre-expansion size. The best performance +is obtained using only the unbound stars (see Fig. 1f). The rea- +son is twofold: First, the velocity vectors of the unbound stars +are rarely altered after gas expulsion. By contrast, bound stars +quickly lose the memory of the pre-gas expulsion phase due to +their ongoing interaction with their fellow cluster members. In +particular, close encounters hinder efficient backtracking for the +bound stars. Second, there is a more significant number of un- +bound than bound stars. Thus, statistical uncertainties are more +easily averaged out. +Figure 4 gives a more quantitative idea of the use of bound vs +unbound stars for backtracking and deriving the pre-expansion +Article number, page 4 of 14 + +Arunima Arunima et al.: Unbound stars hold the key to star cluster history +0 +10 +20 +30 +40 +50 +60 +70 +d [pc] +0.0 +0.2 +0.4 +0.6 +0.8 +Frequency +Time= 1.8 Myr +0 +10 +20 +30 +40 +50 +60 +70 +d [pc] +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +Frequency +Time= 2.3 Myr +0 +10 +20 +30 +40 +50 +60 +70 +d [pc] +0.000 +0.025 +0.050 +0.075 +0.100 +0.125 +0.150 +0.175 +0.200 +Frequency +Time= 5.0 Myr +0 +10 +20 +30 +40 +50 +60 +70 +d [pc] +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +0.12 +Frequency +Time= 10.0 Myr +0 +10 +20 +30 +40 +50 +60 +70 +d [pc] +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +Frequency +Time= 20.0 Myr +0 +1 +2 +3 +4 +5 +6 +v [km/s] +0.00 +0.02 +0.04 +0.06 +0.08 +Frequency +Time= 1.8 Myr +0 +1 +2 +3 +4 +5 +6 +v [km/s] +0.00 +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 +0.07 +0.08 +Frequency +Time= 2.3 Myr +0 +1 +2 +3 +4 +5 +6 +v [km/s] +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +Frequency +Time= 5.0 Myr +0 +1 +2 +3 +4 +5 +6 +v [km/s] +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +0.12 +Frequency +Time= 10.0 Myr +0 +1 +2 +3 +4 +5 +6 +v [km/s] +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +Frequency +Time= 20.0 Myr +0.3 +1.0 +2.0 +5.0 10.0 20.0 40.0 80.0 +d [pc] +0.1 +0.2 +0.5 +1.0 +2.0 +4.0 +8.0 +v [km/s] +Time= 1.8 Myr +(a) +0.3 +1.0 +2.0 +5.0 +10.0 20.0 40.0 80.0 +d [pc] +0.1 +0.2 +0.5 +1.0 +2.0 +4.0 +8.0 +v [km/s] +Time= 2.3 Myr +(b) +0.3 +1.0 +2.0 +5.0 +10.0 20.0 40.0 80.0 +d [pc] +0.1 +0.2 +0.5 +1.0 +2.0 +4.0 +8.0 +v [km/s] +Time= 5.0 Myr +(c) +0.3 +1.0 +2.0 +5.0 +10.0 +20.0 +40.0 +80.0 +d [pc] +0.1 +0.2 +0.5 +1.0 +2.0 +4.0 +8.0 +v [km/s] +Time= 10.0 Myr +(d) +0.3 +1.0 +2.0 +5.0 +10.0 +20.0 +40.0 +80.0 +d [pc] +0.1 +0.2 +0.5 +1.0 +2.0 +4.0 +8.0 +v [km/s] +Time= 20.0 Myr +(e) +Fig. 2. Snapshot of distance (top) and velocity distribution (middle), and distance vs velocity scatter plot (bottom) (a) before gas expulsion (t = +1.8 Myr), (b) just after gas expulsion (t = 2.3 Myr), (c) at t = 5 Myr, (d) at t = 10 Myr, and (e) at the end of our simulation (t =20 Myr). All plots +show the bound stars in blue and the unbound stars in red. A simulation of N = 1000 stars is used here. +. +0.3 +1.0 +2.0 +5.0 +10.0 20.0 40.0 +d [pc] +0.1 +0.2 +0.5 +1.0 +2.0 +4.0 +v [km/s] +Time= 10.0 Myr +Fig. 3. Phase space diagram for an N = 1000 star cluster simulation at +t = 10 Myr. The bound and unbound members are shown in blue and +red colours respectively. Vertical and horizontal red lines indicate dis- +tance and velocity cutoffs respectively for unbound stars. The light blue +line represents the analytical escape velocity dependence on distance +from the cluster centre derived assuming a Plummer distribution for the +members. The black crosses show the stars that underwent a strong en- +counter. +cluster size. All the simulations of N = 1000 cluster have been +used to obtain these distributions. It can be seen that the size +distribution obtained using unbound stars is closer to the real size +distribution than the size distribution obtained using bound stars. +Performing a t-test on the two size distributions with the null +hypothesis being that the distributions have the same mean— +while the alternative hypothesis is that bound stars have a larger +mean than unbound stars—results in a p-value much lower than +the significance level α = 0.01. Hence, unbound stars are clearly +better at recovering the size of the cluster before gas expulsion +than bound stars. +4.1.3. Distance and velocity cutoffs for bound-unbound +classification +While distinguishing between the bound and unbound popula- +tion is straightforward in simulations, it is very challenging in +observations. Often a cut in the distance to the cluster centre or +the velocity is used to distinguish between bound and unbound +stars. Here we want to test when such a method is successful. +In our simulation, the relevant time frame starts at 2 Myr, +when the gas expulsion happens, and many stars become un- +bound. Figure 2 shows snapshots of the distributions of the stel- +lar distance to the cluster centre and velocity distribution before +(1.8 Myr), just after gas expulsion at 2.3 Myr, during the expan- +sion process (5 and 10 Myr) and towards the end (20 Myr) of the +expansion phase for an example cluster. The distributions for the +bound (blue) and unbound (red) stars are shown separately. As +we chose the cluster to be in virial equilibrium, very few stars +become unbound before gas expulsion (see Fig. 2a). The few +unbound stars during this phase result from close encounters +leading to ejections. However, after gas expulsion, many stars +become unbound. Bound and unbound stars share considerable +parts of the phase space for quite some time, as seen in the bot- +tom row of Fig. 2. This increases the complexity of making the +distinction. +In observations, usually, a velocity cutoff is chosen as a given +deviation from the mean for making this distinction (e.g. Luh- +man 2018; Bastian 2019; Esplin & Luhman 2019). However, the +location of these cutoffs is not apparent. Thus, there is some ele- +Article number, page 5 of 14 + +A&A proofs: manuscript no. main_new +0 +1 +2 +3 +4 +5 +Half-mass radius [pc] +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +Real: +r = 0.67, +r = 0.13 +Unbound: +u = 1.47, +u = 0.12 +Bound: +b = 2.70, +b = 0.61 +Real +Unbound +Bound +1 +2 +3 +4 +5 +Half-mass radius [pc] +Fig. 4. Distributions of sizes derived using actual positions of all stars +(Real, shown in green), using backtraced positions of unbound stars +(Unbound, shown in orange), and using backtraced positions of bound +stars (Bound, shown in blue) shown with histograms (top) and boxplots +(bottom). The box extends from the lower to upper quartile values of +the data, with a line at the median while the whiskers reach 1.5 times +the interquartile range from the box. +ment of arbitrariness here, and this is even more so for distance +cutoffs. However, in our simulations, we are in the ideal situation +where we can determine where to apply the cutoff in distance +and velocity. These experiences can be used to provide guide- +lines for both types of cutoffs. Figure 5 shows suggestions for +the choice of distance and velocity cutoff for clusters older than +5 Myr. These have been calculated to minimise the sum of the +false positive rate (FPR) and false negative rate (FNR) for all the +simulations. +It does not make much sense to make distance and velocity +cutoffs in clusters younger than at least 5 Myr to avoid substan- +tial errors in the classification of the members. However, even +at 5 Myr, the FPR and FNR introduced by a cutoff can be of the +order of 15% – 30%. Generally, the percentage of stars identified +as bound members while being unbound is higher than the oppo- +site situation. Only for clusters older than 10 Myr, this method is +relatively robust as the overlap in phase space is of the order of +5% – 10%. Figure 3 shows the phase space diagram for a simu- +lated cluster of 1000 stars with red lines at a distance of 8.09 pc +and a velocity of 0.78 km/s representing the distance and veloc- +ity cutoffs shown in Fig. 5. Applying these to the distribution of +all simulations of 1000 stars leads to a median FNR of 9.7%. The +25th and 75th percentile of the distribution of FNR are 7.5% and +11.4%, respectively. We represent this as an FNR of 9.7+1.7 +−2.2%. +Similarly, an FPR of 0 ± 0% is obtained. The percentage of cor- +rectly identified stars is found to be 94.1 ± 1.1%. +Combining distance and velocity cutoffs gives the best dis- +tinction. This can be done by analytically determining the de- +pendence of the escape velocity of the stars on the distance from +the cluster’s centre. Although the distribution of the stars in the +simulations follows a King (1966b) profile, we use an approxi- +mation of a Plummer (1911) profile to obtain an analytical solu- +tion. The escape velocity vesc(r) at any point in the cluster is then +described by +vesc(r) = +� +2GMcl +√ +a2 + r2 , +(3) +where Mcl is the cluster mass, and a is the initial half-mass +radius. This analytical cutoff can be seen in Fig. 3 as the blue +curve. Applying this as the cutoff for bound-unbound star dis- +tinction leads to an FPR of 0.74+0.91 +−0.47% and an FNR of 4.80+0.88 +−0.10%. +The median of the distribution of the correctly identified stars’ +percentage is found to be 96.7+0.5 +−0.7%. Hence, this analytical cutoff +is an improvement over the distance and velocity cutoffs in the +case of our simulations. +4.2. Backtracking +In the following, we use our simulations of the cluster dynam- +ics to develop guidelines for backtracking depending on cluster +type, age, and mass. We subsequently demonstrate that using the +right subset of stars for backtracking is the key to making the +most of the available information. Here, we employ the simplest +form of backtracking, namely, taking present-day positions and +velocities as constant values and just reversing the arrow of time +(i.e. neglecting any source of acceleration acting upon the stars). +The high quality of the recent Gaia data allows backtrack- +ing from the observed present situation holding the promise to +reveal information about a cluster’s past. So far, unbound stars +are chiefly analysed as ‘runaway’ (v > 30 km/s) stars and ‘walk- +away’ (5 km/s < v < 30 km/s) stars (Eldridge 2011; Schoettler +et al. 2020). The idea is that both types of high-velocity stars +have been ejected from their star-forming regions, and back- +tracking will allow us to determine their origins and characterise +their parent star cluster (e.g. Olczak et al. 2008; Farias et al. +2020; Schoettler et al. 2022). Schoettler et al. (2022) search for +runaway and walkaway stars within 100 pc of the 3–5 Myr old +cluster NGC 2264 using Gaia DR2. They compare the num- +ber of the runaway and walkaway stars (17) to a range of N- +body simulations with different initial conditions and find con- +sistency with initial conditions with a high initial stellar density +(≈ 10 000 M⊙ pc−3) and a high initial amount of spatial substruc- +ture. +However, our simulations find that high-velocity ejec- +tions are rare for short-lived clusters. We found no ejections +with v > 30 km/s and only a few with v > 5 km/s. Thus, back- +tracking based on runaway and walkaway stars suffers from low- +number statistics for young clusters (< 20 Myr) typical for the +solar neighbourhood. As the ejection happens mainly from the +highest-density regions of the cluster, the derived age at gas ex- +pulsion is too short, and the cluster size is also too small. For the +much denser clusters that turn into long-lived open clusters, the +Article number, page 6 of 14 + +Arunima Arunima et al.: Unbound stars hold the key to star cluster history +n200 +n1000 +n4000 +0 +2 +4 +6 +8 +10 +12 +14 +16 +Distance cutoff [pc] +n200 +n1000 +n4000 +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +2.00 +Velocity cutoff [km/s] +Fig. 5. Distance (top) and velocity (bottom) cutoffs for selection of un- +bound members for clusters with different number of members: N = +200, 1000, 4000. The box extends from the lower to upper quartile val- +ues of the data, with a line at the median while the whiskers reach 1.5 +times the interquartile range from the box. +backtracking of cluster sizes is of higher quality as the number of +ejected stars is higher and the ejection happens over larger areas +of the cluster (Pfalzner & Kaczmarek 2013). +4.2.1. Pre-expansion cluster size +Using our simulation results as a starting point for backtracking, +we find that the restriction to the unbound stars gives the best +result in determining the pre-expansion cluster size. This can be +seen clearly in Fig. 6 (top panel), where backtracked half-mass +radius has been plotted against time. Backtracking the bound +members provides no information, whereas using just unbound +members fares much better. It recovers the half-mass radius (rhm) +of the cluster at the time of gas expulsion with a relative error of +121.4+16.3 +−15.0% to the relative error of 298.9+48.1 +−46.7% obtained using +bound members. +It is equally important to include the unbound stars from a +sufficiently large area. Fig. 6 (bottom panel) shows a compari- +son of the backtracked half-mass radius determined by consid- +ering different areas for the member sampling. The horizontal +lines show the derived pre-gas expulsion half-mass radii. It can +be seen that the half-mass radius derived from the unbound stars +sampled from a relatively small area (10 pc) results in a consider- +ably larger error than those derived from including the unbound +0 +2 +4 +6 +8 +10 +Time [Myr] +0.0 +2.5 +5.0 +7.5 +10.0 +12.5 +15.0 +17.5 +20.0 + Half mass radius [pc] +Bound +Unbound +1.80 Myr, 1.40 pc +2 Myr, 0.74 pc +0 +2 +4 +6 +8 +10 +Time [Myr] +0 +2 +4 +6 +8 +10 +12 + Half mass radius [pc] +1.72 Myr, 1.19 pc +1.59 Myr, 1.41 pc +1.27 Myr, 1.49 pc +2 Myr, 0.82 pc +Fig. 6. Backtracked half-mass radii for a simulation with 1000 stars, +Top: using bound (blue) and unbound (red) members only. Red dashed +lines show temb and rhm at the time of gas expulsion determined using +unbound stars whereas black dashed lines show the actual values of the +same. Bottom: using unbound stars within 10 pc (blue), 20 pc (red) and +40 pc (green) from the cluster centre. The actual values of temb and rhm +at the time of gas expulsion are shown in cyan. +stars from larger areas. In relative error terms, the error decreases +from 248.9+41.6 +−27.4% to 149.1+16.1 +−14.2% to finally, 121.4+16.3 +−15.0% as the +search area around the cluster centre increases from 10 pc to +20 pc to 40 pc. The actual size of the ideal backtracking area +depends, among others, on the cluster’s mass. Details on this de- +pendence can be found in Pfalzner et al. (in preparation). +Our simulations work with the idealised situation, where the +search areas are uncontaminated by the presence of a population +of foreground and background stars. In an actual application, +extending the field increases the contamination by these fore- +ground and background stars. A more significant fraction of con- +taminants yields a larger half-mass radius estimate and a shorter +age estimate. As the ideal search radius increases as a function +of cluster age, so do the errors due to the background population. +However, the advent of Gaia again improved the situation; nev- +ertheless, it is still a point to consider in real applications. While +Rizzuto et al. (2012) found ten years ago that the disc fractions +in Upper Sco depend very much on cluster membership proba- +bility and distance to the cluster centre, nowadays, a search area +Article number, page 7 of 14 + +A&A proofs: manuscript no. main_new +of > 100 pc is regarded as giving reliable data (Luhman & Esplin +2020). +4.2.2. Time of gas expulsion +Backtracking can also be used to obtain information concern- +ing the time when gas expulsion happened. Here the same rules +apply as for determining the pre-gas expulsion size: restricting +to unbound stars and including sufficiently large sampling areas +improve the results. In the example shown in Fig. 6, the sim- +ulated and the backtracked time of gas expulsion are shown as +vertical lines. The backtracking of unbound members determines +temb to be 1.8 Myr, which is in excellent agreement with the ac- +tual value from the simulations (2 Myr, see Fig. 6 top panel). The +relative error in gas expulsion time derived using unbound stars +is 40 ± 4% which is much better than that derived using bound +stars (826+45 +−84%). Moreover, including only the unbound particles +within 10 pc is not advisable with its relative error of 88+11 +−32% +in the recovery of temb. The error is reduced to 63+11 +−8 % when +the search area increases to 20 pc. Although the results derived +by including the unbound particles within 20 pc and 40 pc of +the cluster’s centre give nearly identical results for this example +cluster (see Fig. 6 bottom panel), the relative error in the derived +temb decreases significantly to 40 ± 4% when all the N = 1000 +simulations are considered for the 40 pc case. The derived gas +expulsion times tend to underestimate the time of gas expulsion +by a 32+7 +−6%. Given the general uncertainty of cluster ages, this +can be considered a minimal error. Again, it is the stars that un- +derwent close encounters that are responsible for the derived too +short times. +4.2.3. Further improvements +We saw that using the unbound stars from a sufficiently large +area gives the best backtracking results for the pre-gas expul- +sion half-mass radius. However, the value can still be a factor of +two too large. One reason is that even some of the unbound stars +have a relatively strong encounter before leaving the cluster (see +Fig. 3). However, the main reason is that backtracking the un- +bound stars gives the half-mass radius of the unbound, not that +of the entire cluster sample. The stars that become unbound are +predominantly located at the outskirts of the cluster at the mo- +ment of gas expulsion. Therefore, backtracking them, one ob- +tains a value that is larger than the complete half-mass radius. +The actual pre-gas expulsion half-mass radius includes the un- +bound stars. However, simply multiplying the determined value +by a factor of 0.5 recovers the half-mass radius in our case quite +well. For our simulations, the empirical scaling factor has a value +of 0.46+0.06 +−0.04. There does not seem to be any correlation between +the cluster mass and the scaling factor. Although the Spearman +correlation coefficient is calculated to be −0.0133, the p-value +for the hypothesis test of their correlation is found to be 0.48 +which is greater than the significance level α = 0.05. Hence, the +null hypothesis that the cluster mass and the scaling factor are +unrelated can not be rejected. To some degree, the actual cor- +rection value might depend on the star formation efficiency in +the clusters, however, new sets of simulations with varying star +formation efficiencies need to be analysed to establish the depen- +dence. The gas dispersion timescale, on the other hand, should +not affect the factor. +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +Time [Myr] +0 +1 +2 +3 +4 +5 + Half mass radius [pc] +0.2 M : 1.54 Myr, 2.98 pc +0.3 M : 1.54 Myr, 2.78 pc +0.5 M : 1.55 Myr, 2.57 pc +all stars: 1.57 Myr, 2.52 pc +2 Myr, 1.32 pc +0 +2 +4 +6 +8 +10 +Time [Myr] +0 +5 +10 +15 +20 +25 + Half mass radius [pc] +1.51 Myr, 2.54 pc +1.40 Myr, 9.22 pc +1.58 Myr, 3.44 pc +2.68 Myr, 7.67 pc +2 Myr, 1.19 pc +Fig. 7. Backtracked half-mass radii for a simulation with 4000 stars, +Top: calculated using actual masses (green), 0.2 M⊙ (red), 0.3 M⊙ (blue) +and 0.5 M⊙ (yellow). Bottom: calculated using exact velocity values +(green), using vz = 0 (red), using velocities values with systematic er- +rors as well as different levels of statistical uncertainty (blue: 0.27 km/s +& yellow: 1 km/s). The actual values of temb and rhm at the time of gas +expulsion from the simulation are shown in cyan. +4.2.4. Mass of stars +When we determine bound and unbound stars in a cluster, the +mass of the stars plays a role. However, in observations, the stel- +lar classification is often known but not the actual mass of the +stars. Especially for young clusters, there are large uncertainties +between these two properties, and the assumption of different +evolutionary models leads to significant differences. Here, we +test to what extent this uncertainty in classification as bound or +bound due to missing mass information influences backtracking. +To mimic this problem, we assign the same mass to all stars, +determine the bound and unbound stars and then perform the +same backtracking procedure as before. Figure 7 (top) shows the +result of backtracking with the fully known IMF (green) and with +the assumption that all stars have the same mass (Ms = 0.2 M⊙, +0.3 M⊙ and 0.5 M⊙). It can be seen that not knowing the actual +masses of the stars does not influence the derived time of gas +expulsion. In all cases, it is too low. The relative error for the +derived temb is 46+3 +−2% for the case of using actual stellar masses +Article number, page 8 of 14 + +Arunima Arunima et al.: Unbound stars hold the key to star cluster history +(green curve). Using the same stellar mass for all stars increases +this error only marginally to 52+3 +−4%, 49+3 +−2%, and 47+3 +−2% for the +case of Ms = 0.2 M⊙ (red), 0.3 M⊙ (blue), and 0.5 M⊙ (yellow) +respectively. The situation is different for the cluster size at the +moment of gas expulsion. Here, assuming that all stars have the +same mass leads to up to a factor of 1.2 larger sizes than using +the actual stellar masses in the case shown in Fig. 7 (top). The +smaller the assumed mass, the error is larger. The relative error +for the derived rhm is 124.4+8.5 +−5.3% for the case of using actual +stellar masses (green curve). This error increases to 130.6+9.6 +−7.0% +when using stellar mass as 0.5 M⊙ (yellow), to 155.6+11.2 +−9.4 % for +0.3 M⊙ (blue), and to 180.1+15.0 +−12.1% for 0.2 M⊙ (red).2 We find that +assuming all stars to have a mass of 0.5 M⊙, which corresponds +to the mean stellar mass in the cluster, is the best alternative to +knowing the actual stellar masses. +4.2.5. Velocity in the z direction +We also consider the effects of errors in the vz values on the back- +tracking in Fig. 7 (bottom). The velocity component along the z +axis, corresponding with close approximation to the radial ve- +locity component, constitutes the main source of uncertainty in +the total velocity vector (Krolikowski et al. 2021). As a starting +point, we consider the effect induced by the existence of non-null +proper motion uncertainties; the error on radial velocity is for the +moment assumed to be null. Gaia DR2 data have systematic un- +certainties in the measurement of parallax and proper motions +(Lindegren et al. 2018; Vasiliev 2019). The 2D random error is +considered to be of the order of 0.27 km/s, equivalent to the er- +ror in 2D proper motion (0.28 mas yr−1) for sources with G = 17 +mag at a distance of 200 pc in Gaia DR2. Using this error, blue +curve is obtained for backtracked radii. The pre-expansion size is +derived to be about 1.5 times the size obtained compared to the +velocities having no error (green curve in Fig. 7, bottom). The +relative error distributions (with respect to the actual rhm) are de- +termined for rhm obtained using velocities with no error (green) +and using velocities with error (blue). The relative error in rhm +goes from 124.4+8.5 +−5.3% for the green curve to 213.7+12.7 +−10.5% for the +blue curve. An accuracy improvement is seen for the value of +the cluster’s age at the time of gas expulsion. The relative er- +ror decreases from 46+3 +−2% for the green curve to 35+4 +−5% for the +blue curve. However, this improvement is less due to recovering +more information about the cluster’s past, but more with a gen- +eral move of the curve towards the right on the time axis with an +increase in the standard deviation in random errors. +The impact of radial velocity errors results in an even shorter +estimate of the expansion timescale. Krolikowski et al. (2021) +point out that the radial velocity (RV) uncertainty is roughly an +order of magnitude larger than the reported projected proper mo- +tion uncertainty, even when collecting RV measurements from +more precise catalogues than Gaia.Ma et al. (2022) also point +out that even with future Gaia releases, the precision of RV +would be ∼ 1 km/s. The yellow curve in Fig. 7 (bottom) cor- +responds to the backtracked radii determined using the same +systematic error but a random error of 1 km/s. This increases +the relative error in temb and rhm at the time of gas expulsion to +60+8 +−13.5% and 639.0+35.9 +−41.1% respectively. +Only 0.54% of the sources with astrometric data have the RV +measurements available in Gaia DR2. For the extreme situation +of zero information on vz, the red curve in Fig. 7 (bottom) is +obtained. The relative error for the determined size in this case +2 The distributions of sizes and gas expulsion times derived using dif- +ferent masses can be seen in Appendix A. +is the highest of all previously discussed cases at 821.6+47.6 +−55.5% +whereas the relative error in derived time of gas expulsion is +40+10 +−12%3. In reality, for Gaia DR2, the deviation from the actual +parameter values will be somewhere between the cases of vz = 0 +and the added systematic error along with statistical uncertainty. +5. Application to observational data +So far, we have dealt exclusively with the idealised situation that +simulations provide. In the following, we want to show two ex- +amples of applying backtracking procedures to observed clus- +ters. The aim is not so much the age and initial size determination +of these specific clusters, but to show which additional problems +can be expected in real applications. Therefore, we choose two +clusters that differ considerably in age and geometry. When re- +ferring to the age of the cluster, we quote the time elapsed since +the gas started to be expelled and refer to the cluster age as the +median age of all the stars in the cluster. This differs from the +time elapsed since the molecular cloud started producing stars +(Pecaut & Mamajek 2016; Kim et al. 2021; Fujii et al. 2021). +5.1. NGC 6530 +We first apply the before-described backtracking method to NGC +6530, which is a young cluster within Lagoon Nebula. Its age +has been estimated to be 1–2.3 Myr (Prisinzano, L. et al. 2005; +Mayne et al. 2007; Bell et al. 2013) and its distance to be 1326+77 +−69 +pc (Wright et al. 2019; Damiani et al. 2019). We use the cat- +alogue of members provided by Wright et al. (2019), who use +GES spectroscopy, Gaia DR2 astrometry, and ancillary member- +ship information from X-ray, infrared, and Hα surveys to com- +pile the said catalogue. 691 of these cluster members have Gaia +DR2 data and have been used in the following analyses. We as- +sume that all the stars have a mass of 0.5 M⊙. Using the radial +velocity for individual sources when available and assuming it to +be equal to the bulk radial velocity of the cluster when not, 3D +positions and velocities of the stars are calculated in the stan- +dard right-handed Cartesian Galactic frame using the conversion +equations prescribed by the Gaia DR2 documentation. These are +then used to determine the bound and unbound members of the +cluster. +For backtracking the stars’ trajectories, we backtrack the po- +sitions in the plane of the sky using the velocities along α and +δ. Radial velocity is used to backtrack along the line-of-sight +and change the distance of the stars which is assumed to be the +same for all stars at the present time (1326 pc). Although indi- +vidual distances are available for all the stars (Bailer-Jones et al. +2018), the uncertainty is extremely high (fractional uncertainty +is 0.20+0.43 +−0.09 as compared to 0.02 ± 0.01 for the distance data- +set of member stars of Upper Sco in Sec.5.2) and leads to very +high half-mass radius along with loss of most information about +the cluster. The calculated coordinates are then converted to the +Cartesian coordinates to calculate the half-mass radii. The result +of this procedure is shown in Fig. 8 (left panel). However, for +considering the uncertainty in astrometry of the member stars, +we run 1000 Monte Carlo simulations, that is to say repeat the +entire procedure while varying astrometric information in a ran- +dom, normal manner according to the uncertainties associated +with each Gaia DR2 source’s parameters. For the distance value +for all the stars, the uncertainty is taken as 73 pc (Wright et al. +2019). The results of these simulations are fitted with a Gaussian +3 The distributions of values of size and gas expulsion time obtained +for all the cases discussed here can be seen in Appendix B +Article number, page 9 of 14 + +A&A proofs: manuscript no. main_new +10.0 +7.5 +5.0 +2.5 +0.0 +2.5 +5.0 +7.5 +10.0 +Time [Myr] +0 +10 +20 +30 +40 + Half mass radius [pc] +0.04 Myr, 4.01 pc +10.0 +7.5 +5.0 +2.5 +0.0 +2.5 +5.0 +7.5 +10.0 +Time [Myr] +10 +15 +20 +25 +30 + Half mass radius [pc] +-0.54 Myr, 13.14 pc +10.0 +7.5 +5.0 +2.5 +0.0 +2.5 +5.0 +7.5 +10.0 +Time [Myr] +10 +12 +14 +16 +18 +20 +22 +24 +26 + Half mass radius [pc] +-0.25 Myr, 12.16 pc +-1.04 Myr, 10.29 pc +Fig. 8. Backtracked (and extrapolated) half-mass radii determined for bound (blue) and unbound (orange) stars 10 Myr into the past and into the +future. The green dashed lines show the minima of the backtracked half-mass radius for unbound stars. Left: For NGC 6530 members. Middle: For +Upper Sco members. Right: Backtracked (and extrapolated) half-mass radii determined for the unbound members of subclusters of Upper Sco. +to obtain the parameters of the cluster along with their errors. +Hence, we find the gas expulsion to have happened 0.03 ± 0.03 +Myr ago and the size of the cluster at the time of gas expulsion +is found to be 4.16 ± 0.23 pc. This agrees well with the current +age estimate of the cluster. However, the half-mass radius might +be underestimated by the assumption of a fixed distance of the +stars. A more realistic estimate might be obtained by multiplying +it by a factor √3/2, which would yield a limit of 5.09 pc on the +cluster size at the time of gas expulsion. +Despite obtaining a reasonable fit, the reservations pointed +out in Section 4.2.5 also hold here. The median uncertainty in +proper motion amount to 2 km/s (Wright et al. 2019). Any un- +certainty added to the true velocity acts to reduce the best fit. +This uncertainty is the most problematic issue in applying the +backtracking method for determining the age of NGC 6530. +5.2. Upper Scorpius +Upper Sco is a sub-group of Sco-Cen that has been widely stud- +ied with the Gaia data, identifying the cluster’s members (Galli +et al. 2018; Wilkinson et al. 2018; Luhman & Esplin 2020; +Damiani et al. 2019; Žerjal et al. 2021; Squicciarini et al. 2021; +Kerr et al. 2021) and an isochronal age of around 10 Myr has +been recently accepted (Feiden 2016; David et al. 2019; Luh- +man & Esplin 2020; Sullivan & Kraus 2021). We test the quality +of the backtracking for clusters with a more complex morphol- +ogy using Upper Sco as an example. We use the list of mem- +bers compiled by Luhman & Esplin (2020) using optical and IR +spectra to confirm the stars’ youth while parallax and proper mo- +tion offsets to get the kinematic criteria for these candidates. The +list contains 1761 member candidates, 1682 of which have Gaia +DR2 data available and have been used in the following analy- +ses. We apply the same method described for NGC 6530 with +the exception of considering individual distances for the stars in +this case as the uncertainty in distance is much lower. +Despite its complex morphology, we first work with the as- +sumption that Upper Sco was a centrally condensed spherical +structure in the past. In this case, we find that the cluster went +through gas expulsion 0.54 Myr ago and had a half-mass radius +of 13.14 pc at this time as shown in Fig. 8 (middle panel). How- +ever, the Monte Carlo simulations for error propagation estima- +tion provide the gas expulsion time to be 0.80 ± 0.21 Myr ago +while the cluster size is found to be 13.11 ± 0.11 pc. +This value agrees with other backtracking results for Upper +Sco. For example, Žerjal et al. (2021) determine the kinematic +age of the population in the Upper Sco region as 4 ± 4 Myr, +whereas Squicciarini et al. (2021) find 8 subclusters with kine- +matic ages varying from 0.0 ± 0.1 Myr to 3.8 ± 0.4 Myr. How- +ever, this cluster age deviates considerably from that of 10 Myr +obtained by applying corrections, for undetected binaries (Sulli- +van & Kraus 2021) or strong magnetic fields impeding convec- +tion in low-mass stars (Feiden 2016; David et al. 2019), to the +isochronal age determination of Upper Sco. One possible expla- +nation for this discrepancy would be that the backtracking yields +the time elapsed since gas was expelled and refers to the age of +the youngest stars in the association. Taking into account a star +formation history lasting 6-7 Myr, most stars might be about 11 +Myr old and the median age of the association ≈ 7 Myr. These +values are more similar to the ones obtained through stellar evo- +lution models. +Additional complications arise from Upper Sco, unlike NGC +6530, being highly substructured (Kerr et al. 2021; Squicciarini +et al. 2021). Likely, star formation did not happen as a single +burst, but was rather characterised by several formation episodes +(Galli et al. 2018). Thus, the assumption of a centrally condensed +spherical structure in the past is oversimplifying the situation. +Hence, we try to improve our analysis by considering Upper Sco +to consist of subclusters. A density distribution of the cluster +members on the plane of the sky at the present time is plotted +(see Appendix C for more details and plots). Two dense areas +seem to emerge and we consider two rectangles in these areas. +The members’ positions are traced back using the same method +as described above. When a member star enters one of the said +rectangles, it is assigned to the corresponding subcluster. After +the assignment of subcluster membership using this simplified +method, the backtracked and extrapolated half-mass radii are de- +termined using unbound stars for both subclusters. The result is +shown in Fig. 8 (right). To determine the errors, the Monte Carlo +simulations are used which provide the time of gas expulsion in +the two subclusters as −1.09±0.29 Myr and −0.25±0.17 Myr ago +respectively. Similarly, the half-mass radii at the time of gas ex- +pulsion is found to be 10.15±0.20 pc and 12.10±0.23 pc. Various +characterisations of the subclusters are summarised in Table C.1. +There is a slight improvement in the determination of the size +and time of gas expulsion when considering Upper Sco to have +subclusters rather than being one coeval population. However, it +must be reiterated that ours is a simplified method. More robust +clustering methods can be used in the future to get better results +on the subcluster membership and hence, their parameters. For +example, Kerr et al. (2021) use HDBSCAN clustering algorithm +on Gaia DR2 data and find 9 subclusters in the Upper Sco re- +gion. Two of these (Group H and Group I) have more than 100 +members. We analyse these subclusters and find the time of gas +expulsion and their sizes at that time. According to our results, +Article number, page 10 of 14 + +Arunima Arunima et al.: Unbound stars hold the key to star cluster history +gas expulsion in Group H happened 3.40 ± 0.42 Myr ago and its +half-mass radius was 3.96 ± 0.215 pc at the time. For Group I, +the gas expulsion happened 0.78±0.91 Myr ago and its size was +3.73 ± 0.37 pc. The age found by Kerr et al. (2021), using Gaia +DR2’s photometric data, for the groups is 10.2 ± 0.7 Myr and +5.7 ± 0.4 Myr respectively. So, even though there is an improve- +ment in the age and size estimates when using a more robust +clustering algorithm, the kinematic age estimates still show con- +siderable deviation from the photometric estimates. Availability +of accurate radial velocities and distances for the member candi- +dates to use in the subclustering analysis in future would improve +the situation further. +6. Discussion +The improvement in the cluster size, when considering sub- +clusters, already shows that backtracking is more complex for +substructured clusters like Upper Sco. Thus, the less substruc- +tured a cluster is, the more straightforward the backtracking. The +substructured clusters require backtracking to multiple centres, +which is the more complex the more subcluster centres exist. +Another potential difficulty could be the presence of multi- +ple differently aged populations in the Upper Sco region leading +to the miscalculation of the cluster’s age (Wright & Mamajek +2018; Žerjal et al. 2021; Squicciarini et al. 2021). However, this +would require large subgroups to be well over 15 Myr to intro- +duce such a substantial error. This seems unlikely as an expla- +nation. We suspect that the real reason is a different one. The +arguments based on kinematic analysis of a cluster for its his- +tory can not be considered on their own due to the significant +errors in radial velocity and its unavailability for most stars in +Gaia. Large uncertainty in the velocities of the stars can lead to +a significant loss of information about the past of the cluster (see +Fig. 7, bottom panel). This might be the reason for underesti- +mating the cluster age and overestimating the size at the time of +gas expulsion. Furthermore, the assumptions in the backtrack- +ing analysis are numerous. The exact masses of the stars are +unknown, so the distinction between bound and unbound stars +could be highly inaccurate when combined with astrometric un- +certainties and incomplete or inaccurate membership of the clus- +ter. In conclusion, the determination of a much younger age, of +the Upper Sco region, by kinematic analysis than the more accu- +rate isochronal determination could be affected by multiple, dif- +ferently aged and kinematically distinct populations; however, +precise radial velocity measurements are needed to rule out the +possibility that the discrepancy in age determination is due to +astrometric errors. +7. Summary and conclusion +Young star clusters (< 10 Myr) are highly dynamical entities. +Therefore, observations provide only snapshots of this highly +dynamic cluster evolution sequence. Nevertheless, in light of the +unprecedented precision of Gaia position and velocity data, it +should be possible to obtain information about a young cluster’s +past using backtracking techniques. In this work, we used simu- +lations of the cluster dynamics as an idealised version to suggest +how to optimise the backtracking method. Under ideal observa- +tional conditions, the following statements should hold: +– For backtracking to be successful, it is essential to distin- +guish between bound and unbound cluster members. Under +ideal conditions, backtracking the unbound members exclu- +sively, the time of gas expulsion can be determined with only +a 32% error. However, the quality of the backtracking de- +pends on the number of cluster stars, with the best results +obtained for clusters containing a few thousand stars. +– While still the best result, the sizes backtracked from un- +bound members are about a factor of two larger than the ac- +tual value. However, this error is systematic and reflects that +unbound members are primarily located at the cluster out- +skirts at the time of gas expulsion. Thus, applying a correc- +tion factor of 0.46 approximates the actual value very well. +– For obtaining this accuracy, it is essential to determine all the +unbound members to > 20 – 40 pc from the cluster centre. +– The classification of bound and unbound stars based on the +direction of their velocity vectors, or ad hoc distance or ve- +locity cutoffs is highly error-prone. We provide analytical +cutoffs based on the escape velocity and the number of clus- +ter members with a success rate of 96% – 97% for distin- +guishing between bound and unbound stars. +– Runaway and walkaway stars are less suitable to determine +past cluster properties because of their low number and their +production by dynamical ejection. Ejection traces only past +locations of high stellar density regions but not actual cluster +sizes or the time of gas expulsion. +Uncertainty in membership and stellar properties provide +additional challenges. Modelling these uncertainties, we find +that the lack of information about the line-of-sight velocity can +severely affect the determination of the pre-expansion size of the +cluster. Nevertheless, the time of gas expulsion can still be esti- +mated with an error of 40% − 60% due to the unavailability of +radial velocities and uncertainty in the value even when avail- +able. The uncertainty in the mass of the members seems to af- +fect the results much less. Similarly, larger search areas often +struggle with higher false-positive and -negative rates in mem- +bership. Applying our results to observational data, the method +works reasonably for centrally concentrated clusters, but less for +very substructured clusters like Upper Sco. For such substruc- +tured clusters, backtracking to the individual subcluster centres +would be the next step to pursue. +In summary, restricting backtracking to the unbound stars al- +lows deducing the times of gas expulsion and the pre-expansion +cluster size values with relatively high accuracy. Analysing a +large number of clusters with the presented method will allow +drawing valuable conclusions about the clustered star formation +process in the future. +Acknowledgements. We thank the referee for a very detailed report that +made this article significantly better. This work has made use of data +from the European Space Agency (ESA) mission Gaia (https://www. +cosmos.esa.int/gaia), processed by the Gaia Data Processing and Anal- +ysis Consortium (DPAC, https://www.cosmos.esa.int/web/gaia/dpac/ +consortium). Funding for the DPAC has been provided by national institutions, +in particular the institutions participating in the Gaia Multilateral Agreement. +References +Aarseth, S. 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D. 2021, +in Star Clusters: the Gaia Revolution; Online Workshop, 29 +Article number, page 12 of 14 + +Arunima Arunima et al.: Unbound stars hold the key to star cluster history +Appendix A: Mass of stars +We discussed how the unavailability of the mass of stars in ob- +servations affects the determination of gas expulsion time and +cluster size at the time of gas expulsion using backtracking anal- +ysis. Here, we provide the distributions of the derived sizes and +gas expulsion time (Fig. A.1) for all the cases discussed in Sec. +4.2.4. +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +Half-mass radius [pc] +0 +1 +2 +3 +4 + = 1.14, = 0.09 + = 2.56, = 0.10 + = 2.92, = 0.16 + = 2.64, = 0.13 + = 3.21, = 0.18 +1.3 +1.4 +1.5 +1.6 +1.7 +1.8 +1.9 +2.0 +Time [Myr] +0 +2 +4 +6 +8 +10 +12 +green: = 1.54, = 0.04 +blue: = 1.51, = 0.04 +red: = 1.48, = 0.05 +yellow: = 1.53, = 0.04 +actual value of gas expulsion time +Fig. A.1. Distributions of the backtracked half-mass radii (top) and the +time of gas expulsion (bottom) obtained using actual masses (green), +0.2 M⊙ (red), 0.3 M⊙ (blue) and 0.5 M⊙ (yellow). The actual values of +rhm at the time of gas expulsion (as a distribution) and temb from all the +simulations (of N=4000 clusters) are shown in cyan. +Appendix B: Velocity in the z direction +Similarly, we provide the distributions of the derived sizes and +gas expulsion time for all the cases in Sec. 4.2.5 to supplement +the discussion of the effects of errors in the vz values on the back- +tracking analysis and derived parameters (Fig. B.1). +Appendix C: Upper Sco subclusters +The density distribution of the Upper Sco members is shown in +Fig. C.1 (top) along with the rectangles showing the subclus- +ter areas used for the subcluster membership assignment. Figure +C.1 (bottom) shows the scatter plot of the member stars with the +0 +2 +4 +6 +8 +10 +12 +Half-mass radius [pc] +0 +1 +2 +3 +4 +cyan: = 1.14, = 0.09 +green: = 2.56, = 0.10 +blue: = 3.57, = 0.10 +yellow: = 8.37, = 0.17 +red: = 10.44, = 0.42 +1.00 +1.25 +1.50 +1.75 +2.00 +2.25 +2.50 +2.75 +3.00 +Time [Myr] +0 +2 +4 +6 +8 +10 +12 +green: = 1.54, = 0.04 +blue: = 1.66, = 0.06 +red: = 1.62, = 0.18 +yellow: = 2.58, = 0.17 +actual value of gas expulsion time +Fig. B.1. Distributions of the backtracked half-mass radii (top) and the +time of gas expulsion (bottom) obtained using exact velocity values +(green), using vz = 0 (red), using velocities values with systematic er- +rors as well as different levels of statistical uncertainty (blue: 0.27 km/s +& yellow: 1 km/s). The actual values of rhm at the time of gas expulsion +(as a distribution) and temb from all the simulations (of N=4000 clusters) +are shown in cyan. +same rectangles and the members of the two subclusters in red +and green. The purple points represent the few members which +did not enter any of the rectangles in the 10 Myr up to which the +positions were backtracked and hence, are not assigned to any +subcluster. Furthermore, Table C.1 provides characteristic infor- +mation about the subclusters identified in this work as well as +about Group H and I from Kerr et al. (2021). +Article number, page 13 of 14 + +A&A proofs: manuscript no. main_new +Fig. C.1. Density distribution (top) and scatter plot (bottom) of the Up- +per Sco members at the present time. The two rectangles show the area +selected for the clustering process. Green and red points in the bottom +plot show the members of Group 1 and Group 2, respectively. Purple +points are the ones which were not assigned to any group. +Table C.1. Information about the subclusters identified in this work (ID: +1,2) and the groups from Kerr et al. (2021) (ID: H, I). +ID +N +RA +Dec +tK +rhm +[deg] +[deg] +[Myr] +[pc] +1 +1102 +241.60 +-21.93 +−1.09 ± 0.29 +10.15 ± 0.20 +2 +454 +245.68 +-25.12 +−0.25 ± 0.17 +12.10 ± 0.23 +H +102 +240.6 +-22.4 +−3.40 ± 0.42 +3.96 ± 0.21 +I +110 +246.4 +-23.9 +−0.78 ± 0.91 +3.73 ± 0.37 +Notes. Number of stars (N) and mean positions (RA, Dec) are provided +along with the time of gas expulsion (tK, kinematic age) and half-mass +radius of subcluster at the time of gas expulsion (rhm). +Article number, page 14 of 14 + +20.0 +-18 +17.5 +-20 +15.0 +Number +-22 +12.5 +(。)9 +-24 +10.0 +of +sources +-26 +7.5 +-28 +5.0 +-30 +2.5 +0.0 +235 +240 +245 +250 +α(°)-16 +-18 +-20 +-22 +。 +-24 +-26 +-28 +-30 +-32 +232.5 235.0 237.5 240.0 242.5 245.0 247.5 250.0 252.5 +α(° \ No newline at end of file diff --git a/8tE1T4oBgHgl3EQfngT9/content/tmp_files/load_file.txt b/8tE1T4oBgHgl3EQfngT9/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..54d1af60d09a53d22d054f940565a4945013ace4 --- /dev/null +++ b/8tE1T4oBgHgl3EQfngT9/content/tmp_files/load_file.txt @@ -0,0 +1,2008 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf,len=2007 +page_content='Astronomy & Astrophysics manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' main_new ©ESO 2023 January 10, 2023 Unbound stars hold the key to young star cluster history Arunima Arunima1,2, Susanne Pfalzner1,2,3, and Amith Govind1,2 1 Jülich Supercomputing Center, Forschungszentrum Jülich, 52428 Jülich, Germany e-mail: s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='pfalzner@fz-juelich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='de 2 Physics Department, University of Cologne, Cologne, Germany 3 Max Planck Institute for Radio Astronomy, Auf dem Hügel 69, 53121 Bonn, Germany Received .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' ABSTRACT Aims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' GAIA delivers the positions and velocities of stars at an unprecedented precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Therefore, for star clusters, there exists much higher confidence in whether a specific star is a member of a particular cluster or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' However, membership determination is still especially challenging for young star clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' At ages 2–10 Myr, the gas is expelled, ending the star formation process and leading to their expansion, while at the same time, many former members become unbound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' As a first step, we aim to assess the accuracy of the methods commonly used to distinguish between bound and unbound cluster members;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' after identifying the most suitable technique for this task, we wish to understand which of the two populations is more suited to provide insights into the initial configuration and the dynamical history of a cluster starting from its currently observed properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Here, we perform N-body simulations of the dynamics of such young star clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' We investigate how cluster dynamics and observational limitations affect the recovered information about the cluster from a theoretical perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' We find that the much-used method of distance and velocity cutoffs for membership determination often leads to false negatives and positives alike.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Often observational studies focus on the stars remaining bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' However, bound stars quickly lose the memory of the pre-gas expulsion phase due to their ongoing interaction with their fellow cluster members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Our study shows that it is the unbound stars that hold the key to charting a cluster’s dynamic history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Backtracking unbound stars can provide the original cluster size and determine the time of gas expulsion – two parameters that are currently still poorly constrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' This information is lost in the bound population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' In addition, former members are often better indicators for disc lifetimes or initial binary fractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' We apply the backtracking analysis, with varying success, to the clusters: Upper Scorpius and NGC 6530.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' For highly substructured clusters such as Upper Scorpius, backtracking to the individual subcluster centres will provide better results in future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Key words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' stars: formation – open clusters and associations: general – ISM: clouds – solar neighbourhood 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Introduction Star clusters are the nurseries for most stars (Porras et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Lada & Lada 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' As such, young star clusters play a vital role in our understanding of how young stars form and develop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' They signify the starting point for all that happens later on, as they pro- vide the initial stellar mass distribution (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Kroupa 2002) and the fraction of stars forming as a single-, binary-, or multiple-star system (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Duchêne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' It is a standard procedure to use properties of clusters of different ages to obtain information on the dynamical development of young binary stars or the dis- persal time of discs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Haisch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Ansdell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Marks et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Ribas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Richert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Michel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Often the task of determining cluster membership and deriving the temporal development of specific properties are separate endeavours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' While distinguishing members is a chal- lenge in itself, any bias in membership determination (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' false positives and false negatives) feeds through to the derived pa- rameters used in other applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' This study’s central aim is to utilise cluster dynamics simula- tions to optimise the data used to determine a cluster’s past.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Un- til recently, the role of dynamics during the formation history of young clusters was highly uncertain (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Elmegreen 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Fujii et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Ward et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Banerjee & Kroupa 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Dib et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2018), mainly because observational limitations hampered pre- cise velocity determination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' The precision of data coming from the Gaia satellite (Gaia Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2016, 2018, 2021) helped shed light on this issue since a complete understanding of the dynamical evolution of present-day clusters has not been attained yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Investigating a sample of 28 clusters and associa- tions with ages ≈ 1–5 Myr, Kuhn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' (2019) found that at least 75% of these systems are expanding at typical expansion veloc- ities of the order of ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='5 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Cluster expansion was pre- dicted by the gas expulsion scenario (Mathieu 1983;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Lada et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 1984;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Adams 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Kroupa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Baumgardt & Kroupa 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Pelupessy & Portegies Zwart 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Pfalzner & Kaczmarek 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Brinkmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Pfalzner & Govind 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' During the star formation phase, the stars are embedded in the gas and dust reservoir from which they are forming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' However, after ap- proximately 1–2 Myr, the gas starts to be expelled from the clus- ters by various mechanisms (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Krumholz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Fujii et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Due to loss in gas and dust mass, the system is no longer in equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Therefore, a considerable portion of the stars, bound in the embedded phase, become unbound in the gas expulsion phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' The three-dimensional information available from the Gaia data has been a tremendous step forward in this field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Neverthe- less, discriminating the members of star clusters and associations from the foreground and background population is still challeng- ing (Gagné et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Many new methods have been devel- oped for determining the members of open and globular clusters (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Sollima et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Garro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Vitral 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Cluster Article number, page 1 of 14 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='03311v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='GA] 9 Jan 2023 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' main_new membership determination is challenging in the early expansion phase (< 10 Myr), especially if a clear-cut distinction between currently bound and formerly bound (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' unbound) members is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' In this case, there are additional difficulties to over- come compared to older clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' First, the earliest stages of the formation of star clusters are hidden from view by gas and dust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Thus, at this young age, veiling is a severe problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Second, the young clusters’ expansion requires additional attention in mem- bership determination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Third, short- and long-lived clusters co- exist during a 10 Myr timespan (Lada & Lada 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' They un- dergo very different cluster dynamics (Pfalzner & Kaczmarek 2013), and it is not always straightforward whether a specific cluster will remain bound for a long time or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Here, we concentrate on these dynamical aspects of young short-lived clusters1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Any cluster observation is just a snapshot in time of the sequence of its dynamical evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Based on simulations of the cluster dynamics, we show the importance of cluster dynamics in membership determination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' We investigate the efficiency of backtracking cluster expansion and find that dis- tinguishing between bound and unbound stars in the expansion phase is vital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Finally, we show that the unbound stars hold the key to determining a cluster’s past.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Cluster observation techniques Historically, star clusters have been identified visually as stel- lar density enhancements (Dreyer 1888;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Trumpler 1930;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Bailey 1908;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Collinder 1931).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Surveys like Hipparcos (Perryman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 1997), 2MASS (Skrutskie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2006), and Gaia have each in- creased the samples by hundreds of candidate clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Due to Gaia’s high-precision parallax measurements, the clustering of stars can be analysed in a higher dimensional space by combin- ing their positions in the sky, proper motions, parallaxes, and radial velocities (when available).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' For studies which do auto- mated blind searches with clustering algorithms, the youth of the stars is used as a confirmation of membership.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Such youth indi- cators can be X-ray activity, infrared excess (Broos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Feigelson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Getman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2017), lithium abundance (Soderblom 2010), and gravity-sensitive spectral indices such as TiO molecular lines (Wilking et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2005), empirically con- structed spectral indices (Damiani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2014), or the shape of the H-band peak (Scholz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Among the clustering algorithms, one can distinguish differ- ent classes: Density-based spatial clustering like DBSCAN (Es- ter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Wilkinson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Zari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Castro- Ginard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2019, 2020, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Hunt & Reffert 2021), HDB- SCAN (Campello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2013), and OPTICS (Ordering Points To Identify the Clustering Structure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Ankerst et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 1999), mul- tidimensional Gaussian-based methods (Vasiliev 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Cantat- Gaudin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Kuhn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2020), k-means clustering (Mac- Queen 1967;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Hunt & Reffert 2021), and Friend of Friend algo- rithm (FoF;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Liu & Pang 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' In addition, there exist several un- supervised algorithms like UPMASK (Krone-Martins & Moit- inho 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Cantat-Gaudin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Cantat-Gaudin & Anders 2020), the nearest neighbour-based method by He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' (2021), and STARGO (Tang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Pang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 1 The nomenclature of short-lived clusters is not unequivocal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' While referred to as clusters while embedded, they are often classified as as- sociations when the gas is expelled, and most of their stars become un- bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Here, we refer to short-lived clusters as clusters and point out expressly when talking about long-lived clusters, that is, open and glob- ular clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Young star clusters pose additional challenges compared to open or globular clusters due to their highly dynamic nature after gas expulsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Although space velocity is used to iden- tify clusters, algorithms rarely consider dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Observations only provide a snapshot in the dynamic evolution of the clus- ter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Hence, even clustering in the velocity space at the present moment might be a chance alignment as the velocity changes rapidly in young star cluster members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' More limitations in iden- tifying clusters come from Gaia’s poor completeness in crowded fields and no particular regard for binarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Moreover, young clusters are still embedded in natal gas and dust that can not be penetrated by optical wavelengths, which presents another diffi- culty in identifying and analysing young clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Blaauw (1964) first gave the notion of linear expansion in associations, assuming that all members move away from their birthplace without any forces acting on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Then, the recipro- cal of the expansion coefficient can provide an estimate of the as- sociation’s kinematic age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Alternatively, the individual motions of the stars can be traced back until they reach the smallest con- figuration at a past time, and the kinematic age, as well as the initial configuration of the association, can be possibly obtained (Blaauw 1978).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Most studies apply cutoffs to remove objects with low- quality astrometry and outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' The sigma-clipping method aims to reduce the chances of contaminants or uninformative stars and improve clusters’ signal-to-noise ratio (S/N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Alternatively, out- liers can be modelled in the fitting procedure without rejecting points a priori (see Hogg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Before Gaia, the significant errors in astrometry and the low number of confirmed members with available radial velocities were the main hindrances in the analysis (Fernández et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' The higher precision of the Gaia data allows for better trace- back analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' For example, recent studies by Heyl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' (2022, 2021) trace back the stars of clusters aged 40–200 Myr using Gaia EDR3 data and determine their kinematic ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Similarly, Schoettler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' (2022) trace back runaway (RW) and slower walkaway (WW) stars within a distance of 100 pc of NGC 2264 to the three subclusters S Mon, IRS 1 and IRS 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' The study by Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' (2022) uses Gaia DR2 data to trace back (and extrapo- late) the trajectories of members of the Scorpius-Centaurus (Sco- Cen) association and find evidence of past and future close stellar flybys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Observational challenges like distinguishing the cluster pop- ulation from the back and foreground stars, limiting magnitudes, imprecision of derived properties like age and mass, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=', com- plicate backtracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Here we apply backtracking to snapshots in the simulations of the cluster dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Under these idealised conditions, membership is certain, the exact positions and ve- locities of the stars are known at all times, and last, but not least, we know what the result should be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' This certainty allows us to determine the most expedient method and suggest measures to optimise the backtracking technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Cluster simulation method We use a sub-set of simulations of the dynamics of clusters containing N stars we performed recently (Pfalzner & Govind 2021), using the simulation code NBODY6++GPU (Aarseth 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' The simulations try to represent the situation in real clusters as closely as possible by adopting initial conditions backed by re- cent observations and following the observed cluster expansion derived from the sizes of clusters in the age range of 1–10 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Here we give only a summary of the assumptions, and the nu- merical method we applied in Pfalzner & Govind (2021), as the Article number, page 2 of 14 Arunima Arunima et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' : Unbound stars hold the key to star cluster history actual choice of simulation parameters is uncritical for the gen- eral challenges in membership determination and backtracking of the cluster history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' We model the dynamics of the young clusters covering all the phases: Starting from the embedded phase, we simulate the subsequent gas expulsion that leaves the cluster in a super-virial state and results in the cluster expanding until it reaches a new equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' It is assumed that all stars are already formed and that the gas expulsion occurs at temb = 2 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Observations in- dicate that the entire gas expulsion process takes ≈ 1 – 2 Myr (Kuhn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Simulations investigating the dependence of the cluster dynamics on the gas expulsion time found that the gas expulsion can be modelled as being instantaneous (Geyer & Burkert 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Portegies Zwart et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Stellar evolution has not been included in this work as it has little influence on the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' We analyse the dynamics of clusters with different numbers of cluster members N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' The corresponding clusters’ masses Mc and sizes, illustrated by their half-mass radius rhm, are given in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Low-mass clusters are usually smaller than high-mass clusters of the same age (Lada & Lada 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Adams 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Pfalzner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' This relation between the cluster’s mass and its half-mass radius can be approximated by a power law: Mc = Crhm γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' (1) The values of the constant C and scaling exponent γ differ in different observational studies due to the involved observational uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' The clusters’ sizes given in Table 1 are based on the mass-radius relation by Pfalzner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' (2016) where C = 717.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='794 and γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' We assume that the star formation efficiency in the system is 30 % (Lada & Lada 2003), which sets the gas mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' The gas and dust component of the embedded phase is implemented as a background potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' In our simulations, a test particle represents a star with a given mass, position, and velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' The particles’ positions are chosen so that the resulting stellar number density distribution obeys a King profile with King parameter, W0 = 9 (King 1966a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' The King model is an empirical law that can not be defined ana- lytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' It consists of an energy distribution function of the form fK(E) = �ρ1(2πσ2 K)−3/2(eE/σ2 K − 1) : E > 0, 0 : E ≤ 0, (2) with E = Ψ− 1 2ν2 and Ψ = −Φ+Φ0 being the relative energy and relative potential of a particle, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Also, f(E) > 0 for E > 0 and σK is the King velocity dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' The profiles are characterised by the King parameter W0 = Ψ/σ2 K, an increase of which signifies decrease in the relative size of the cluster core Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Initial cluster parameters for the simulation campaign using mass-radius dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' N Nsim Mc [M⊙] rhm [pc] Mt [M⊙] temb [Myr] 200 1941 117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='26 393.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='31 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='0 1000 497 589.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='67 1966.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='57 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='0 4000 127 2359.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='88 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='3 7866.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='27 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='0 Notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Here N denotes the number of cluster members, Nsim the number of simulations, temb the duration of the embedded phase, Mc the stellar mass of the cluster, rhm the half-mass radius, and Mt the total cluster mass (stars + gas).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' rc/rhm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Observationally, determining the stellar density distribu- tion of young star clusters can be challenging but it has been found that young clusters are best represented by King model with W0 ≥ 7 (Hillenbrand & Hartmann 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Nürnberger & Petr-Gotzens 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' The choice of W0 mainly affects the size of the central high-density area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Hence, the number of expelled stars also depends on the choice of W0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Even for a relatively steep W0 = 9-potential, the number of escapers is < 1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Therefore, the conclusions about membership determination methodology are unaffected by the choice of potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' The individual test par- ticles are assigned masses following the initial mass function (IMF) by Kroupa (2002), with the lower mass limit set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='08 M⊙ (hydrogen-burning limit) and an upper mass limit of 150 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Potentially existing initial mass segregation in the clusters is neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' The cluster members are given velocities following a Maxwellian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' We assume that the cluster is initially in virial equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' We perform (Nsim) simulations for every cluster mass, where the actual distribution of the stars depends on the seed selected in the randomised procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' We analyse all the simulation results in this statistical study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' However, why a specific method works or fails, we illustrate exemplarily for just one specific randomly chosen realisation in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 1 – 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Figures 6 – 8 also show the method applied to randomly chosen specific clusters for visual understanding;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' however, statistical results are mentioned in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' For simplicity, we exclude primordial binaries, modelling all cluster stars as initially being single stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' The absence of pri- mordial binaries can lead to underestimating ejections from the cluster centre (Heggie 1975).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' However, in most clusters, ≪1% of the stars are affected (Olczak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Results Observations investigate one specific cluster at a snapshot of its development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Mimicking this observational situation, we ran- domly choose one of our sets of simulations and investigate it at a specific time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' However, unlike actual observations, we have complete temporal information available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Hence, we know the past and the future of this particular cluster down to the path of each star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Equally, all other observational challenges, like mem- bership uncertainty due to back and foreground populations and limiting magnitudes, are removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' We even know each star’s ex- act properties like its mass, position, and velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' This informa- tion allows us to investigate the fundamental and unavoidable challenges in backtracking caused by the cluster dynamics that exist even without the mentioned additional observational diffi- culties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Bound and unbound stars After gas expulsion, bound and unbound stars coexist in the same spatial area for some time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Distinguishing the two popu- lations is vital for some applications;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' it does not matter or is not even desirable for others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' An example of the latter is the use of clusters in determining disc lifetimes (Haisch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Here, it is best to identify all stars that once formed together in the clus- ter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' However, if one is interested in the long-term development of clusters (≫ 20 Myr), one would be predominantly interested in the portion of stars that remain bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' We subsequently see here that using backtracking to distinguish between bound and unbound stars after gas expulsion is the key to success in ob- taining valuable information concerning a cluster’s past.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' At each Article number, page 3 of 14 A&A proofs: manuscript no.' metadata={'source': 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simulations with N = 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Velocity vectors of bound stars are highlighted in blue, and those of unbound stars in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Counter-intuitive examples of (a) outward-pointing distant bound stars and (b) inward-pointing central unbound stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Snapshot of the temporal development at (c) t=2 Myr and (d) t=10 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Backtracking from the results at 10 Myr to 2 Myr considering only the stars within 6 pc from the cluster centre for (e) bound stars only and (f) unbound stars only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Same backtracking con- sidering all the (g) bound stars and (f) unbound stars of the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' A film of the cluster dynamics and the backtracking can be found at https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='5281/zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='6041920 snapshot of the simulations, bound and unbound stars are de- fined as those having positive and negative total energy respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' However, in observations, distinguishing between these two states is often not straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Velocity vectors Individual stars are sometimes classified as bound or unbound simply because their velocity vectors point towards or away from the cluster centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' In the past, doubts about this approach were usually anchored on the fact that only two-dimensional informa- tion was available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' However, even with three-dimensional infor- mation becoming more accurate, this method is not advisable even for perfectly known 3D velocities for the following reason: The top row of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 1 shows a typical snapshot of a randomly chosen example from our sample of simulated clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' The clus- ter centre is marked as a green dot as a reference point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' As the many outward-pointing velocity vectors indicate, this cluster is in the expansion phase, with many former members becoming unbound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Nevertheless, a considerable fraction of the outward- pointing velocity vectors belongs to stars that remain bound in the long term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Examples of such stars are shown in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Equally, stars that point inwards and are close to the cluster centre can nevertheless be unbound (shown in red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' The dynamics of these example stars can be seen better in the corresponding video at https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='5281/zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='6041920.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Especially among the bound stars with outward-pointing velocity vectors, quite a few are bound despite being located at relatively large distances from the cluster centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' We find that there is a high failure rate in this approach, not only for this specific cluster, but for all clusters in our extensive sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' The situation improves for clusters aged more than 15 Myr as many of the unbound stars are better identifiable by their larger distances to the cluster cen- tre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Advantage of using unbound stars for backtracking The size of a cluster before expansion sets in is an essential pa- rameter for constraining the cluster formation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Besides the density profile, the size of the cluster core and half-mass radius are good indicators of the cluster density and, thus, the importance of the environment in the star and planet formation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' The environment’s influence includes close stellar fly- bys and external photo-evaporation that can truncate protoplan- etary discs or completely destroy them (Vincke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Win- ter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Concha-Ramírez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' These processes influence the type and frequency of the formed planetary sys- tems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Another example is binary capture and destruction pro- cesses which can alter the binary fraction in clusters (Kaczmarek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Marks et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Guszejnov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' We find that using just the unbound stars gives the best re- sult in determining the pre-expansion cluster size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' As an exam- ple, the second row in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 1 illustrates the cluster expansion by showing the bound and unbound stars, including their velocity vectors, (a) shortly after gas expulsion and (b) at 10 Myr for a cluster with N = 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' We note the different scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Using only the bound stars for backtracking (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 1g) results in a relatively poor constraint on the pre-expansion size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' The best performance is obtained using only the unbound stars (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 1f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' The rea- son is twofold: First, the velocity vectors of the unbound stars are rarely altered after gas expulsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' By contrast, bound stars quickly lose the memory of the pre-gas expulsion phase due to their ongoing interaction with their fellow cluster members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' In particular, close encounters hinder efficient backtracking for the bound stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Second, there is a more significant number of un- bound than bound stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Thus, statistical uncertainties are more easily averaged out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' All plots show the bound stars in blue and the unbound stars in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' A simulation of N = 1000 stars is used here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='0 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='0 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='0 d [pc] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='0 v [km/s] Time= 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='0 Myr Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Phase space diagram for an N = 1000 star cluster simulation at t = 10 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' The bound and unbound members are shown in blue and red colours respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Vertical and horizontal red lines indicate dis- tance and velocity cutoffs respectively for unbound stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' The light blue line represents the analytical escape velocity dependence on distance from the cluster centre derived assuming a Plummer distribution for the members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' The black crosses show the stars that underwent a strong en- counter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' cluster size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' All the simulations of N = 1000 cluster have been used to obtain these distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' It can be seen that the size distribution obtained using unbound stars is closer to the real size distribution than the size distribution obtained using bound stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Performing a t-test on the two size distributions with the null hypothesis being that the distributions have the same mean— while the alternative hypothesis is that bound stars have a larger mean than unbound stars—results in a p-value much lower than the significance level α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Hence, unbound stars are clearly better at recovering the size of the cluster before gas expulsion than bound stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Distance and velocity cutoffs for bound-unbound classification While distinguishing between the bound and unbound popula- tion is straightforward in simulations, it is very challenging in observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Often a cut in the distance to the cluster centre or the velocity is used to distinguish between bound and unbound stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Here we want to test when such a method is successful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' In our simulation, the relevant time frame starts at 2 Myr, when the gas expulsion happens, and many stars become un- bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Figure 2 shows snapshots of the distributions of the stel- lar distance to the cluster centre and velocity distribution before (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='8 Myr), just after gas expulsion at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='3 Myr, during the expan- sion process (5 and 10 Myr) and towards the end (20 Myr) of the expansion phase for an example cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' The distributions for the bound (blue) and unbound (red) stars are shown separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' As we chose the cluster to be in virial equilibrium, very few stars become unbound before gas expulsion (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' The few unbound stars during this phase result from close encounters leading to ejections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' However, after gas expulsion, many stars become unbound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Bound and unbound stars share considerable parts of the phase space for quite some time, as seen in the bot- tom row of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' This increases the complexity of making the distinction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' In observations, usually, a velocity cutoff is chosen as a given deviation from the mean for making this distinction (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Luh- man 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Bastian 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Esplin & Luhman 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' However, the location of these cutoffs is not apparent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Thus, there is some ele- Article number, page 5 of 14 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' main_new 0 1 2 3 4 5 Half-mass radius [pc] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='5 Real: r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='67, r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='13 Unbound: u = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='47, u = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='12 Bound: b = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='70, b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='61 Real Unbound Bound 1 2 3 4 5 Half-mass radius [pc] Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Distributions of sizes derived using actual positions of all stars (Real, shown in green), using backtraced positions of unbound stars (Unbound, shown in orange), and using backtraced positions of bound stars (Bound, shown in blue) shown with histograms (top) and boxplots (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' The box extends from the lower to upper quartile values of the data, with a line at the median while the whiskers reach 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='5 times the interquartile range from the box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' ment of arbitrariness here, and this is even more so for distance cutoffs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' However, in our simulations, we are in the ideal situation where we can determine where to apply the cutoff in distance and velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' These experiences can be used to provide guide- lines for both types of cutoffs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Figure 5 shows suggestions for the choice of distance and velocity cutoff for clusters older than 5 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' These have been calculated to minimise the sum of the false positive rate (FPR) and false negative rate (FNR) for all the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' It does not make much sense to make distance and velocity cutoffs in clusters younger than at least 5 Myr to avoid substan- tial errors in the classification of the members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' However, even at 5 Myr, the FPR and FNR introduced by a cutoff can be of the order of 15% – 30%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Generally, the percentage of stars identified as bound members while being unbound is higher than the oppo- site situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Only for clusters older than 10 Myr, this method is relatively robust as the overlap in phase space is of the order of 5% – 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Figure 3 shows the phase space diagram for a simu- lated cluster of 1000 stars with red lines at a distance of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='09 pc and a velocity of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='78 km/s representing the distance and veloc- ity cutoffs shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Applying these to the distribution of all simulations of 1000 stars leads to a median FNR of 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='7%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' The 25th and 75th percentile of the distribution of FNR are 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='5% and 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='4%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' We represent this as an FNR of 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='7+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='7 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='2%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Similarly, an FPR of 0 ± 0% is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' The percentage of cor- rectly identified stars is found to be 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='1 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Combining distance and velocity cutoffs gives the best dis- tinction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' This can be done by analytically determining the de- pendence of the escape velocity of the stars on the distance from the cluster’s centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Although the distribution of the stars in the simulations follows a King (1966b) profile, we use an approxi- mation of a Plummer (1911) profile to obtain an analytical solu- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' The escape velocity vesc(r) at any point in the cluster is then described by vesc(r) = � 2GMcl √ a2 + r2 , (3) where Mcl is the cluster mass, and a is the initial half-mass radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' This analytical cutoff can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 3 as the blue curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Applying this as the cutoff for bound-unbound star dis- tinction leads to an FPR of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='74+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='91 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='47% and an FNR of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='80+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='88 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' The median of the distribution of the correctly identified stars’ percentage is found to be 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='7+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='5 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='7%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Hence, this analytical cutoff is an improvement over the distance and velocity cutoffs in the case of our simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Backtracking In the following, we use our simulations of the cluster dynam- ics to develop guidelines for backtracking depending on cluster type, age, and mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' We subsequently demonstrate that using the right subset of stars for backtracking is the key to making the most of the available information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Here, we employ the simplest form of backtracking, namely, taking present-day positions and velocities as constant values and just reversing the arrow of time (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' neglecting any source of acceleration acting upon the stars).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' The high quality of the recent Gaia data allows backtrack- ing from the observed present situation holding the promise to reveal information about a cluster’s past.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' So far, unbound stars are chiefly analysed as ‘runaway’ (v > 30 km/s) stars and ‘walk- away’ (5 km/s < v < 30 km/s) stars (Eldridge 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Schoettler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' The idea is that both types of high-velocity stars have been ejected from their star-forming regions, and back- tracking will allow us to determine their origins and characterise their parent star cluster (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Olczak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Farias et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Schoettler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Schoettler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' (2022) search for runaway and walkaway stars within 100 pc of the 3–5 Myr old cluster NGC 2264 using Gaia DR2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' They compare the num- ber of the runaway and walkaway stars (17) to a range of N- body simulations with different initial conditions and find con- sistency with initial conditions with a high initial stellar density (≈ 10 000 M⊙ pc−3) and a high initial amount of spatial substruc- ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' However, our simulations find that high-velocity ejec- tions are rare for short-lived clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' We found no ejections with v > 30 km/s and only a few with v > 5 km/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Thus, back- tracking based on runaway and walkaway stars suffers from low- number statistics for young clusters (< 20 Myr) typical for the solar neighbourhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' As the ejection happens mainly from the highest-density regions of the cluster, the derived age at gas ex- pulsion is too short, and the cluster size is also too small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' For the much denser clusters that turn into long-lived open clusters, the Article number, page 6 of 14 Arunima Arunima et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' : Unbound stars hold the key to star cluster history n200 n1000 n4000 0 2 4 6 8 10 12 14 16 Distance cutoff [pc] n200 n1000 n4000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='00 Velocity cutoff [km/s] Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Distance (top) and velocity (bottom) cutoffs for selection of un- bound members for clusters with different number of members: N = 200, 1000, 4000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' The box extends from the lower to upper quartile val- ues of the data, with a line at the median while the whiskers reach 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='5 times the interquartile range from the box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' backtracking of cluster sizes is of higher quality as the number of ejected stars is higher and the ejection happens over larger areas of the cluster (Pfalzner & Kaczmarek 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Pre-expansion cluster size Using our simulation results as a starting point for backtracking, we find that the restriction to the unbound stars gives the best result in determining the pre-expansion cluster size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' This can be seen clearly in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 6 (top panel), where backtracked half-mass radius has been plotted against time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Backtracking the bound members provides no information, whereas using just unbound members fares much better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' It recovers the half-mass radius (rhm) of the cluster at the time of gas expulsion with a relative error of 121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='4+16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='3 −15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='0% to the relative error of 298.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='9+48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='1 −46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='7% obtained using bound members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' It is equally important to include the unbound stars from a sufficiently large area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 6 (bottom panel) shows a compari- son of the backtracked half-mass radius determined by consid- ering different areas for the member sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' The horizontal lines show the derived pre-gas expulsion half-mass radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' It can be seen that the half-mass radius derived from the unbound stars sampled from a relatively small area (10 pc) results in a consider- ably larger error than those derived from including the unbound 0 2 4 6 8 10 Time [Myr] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='5 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='0 Half mass radius [pc] Bound Unbound 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='80 Myr, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='40 pc 2 Myr, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='74 pc 0 2 4 6 8 10 Time [Myr] 0 2 4 6 8 10 12 Half mass radius [pc] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='72 Myr, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='19 pc 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='59 Myr, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='41 pc 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='27 Myr, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='49 pc 2 Myr, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='82 pc Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Backtracked half-mass radii for a simulation with 1000 stars, Top: using bound (blue) and unbound (red) members only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Red dashed lines show temb and rhm at the time of gas expulsion determined using unbound stars whereas black dashed lines show the actual values of the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Bottom: using unbound stars within 10 pc (blue), 20 pc (red) and 40 pc (green) from the cluster centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' The actual values of temb and rhm at the time of gas expulsion are shown in cyan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' stars from larger areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' In relative error terms, the error decreases from 248.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='9+41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='6 −27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='4% to 149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='1+16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='1 −14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='2% to finally, 121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='4+16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='3 −15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='0% as the search area around the cluster centre increases from 10 pc to 20 pc to 40 pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' The actual size of the ideal backtracking area depends, among others, on the cluster’s mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Details on this de- pendence can be found in Pfalzner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' (in preparation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Our simulations work with the idealised situation, where the search areas are uncontaminated by the presence of a population of foreground and background stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' In an actual application, extending the field increases the contamination by these fore- ground and background stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' A more significant fraction of con- taminants yields a larger half-mass radius estimate and a shorter age estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' As the ideal search radius increases as a function of cluster age, so do the errors due to the background population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' However, the advent of Gaia again improved the situation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' nev- ertheless, it is still a point to consider in real applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' While Rizzuto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' (2012) found ten years ago that the disc fractions in Upper Sco depend very much on cluster membership proba- bility and distance to the cluster centre, nowadays, a search area Article number, page 7 of 14 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' main_new of > 100 pc is regarded as giving reliable data (Luhman & Esplin 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Time of gas expulsion Backtracking can also be used to obtain information concern- ing the time when gas expulsion happened.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Here the same rules apply as for determining the pre-gas expulsion size: restricting to unbound stars and including sufficiently large sampling areas improve the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' In the example shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 6, the sim- ulated and the backtracked time of gas expulsion are shown as vertical lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' The backtracking of unbound members determines temb to be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='8 Myr, which is in excellent agreement with the ac- tual value from the simulations (2 Myr, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 6 top panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' The relative error in gas expulsion time derived using unbound stars is 40 ± 4% which is much better than that derived using bound stars (826+45 −84%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Moreover, including only the unbound particles within 10 pc is not advisable with its relative error of 88+11 −32% in the recovery of temb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' The error is reduced to 63+11 −8 % when the search area increases to 20 pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Although the results derived by including the unbound particles within 20 pc and 40 pc of the cluster’s centre give nearly identical results for this example cluster (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 6 bottom panel), the relative error in the derived temb decreases significantly to 40 ± 4% when all the N = 1000 simulations are considered for the 40 pc case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' The derived gas expulsion times tend to underestimate the time of gas expulsion by a 32+7 −6%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Given the general uncertainty of cluster ages, this can be considered a minimal error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Again, it is the stars that un- derwent close encounters that are responsible for the derived too short times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Further improvements We saw that using the unbound stars from a sufficiently large area gives the best backtracking results for the pre-gas expul- sion half-mass radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' However, the value can still be a factor of two too large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' One reason is that even some of the unbound stars have a relatively strong encounter before leaving the cluster (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' However, the main reason is that backtracking the un- bound stars gives the half-mass radius of the unbound, not that of the entire cluster sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' The stars that become unbound are predominantly located at the outskirts of the cluster at the mo- ment of gas expulsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Therefore, backtracking them, one ob- tains a value that is larger than the complete half-mass radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' The actual pre-gas expulsion half-mass radius includes the un- bound stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' However, simply multiplying the determined value by a factor of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='5 recovers the half-mass radius in our case quite well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' For our simulations, the empirical scaling factor has a value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='46+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='06 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' There does not seem to be any correlation between the cluster mass and the scaling factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Although the Spearman correlation coefficient is calculated to be −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='0133, the p-value for the hypothesis test of their correlation is found to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='48 which is greater than the significance level α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Hence, the null hypothesis that the cluster mass and the scaling factor are unrelated can not be rejected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' To some degree, the actual cor- rection value might depend on the star formation efficiency in the clusters, however, new sets of simulations with varying star formation efficiencies need to be analysed to establish the depen- dence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' The gas dispersion timescale, on the other hand, should not affect the factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='5 Time [Myr] 0 1 2 3 4 5 Half mass radius [pc] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='2 M : 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='54 Myr, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='98 pc 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='3 M : 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='54 Myr, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='78 pc 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='5 M : 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='55 Myr, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='57 pc all stars: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='57 Myr, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='52 pc 2 Myr, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='32 pc 0 2 4 6 8 10 Time [Myr] 0 5 10 15 20 25 Half mass radius [pc] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='51 Myr, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='54 pc 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='40 Myr, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='22 pc 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='58 Myr, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='44 pc 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='68 Myr, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='67 pc 2 Myr, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='19 pc Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Backtracked half-mass radii for a simulation with 4000 stars, Top: calculated using actual masses (green), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='2 M⊙ (red), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='3 M⊙ (blue) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='5 M⊙ (yellow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Bottom: calculated using exact velocity values (green), using vz = 0 (red), using velocities values with systematic er- rors as well as different levels of statistical uncertainty (blue: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='27 km/s & yellow: 1 km/s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' The actual values of temb and rhm at the time of gas expulsion from the simulation are shown in cyan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Mass of stars When we determine bound and unbound stars in a cluster, the mass of the stars plays a role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' However, in observations, the stel- lar classification is often known but not the actual mass of the stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Especially for young clusters, there are large uncertainties between these two properties, and the assumption of different evolutionary models leads to significant differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Here, we test to what extent this uncertainty in classification as bound or bound due to missing mass information influences backtracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' To mimic this problem, we assign the same mass to all stars, determine the bound and unbound stars and then perform the same backtracking procedure as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Figure 7 (top) shows the result of backtracking with the fully known IMF (green) and with the assumption that all stars have the same mass (Ms = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='2 M⊙, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='3 M⊙ and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='5 M⊙).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' It can be seen that not knowing the actual masses of the stars does not influence the derived time of gas expulsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' In all cases, it is too low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' The relative error for the derived temb is 46+3 −2% for the case of using actual stellar masses Article number, page 8 of 14 Arunima Arunima et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' : Unbound stars hold the key to star cluster history (green curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Using the same stellar mass for all stars increases this error only marginally to 52+3 −4%, 49+3 −2%, and 47+3 −2% for the case of Ms = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='2 M⊙ (red), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='3 M⊙ (blue), and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='5 M⊙ (yellow) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' The situation is different for the cluster size at the moment of gas expulsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Here, assuming that all stars have the same mass leads to up to a factor of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='2 larger sizes than using the actual stellar masses in the case shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 7 (top).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' The smaller the assumed mass, the error is larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' The relative error for the derived rhm is 124.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='4+8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='5 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='3% for the case of using actual stellar masses (green curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' This error increases to 130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='6+9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='6 −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='0% when using stellar mass as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='5 M⊙ (yellow), to 155.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='6+11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='2 −9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='4 % for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='3 M⊙ (blue), and to 180.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='1+15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='0 −12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='1% for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='2 M⊙ (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='2 We find that assuming all stars to have a mass of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='5 M⊙, which corresponds to the mean stellar mass in the cluster, is the best alternative to knowing the actual stellar masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Velocity in the z direction We also consider the effects of errors in the vz values on the back- tracking in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 7 (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' The velocity component along the z axis, corresponding with close approximation to the radial ve- locity component, constitutes the main source of uncertainty in the total velocity vector (Krolikowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' As a starting point, we consider the effect induced by the existence of non-null proper motion uncertainties;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' the error on radial velocity is for the moment assumed to be null.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Gaia DR2 data have systematic un- certainties in the measurement of parallax and proper motions (Lindegren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Vasiliev 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' The 2D random error is considered to be of the order of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='27 km/s, equivalent to the er- ror in 2D proper motion (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='28 mas yr−1) for sources with G = 17 mag at a distance of 200 pc in Gaia DR2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Using this error, blue curve is obtained for backtracked radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' The pre-expansion size is derived to be about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='5 times the size obtained compared to the velocities having no error (green curve in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 7, bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' The relative error distributions (with respect to the actual rhm) are de- termined for rhm obtained using velocities with no error (green) and using velocities with error (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' The relative error in rhm goes from 124.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='4+8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='5 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='3% for the green curve to 213.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='7+12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='7 −10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='5% for the blue curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' An accuracy improvement is seen for the value of the cluster’s age at the time of gas expulsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' The relative er- ror decreases from 46+3 −2% for the green curve to 35+4 −5% for the blue curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' However, this improvement is less due to recovering more information about the cluster’s past, but more with a gen- eral move of the curve towards the right on the time axis with an increase in the standard deviation in random errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' The impact of radial velocity errors results in an even shorter estimate of the expansion timescale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Krolikowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' (2021) point out that the radial velocity (RV) uncertainty is roughly an order of magnitude larger than the reported projected proper mo- tion uncertainty, even when collecting RV measurements from more precise catalogues than Gaia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' (2022) also point out that even with future Gaia releases, the precision of RV would be ∼ 1 km/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' The yellow curve in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 7 (bottom) cor- responds to the backtracked radii determined using the same systematic error but a random error of 1 km/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' This increases the relative error in temb and rhm at the time of gas expulsion to 60+8 −13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='5% and 639.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='0+35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='9 −41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='1% respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='54% of the sources with astrometric data have the RV measurements available in Gaia DR2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' For the extreme situation of zero information on vz, the red curve in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 7 (bottom) is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' The relative error for the determined size in this case 2 The distributions of sizes and gas expulsion times derived using dif- ferent masses can be seen in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' is the highest of all previously discussed cases at 821.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='6+47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='6 −55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='5% whereas the relative error in derived time of gas expulsion is 40+10 −12%3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' In reality, for Gaia DR2, the deviation from the actual parameter values will be somewhere between the cases of vz = 0 and the added systematic error along with statistical uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Application to observational data So far, we have dealt exclusively with the idealised situation that simulations provide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' In the following, we want to show two ex- amples of applying backtracking procedures to observed clus- ters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' The aim is not so much the age and initial size determination of these specific clusters, but to show which additional problems can be expected in real applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Therefore, we choose two clusters that differ considerably in age and geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' When re- ferring to the age of the cluster, we quote the time elapsed since the gas started to be expelled and refer to the cluster age as the median age of all the stars in the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' This differs from the time elapsed since the molecular cloud started producing stars (Pecaut & Mamajek 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Fujii et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' NGC 6530 We first apply the before-described backtracking method to NGC 6530, which is a young cluster within Lagoon Nebula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Its age has been estimated to be 1–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='3 Myr (Prisinzano, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Mayne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Bell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2013) and its distance to be 1326+77 −69 pc (Wright et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Damiani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' We use the cat- alogue of members provided by Wright et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' (2019), who use GES spectroscopy, Gaia DR2 astrometry, and ancillary member- ship information from X-ray, infrared, and Hα surveys to com- pile the said catalogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 691 of these cluster members have Gaia DR2 data and have been used in the following analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' We as- sume that all the stars have a mass of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='5 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Using the radial velocity for individual sources when available and assuming it to be equal to the bulk radial velocity of the cluster when not, 3D positions and velocities of the stars are calculated in the stan- dard right-handed Cartesian Galactic frame using the conversion equations prescribed by the Gaia DR2 documentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' These are then used to determine the bound and unbound members of the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' For backtracking the stars’ trajectories, we backtrack the po- sitions in the plane of the sky using the velocities along α and δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Radial velocity is used to backtrack along the line-of-sight and change the distance of the stars which is assumed to be the same for all stars at the present time (1326 pc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Although indi- vidual distances are available for all the stars (Bailer-Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2018), the uncertainty is extremely high (fractional uncertainty is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='20+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='43 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='09 as compared to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='02 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='01 for the distance data- set of member stars of Upper Sco in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='2) and leads to very high half-mass radius along with loss of most information about the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' The calculated coordinates are then converted to the Cartesian coordinates to calculate the half-mass radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' The result of this procedure is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 8 (left panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' However, for considering the uncertainty in astrometry of the member stars, we run 1000 Monte Carlo simulations, that is to say repeat the entire procedure while varying astrometric information in a ran- dom, normal manner according to the uncertainties associated with each Gaia DR2 source’s parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' For the distance value for all the stars, the uncertainty is taken as 73 pc (Wright et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' The results of these simulations are fitted with a Gaussian 3 The distributions of values of size and gas expulsion time obtained for all the cases discussed here can be seen in Appendix B Article number, page 9 of 14 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' main_new 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='0 Time [Myr] 0 10 20 30 40 Half mass radius [pc] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='04 Myr, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='01 pc 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='0 Time [Myr] 10 15 20 25 30 Half mass radius [pc] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='54 Myr, 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='14 pc 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='0 Time [Myr] 10 12 14 16 18 20 22 24 26 Half mass radius [pc] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='25 Myr, 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='16 pc 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='04 Myr, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='29 pc Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Backtracked (and extrapolated) half-mass radii determined for bound (blue) and unbound (orange) stars 10 Myr into the past and into the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' The green dashed lines show the minima of the backtracked half-mass radius for unbound stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Left: For NGC 6530 members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Middle: For Upper Sco members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Right: Backtracked (and extrapolated) half-mass radii determined for the unbound members of subclusters of Upper Sco.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' to obtain the parameters of the cluster along with their errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Hence, we find the gas expulsion to have happened 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='03 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='03 Myr ago and the size of the cluster at the time of gas expulsion is found to be 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='16 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='23 pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' This agrees well with the current age estimate of the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' However, the half-mass radius might be underestimated by the assumption of a fixed distance of the stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' A more realistic estimate might be obtained by multiplying it by a factor √3/2, which would yield a limit of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='09 pc on the cluster size at the time of gas expulsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Despite obtaining a reasonable fit, the reservations pointed out in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='5 also hold here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' The median uncertainty in proper motion amount to 2 km/s (Wright et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Any un- certainty added to the true velocity acts to reduce the best fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' This uncertainty is the most problematic issue in applying the backtracking method for determining the age of NGC 6530.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Upper Scorpius Upper Sco is a sub-group of Sco-Cen that has been widely stud- ied with the Gaia data, identifying the cluster’s members (Galli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Wilkinson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Luhman & Esplin 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Damiani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Žerjal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Squicciarini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Kerr et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2021) and an isochronal age of around 10 Myr has been recently accepted (Feiden 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' David et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Luh- man & Esplin 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Sullivan & Kraus 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' We test the quality of the backtracking for clusters with a more complex morphol- ogy using Upper Sco as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' We use the list of mem- bers compiled by Luhman & Esplin (2020) using optical and IR spectra to confirm the stars’ youth while parallax and proper mo- tion offsets to get the kinematic criteria for these candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' The list contains 1761 member candidates, 1682 of which have Gaia DR2 data available and have been used in the following analy- ses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' We apply the same method described for NGC 6530 with the exception of considering individual distances for the stars in this case as the uncertainty in distance is much lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Despite its complex morphology, we first work with the as- sumption that Upper Sco was a centrally condensed spherical structure in the past.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' In this case, we find that the cluster went through gas expulsion 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='54 Myr ago and had a half-mass radius of 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='14 pc at this time as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 8 (middle panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' How- ever, the Monte Carlo simulations for error propagation estima- tion provide the gas expulsion time to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='80 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='21 Myr ago while the cluster size is found to be 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='11 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='11 pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' This value agrees with other backtracking results for Upper Sco.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' For example, Žerjal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' (2021) determine the kinematic age of the population in the Upper Sco region as 4 ± 4 Myr, whereas Squicciarini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' (2021) find 8 subclusters with kine- matic ages varying from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='1 Myr to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='4 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' How- ever, this cluster age deviates considerably from that of 10 Myr obtained by applying corrections, for undetected binaries (Sulli- van & Kraus 2021) or strong magnetic fields impeding convec- tion in low-mass stars (Feiden 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' David et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2019), to the isochronal age determination of Upper Sco.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' One possible expla- nation for this discrepancy would be that the backtracking yields the time elapsed since gas was expelled and refers to the age of the youngest stars in the association.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Taking into account a star formation history lasting 6-7 Myr, most stars might be about 11 Myr old and the median age of the association ≈ 7 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' These values are more similar to the ones obtained through stellar evo- lution models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Additional complications arise from Upper Sco, unlike NGC 6530, being highly substructured (Kerr et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Squicciarini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Likely, star formation did not happen as a single burst, but was rather characterised by several formation episodes (Galli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Thus, the assumption of a centrally condensed spherical structure in the past is oversimplifying the situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Hence, we try to improve our analysis by considering Upper Sco to consist of subclusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' A density distribution of the cluster members on the plane of the sky at the present time is plotted (see Appendix C for more details and plots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Two dense areas seem to emerge and we consider two rectangles in these areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' The members’ positions are traced back using the same method as described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' When a member star enters one of the said rectangles, it is assigned to the corresponding subcluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' After the assignment of subcluster membership using this simplified method, the backtracked and extrapolated half-mass radii are de- termined using unbound stars for both subclusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' The result is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 8 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' To determine the errors, the Monte Carlo simulations are used which provide the time of gas expulsion in the two subclusters as −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='09±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='29 Myr and −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='25±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='17 Myr ago respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Similarly, the half-mass radii at the time of gas ex- pulsion is found to be 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='15±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='20 pc and 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='10±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='23 pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Various characterisations of the subclusters are summarised in Table C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' There is a slight improvement in the determination of the size and time of gas expulsion when considering Upper Sco to have subclusters rather than being one coeval population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' However, it must be reiterated that ours is a simplified method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' More robust clustering methods can be used in the future to get better results on the subcluster membership and hence, their parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' For example, Kerr et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' (2021) use HDBSCAN clustering algorithm on Gaia DR2 data and find 9 subclusters in the Upper Sco re- gion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Two of these (Group H and Group I) have more than 100 members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' We analyse these subclusters and find the time of gas expulsion and their sizes at that time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' According to our results, Article number, page 10 of 14 Arunima Arunima et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' : Unbound stars hold the key to star cluster history gas expulsion in Group H happened 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='40 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='42 Myr ago and its half-mass radius was 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='96 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='215 pc at the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' For Group I, the gas expulsion happened 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='78±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='91 Myr ago and its size was 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='73 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='37 pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' The age found by Kerr et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' (2021), using Gaia DR2’s photometric data, for the groups is 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='7 Myr and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='4 Myr respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' So, even though there is an improve- ment in the age and size estimates when using a more robust clustering algorithm, the kinematic age estimates still show con- siderable deviation from the photometric estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Availability of accurate radial velocities and distances for the member candi- dates to use in the subclustering analysis in future would improve the situation further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Discussion The improvement in the cluster size, when considering sub- clusters, already shows that backtracking is more complex for substructured clusters like Upper Sco.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Thus, the less substruc- tured a cluster is, the more straightforward the backtracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' The substructured clusters require backtracking to multiple centres, which is the more complex the more subcluster centres exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Another potential difficulty could be the presence of multi- ple differently aged populations in the Upper Sco region leading to the miscalculation of the cluster’s age (Wright & Mamajek 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Žerjal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Squicciarini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' However, this would require large subgroups to be well over 15 Myr to intro- duce such a substantial error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' This seems unlikely as an expla- nation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' We suspect that the real reason is a different one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' The arguments based on kinematic analysis of a cluster for its his- tory can not be considered on their own due to the significant errors in radial velocity and its unavailability for most stars in Gaia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Large uncertainty in the velocities of the stars can lead to a significant loss of information about the past of the cluster (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 7, bottom panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' This might be the reason for underesti- mating the cluster age and overestimating the size at the time of gas expulsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Furthermore, the assumptions in the backtrack- ing analysis are numerous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' The exact masses of the stars are unknown, so the distinction between bound and unbound stars could be highly inaccurate when combined with astrometric un- certainties and incomplete or inaccurate membership of the clus- ter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' In conclusion, the determination of a much younger age, of the Upper Sco region, by kinematic analysis than the more accu- rate isochronal determination could be affected by multiple, dif- ferently aged and kinematically distinct populations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' however, precise radial velocity measurements are needed to rule out the possibility that the discrepancy in age determination is due to astrometric errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Summary and conclusion Young star clusters (< 10 Myr) are highly dynamical entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Therefore, observations provide only snapshots of this highly dynamic cluster evolution sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Nevertheless, in light of the unprecedented precision of Gaia position and velocity data, it should be possible to obtain information about a young cluster’s past using backtracking techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' In this work, we used simu- lations of the cluster dynamics as an idealised version to suggest how to optimise the backtracking method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Under ideal observa- tional conditions, the following statements should hold: – For backtracking to be successful, it is essential to distin- guish between bound and unbound cluster members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Under ideal conditions, backtracking the unbound members exclu- sively, the time of gas expulsion can be determined with only a 32% error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' However, the quality of the backtracking de- pends on the number of cluster stars, with the best results obtained for clusters containing a few thousand stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' – While still the best result, the sizes backtracked from un- bound members are about a factor of two larger than the ac- tual value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' However, this error is systematic and reflects that unbound members are primarily located at the cluster out- skirts at the time of gas expulsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Thus, applying a correc- tion factor of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='46 approximates the actual value very well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' – For obtaining this accuracy, it is essential to determine all the unbound members to > 20 – 40 pc from the cluster centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' – The classification of bound and unbound stars based on the direction of their velocity vectors, or ad hoc distance or ve- locity cutoffs is highly error-prone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' We provide analytical cutoffs based on the escape velocity and the number of clus- ter members with a success rate of 96% – 97% for distin- guishing between bound and unbound stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' – Runaway and walkaway stars are less suitable to determine past cluster properties because of their low number and their production by dynamical ejection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Ejection traces only past locations of high stellar density regions but not actual cluster sizes or the time of gas expulsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Uncertainty in membership and stellar properties provide additional challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Modelling these uncertainties, we find that the lack of information about the line-of-sight velocity can severely affect the determination of the pre-expansion size of the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Nevertheless, the time of gas expulsion can still be esti- mated with an error of 40% − 60% due to the unavailability of radial velocities and uncertainty in the value even when avail- able.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' The uncertainty in the mass of the members seems to af- fect the results much less.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Similarly, larger search areas often struggle with higher false-positive and -negative rates in mem- bership.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Applying our results to observational data, the method works reasonably for centrally concentrated clusters, but less for very substructured clusters like Upper Sco.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' For such substruc- tured clusters, backtracking to the individual subcluster centres would be the next step to pursue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' In summary, restricting backtracking to the unbound stars al- lows deducing the times of gas expulsion and the pre-expansion cluster size values with relatively high accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Analysing a large number of clusters with the presented method will allow drawing valuable conclusions about the clustered star formation process in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' We thank the referee for a very detailed report that made this article significantly better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' This work has made use of data from the European Space Agency (ESA) mission Gaia (https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' cosmos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='esa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='int/gaia), processed by the Gaia Data Processing and Anal- ysis Consortium (DPAC, https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='cosmos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='esa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='int/web/gaia/dpac/ consortium).' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2007, MNRAS, 375, 1220 Michel, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=', van der Marel, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=', & Matthews, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2021, ApJ, 921, 72 Nürnberger, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' & Petr-Gotzens, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2002, A&A, 382, 537 Olczak, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=', Pfalzner, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=', & Eckart, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2008, A&A, 488, 191 Olczak, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=', Pfalzner, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=', & Spurzem, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2006, ApJ, 642, 1140 Pang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=', Li, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=', Tang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=', Pasquato, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=', & Kouwenhoven, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2020, ApJ, 900, L4 Pecaut, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' & Mamajek, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} 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Sciortino, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2005, A&A, 430, 941 Ribas, Á.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=', Merín, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=', Bouy, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=', & Maud, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2014, A&A, 561, A54 Richert, A.' metadata={'source': 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421, L97 Schoettler, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=', de Bruijne, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=', Vaher, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=', & Parker, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2020, MNRAS, 495, 3104 Schoettler, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=', Parker, R.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=', Ireland, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=', Crundall, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=', Krumholz, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=', & Rains, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 2021, in Star Clusters: the Gaia Revolution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Online Workshop, 29 Article number, page 12 of 14 Arunima Arunima et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' : Unbound stars hold the key to star cluster history Appendix A: Mass of stars We discussed how the unavailability of the mass of stars in ob- servations affects the determination of gas expulsion time and cluster size at the time of gas expulsion using backtracking anal- ysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Here, we provide the distributions of the derived sizes and gas expulsion time (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='1) for all the cases discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='0 Half-mass radius [pc] 0 1 2 3 4 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='14, = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='09 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='56, = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='10 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='92, = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='16 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='64, = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='13 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='21, = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='18 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='0 Time [Myr] 0 2 4 6 8 10 12 green: = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='54, = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='04 blue: = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='51, = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='04 red: = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='48, = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='05 yellow: = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='53, = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='04 actual value of gas expulsion time Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Distributions of the backtracked half-mass radii (top) and the time of gas expulsion (bottom) obtained using actual masses (green), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='2 M⊙ (red), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='3 M⊙ (blue) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='5 M⊙ (yellow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' The actual values of rhm at the time of gas expulsion (as a distribution) and temb from all the simulations (of N=4000 clusters) are shown in cyan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Appendix B: Velocity in the z direction Similarly, we provide the distributions of the derived sizes and gas expulsion time for all the cases in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='5 to supplement the discussion of the effects of errors in the vz values on the back- tracking analysis and derived parameters (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Appendix C: Upper Sco subclusters The density distribution of the Upper Sco members is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='1 (top) along with the rectangles showing the subclus- ter areas used for the subcluster membership assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Figure C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='1 (bottom) shows the scatter plot of the member stars with the 0 2 4 6 8 10 12 Half-mass radius [pc] 0 1 2 3 4 cyan: = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='14, = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='09 green: = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='56, = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='10 blue: = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='57, = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='10 yellow: = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='37, = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='17 red: = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='44, = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='42 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='50 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='75 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='00 Time [Myr] 0 2 4 6 8 10 12 green: = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='54, = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='04 blue: = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='66, = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='06 red: = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='62, = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='18 yellow: = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='58, = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='17 actual value of gas expulsion time Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Distributions of the backtracked half-mass radii (top) and the time of gas expulsion (bottom) obtained using exact velocity values (green), using vz = 0 (red), using velocities values with systematic er- rors as well as different levels of statistical uncertainty (blue: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='27 km/s & yellow: 1 km/s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' The actual values of rhm at the time of gas expulsion (as a distribution) and temb from all the simulations (of N=4000 clusters) are shown in cyan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' same rectangles and the members of the two subclusters in red and green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' The purple points represent the few members which did not enter any of the rectangles in the 10 Myr up to which the positions were backtracked and hence, are not assigned to any subcluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Furthermore, Table C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='1 provides characteristic infor- mation about the subclusters identified in this work as well as about Group H and I from Kerr et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Article number, page 13 of 14 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' main_new Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Density distribution (top) and scatter plot (bottom) of the Up- per Sco members at the present time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' The two rectangles show the area selected for the clustering process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Green and red points in the bottom plot show the members of Group 1 and Group 2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Purple points are the ones which were not assigned to any group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Table C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Information about the subclusters identified in this work (ID: 1,2) and the groups from Kerr et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' (2021) (ID: H, I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' ID N RA Dec tK rhm [deg] [deg] [Myr] [pc] 1 1102 241.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='60 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='93 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='09 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='29 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='15 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='20 2 454 245.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='68 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='12 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='25 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='17 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='10 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='23 H 102 240.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='6 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='4 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='40 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='42 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='96 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='21 I 110 246.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='4 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='9 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='78 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='91 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='73 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='37 Notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Number of stars (N) and mean positions (RA, Dec) are provided along with the time of gas expulsion (tK, kinematic age) and half-mass radius of subcluster at the time of gas expulsion (rhm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' Article number, page 14 of 14 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='0 18 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='5 20 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='0 Number 22 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='5 (。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=')9 24 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='0 of sources 26 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='5 28 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='0 30 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content='0 235 240 245 250 α(°)-16 18 20 22 。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} +page_content=' 24 26 28 30 32 232.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfngT9/content/2301.03311v1.pdf'} 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b/D9AzT4oBgHgl3EQfwv5m/content/tmp_files/2301.01727v1.pdf.txt @@ -0,0 +1,1790 @@ +arXiv:2301.01727v1 [nlin.SI] 4 Jan 2023 +Classical Solutions of the Degenerate +Fifth Painlev´e Equation +Peter A. Clarkson +School of Mathematics, Statistics and Actuarial Science, +University of Kent, Canterbury, CT2 7FS, UK +Email: P.A.Clarkson@kent.ac.uk +January 5, 2023 +Abstract +In this paper classical solutions of the degenerate fifth Painlev´e equation are classified, which +include hierarchies of algebraic solutions and solutions expressible in terms of Bessel functions. Solu- +tions of the degenerate fifth Painlev´e equation are known to expressible in terms of the third Painlev´e +equation. Two applications of these classical solutions are discussed, deriving exact solutions of the +complex sine-Gordon equation and of the coefficients in the three-term recurrence relation associated +with generalised Charlier polynomials. +1 +Introduction +In this paper we are concerned with solutions of the equation +d2w +dz2 = +� 1 +2w + +1 +w − 1 +� �dw +dz +�2 +− 1 +z +dw +dz + (w − 1)2(αw2 + β) +z2w ++ γw +z , +(1.1) +with α, β and γ constants. Equation (1.1) is the special case of the fifth Painlev´e equation (PV) +d2w +dz2 = +� 1 +2w + +1 +w − 1 +� �dw +dz +�2 +− 1 +z +dw +dz + (w − 1)2(αw2 + β) +z2w ++ γw +z + δw(w + 1) +w − 1 +. +(1.2) +with α, β, γ and δ constants, when δ = 0 and is known as the degenerate fifth Painlev´e equation (deg- +PV), cf. [42]. +The six Painlev´e equations (PI–PVI), were discovered by Painlev´e, Gambier and their colleagues +whilst studying second order ordinary differential equations of the form +d2w +dz2 = F +� +z, w, dw +dz +� +, +(1.3) +where F is rational in dw/dz and w and analytic in z. The Painlev´e equations can be thought of as +nonlinear analogues of the classical special functions. The general solutions of the Painlev´e equations +are transcendental in the sense that they cannot be expressed in terms of known elementary functions +and so require the introduction of a new transcendental function to describe their solution. However, +it is well known that PII–PVI possess rational solutions, algebraic solutions and solutions expressed in +terms of the classical special functions — Airy, Bessel, parabolic cylinder, Kummer and hypergeometric +functions, respectively — for special values of the parameters, see, e.g. [11, 22] and the references +therein. These hierarchies are usually generated from “seed solutions” using the associated B¨acklund +transformations and frequently can be expressed in the form of determinants. These solutions of the +Painlev´e equations are often called “classical solutions”, cf. [53, 54]. +It is well known that solutions of deg-PV (1.1) are related to solutions of the third Painlev´e equation +d2q +dx2 = 1 +q +� dq +dx +�2 +− 1 +x +dq +dx + Aq2 + B +x ++ Cq3 + D +q , +(1.4) +1 + +with A, B, C and D constants, a result originally due to Gromak [21]; see also [22, §34]. The purpose +of this paper is to give a classification and description of the classical solutions of deg-PV (1.1) directly, +rather than indirectly through (1.4). +In §2, the relationship between deg-PV (1.1) and the third Painlev´e equation (1.4) is discussed. In +§3, classical solutions of the third Painlev´e equation (1.4) are reviewed, the rational solutions in §3.1 +and the Bessel function solutions in §3.2. In §4, B¨acklund transformations of deg-PV (1.1) are given, +which can be used to derive a hierarchy of solutions from a “seed solution”. In §5, classical solutions +of deg-PV (1.1) are classified, the algebraic solutions in §5.1 and the Bessel function solutions in §5.2. +In §6, two applications of classical solutions of deg-PV (1.1) are given to derive exact solutions of the +complex sine-Gordon equation, which is equivalent to the Pohlmeyer-Lund-Regge model, and to derive +explicit representations of the coefficients in the three-term recurrence relation satisfied by generalised +Charlier polynomials, which are discrete orthogonal polynonials. +2 +The relationship between deg-PV and PIII +In the generic case when CD ̸= 0 in the third Painlev´e equation (1.4), we set C = 1 and D = −1, +without loss of generality (by rescaling the variables if necessary), and so consider the equation +d2q +dx2 = 1 +q +� dq +dx +�2 +− 1 +x +dq +dx + Aq2 + B +x ++ q3 − 1 +q . +(2.1) +In the sequel, we shall refer to this equation as PIII since it is the generic case. +Consider the Hamiltonian associated with PIII (2.1) given by +HIII(q, p, x; a, b, ε) = q2p2 − xq2p − (2a + 2b + 1)qp + εxp + 2bxq, +(2.2) +with a and b parameters and ε = ±1, see [28, 46]. Then p(x) and q(x) satisfy the Hamiltonian system +x dq +dx = ∂HIII +∂p += 2q2p − xq2 − (2a + 2b + 1)q + εx, +(2.3a) +x dp +dx = −∂HIII +∂q += −2qp2 + 2xqp + (2a + 2b + 1)p − 2bx. +(2.3b) +Solving (2.3a) for p(x) gives +p(x) = 1 +2q +� +x dq +dx + xq2 + (2a + 2b + 1)q − εx +� +, +and then substituting this in (2.3b) gives +d2q +dx2 = 1 +q +� dq +dx +�2 +− 1 +x +dq +dx + 2(a − b)q2 +x ++ 2ε(a + b + 1) +x ++ q3 − 1 +q . +(2.4) +which is PIII (2.1), with parameters +A = 2(a − b), +B = 2ε(a + b + 1). +(2.5) +Solving (2.3a) for q(x) gives +q(x) = +1 +2p(x − p) +� +x dp +dx − (2a + 2b + 1) + 2bx +� +, +and then substituting this in (2.3a) gives +d2p +dx2 = 1 +2 +�1 +p + +1 +p − x +� � dp +dx +�2 +− +p +x(p − x) +dp +dx + 2εp − 2b2 +p − 4a2 − 1 +2(p − x) + 1 − 4(a2 − b2) − 4εp2 +2x +. +(2.6) +Then making the transformation +p(x) = 2√z w(z) +w(z) − 1 , +x = 2√z, +(2.7) +2 + +in (2.6) gives +d2w +dz2 = +� 1 +2w + +1 +w − 1 +� �dw +dz +�2 +− 1 +z +dw +dz + (w − 1)2(a2w2 − b2) +2z2w ++ εw +z , +(2.8) +which is deg-PV (1.1) with parameters +α = 1 +2a2, +β = − 1 +2b2, +γ = ε. +(2.9) +Hence we have the following result; see also [22, Theorem 34.2]. +Lemma 2.1. If q(x) is a solution of (2.4) then +w(z) = xq′(x) + xq2(x) + (2a + 2b + 1)q(x) − εx +xq′(x) − xq2(x) + (2a + 2b + 1)q(x) − εx, +z = 1 +2x2, +(2.10) +with ′ ≡ d/dx is a solution of (2.8), provided that +x dq +dx − xq2 + (2a + 2b + 1)q − εx ̸= 0. +Conversely, if w(z) is a solution of (2.8), then +q(x) = +1 +2√z w +� +z dw +dz + (w − 1)(aw + b) +� +, +x = +√ +2z, +(2.11) +is a solution of (2.4). +Proof. Solving (2.3a) for p(x), substituting in (2.7) and solving for w(z) gives (2.10). Also solving (2.3b) +for q(x) and substituting (2.7) gives (2.11). +An alternative method of deriving solutions of (2.8) involves the second-order, second-degree equa- +tion satisfied associated with the Hamiltonian (2.2), due to Jimbo and Miwa [28] and Okamoto [46], +which is often called the “σ-equation”. +Theorem 2.2. If HIII(q, p, x; a, b, ε) is given by (2.2), then +σ(x; a, b, ε) = HIII(q, p, x; a, b, ε) + qp − 1 +2εx2 + (a + b)2, +(2.12) +where q(x) and p(x) satisfy the system (2.3), satisfies the second-order, second-degree equation (SIII) +� +xd2σ +dx2 − dσ +dx +�2 ++ 2 +��dσ +dx +�2 +− x2 +� � +xdσ +dx − 2σ +� +− 8ε(a2 − b2)xdσ +dx = 8(a2 + b2)x2. +(2.13) +Conversely, if σ(x; a, b, ε) satisfies (2.13) then the solution of the Hamiltonian system (2.3) is given by +q(x) = εxσ′′(x) − ε(2a + 2b + 1)σ′(x) − 2(a − b)x +x2 − [σ′(x)]2 +, +p(x) = 1 +2εσ′(x) + 1 +2x. +(2.14) +Proof. See Jimbo and Miwa [28] and Okamoto [46]. +Consequently solutions of (2.8) can be expressed in terms of solutions of SIII (2.13). +Corollary 2.3. If σ(x; a, b, ε) is a solution of SIII (2.13), then +w(z; a, b, ε) = σ′(x; a, b, ε) + εx +σ′(x; a, b, ε) − εx, +z = 1 +2x2, +(2.15) +is a solution of (2.8). +Proof. This immediately follows from (2.7) and (2.14). +3 + +3 +Classical solutions of PIII and SIII +3.1 +Rational solutions of PIII and SIII +Rational solutions of PIII (2.1) are classified in the following theorem. +Theorem 3.1. Equation (2.1) has a rational solution if and only if +ε1A + ε2B = 4n, +with n ∈ Z and ε2 +1 = 1, ε2 +2 = 1, independently. +Proof. For details see Lukashevich [32]; see also [39, 40]. +Umemura [55]1 derived special polynomials associated with rational solutions of (2.1), which we +now define; see also [9, 29, 30]. +Definition 3.2. The Umemura polynomial Sn(x; µ) is given by the recursion relation +Sn+1Sn−1 = −x +� +Sn +d2Sn +dx2 − +�dSn +dx +�2� +− Sn +dSn +dx + (x + µ)S2 +n, +(3.1) +where S−1(x; µ) = S0(x; µ) = 1, with µ an arbitrary parameter. +Remark 3.3. The Umemura polynomial Sn(x; µ) has the Wronskian representation +Sn(x; µ) = cnW (ϕ1, ϕ3, . . . , ϕ2n−1) , +cn = +n +� +k=0 +(2k + 1)n−k, +(3.2a) +where +ϕm(x; µ) = L(µ−2m+1) +2m−1 +(−x), +(3.2b) +with L(α) +k (x) the Laguerre polynomial, for details see Kajiwara and Masuda [30]; see also [9, 29]. +Theorem 3.4. The rational function solution of SIII (2.13) is given by +σn(x; µ, ε) = 2x d +dx {ln Sn(x; µ)} − 1 +2x2 − 2µx − 1 +4, +n ≥ 0, +(3.3a) +with Sn(x; µ) the Umemura polynomial, for the parameters +a = n + 1 +2, +b = µ, +ε = 1. +(3.3b) +Proof. See Clarkson [9]. +3.2 +Special function solutions of PIII and SIII +Special function solutions of PIII (2.1), which are expressed in terms of Bessel functions and are classi- +fied in the following Theorem. +Theorem 3.5. Equation (2.1) has solutions expressible in terms of the Riccati equation +x dq +dx = ε1xq2 + (Aε1 − 1)q + ε2x, +(3.4) +if and only if +ε1A + ε2B = 4n + 2, +(3.5) +with n ∈ Z and ε2 +1 = 1, ε2 +2 = 1, independently. Further, the Riccati equation (3.4) has the solution +q(x) = −ε1 +d +dx ln ψν(x), +(3.6) +1The original manuscript was written by Umemura in 1996 for the proceedings of the conference “Theory of nonlinear special +functions: the Painlev´e transcendents” in Montreal, which were not published; for further details see [47]. +4 + +where ψν(x) satisfies +xd2ψν +dx2 + (1 − 2ε1ν)dψν +dx + ε1ε2xψν = 0, +(3.7) +which has solution +ψν(x) = + + + + + + + + + +xν {C1Jν(x) + C2Yν(x)} , +if +ε1 = 1, +ε2 = 1, +x−ν {C1Jν(x) + C2Yν(x)} , +if +ε1 = −1, ε2 = −1, +xν {C1Iν(x) + C2Kν(x)} , +if +ε1 = 1, +ε2 = −1, +x−ν {C1Iν(x) + C2Kν(x)} , +if +ε1 = −1, ε2 = 1, +(3.8) +with C1, C2 arbitrary constants, and Jν(x), Yν(x), Iν(x), Kν(x) Bessel functions. +Proof. For details see Okamoto [46]; see also [11, 22, 36, 39, 40]. +Determinantal representations of special function solutions of PIII (2.1) were given by Okamoto +[46]; see also [19, 38]. +Theorem 3.6. Suppose τn(x; µ, ε) is the determinant given by +τn(x; µ, ε) = det +�� +x d +dx +�j+k +ϕµ(x; ε) +�n−1 +j,k=0 +, +(3.9a) +where +ϕµ(x; ε) = +� +c1Jµ(x) + c2Yµ(x), +if +ε = 1, +c1Iµ(x) + c2Kµ(x), +if +ε = −1, +(3.9b) +with c1, c2 arbitrary constants, and Jµ(z), Yµ(z), Iµ(z), Kµ(z) Bessel functions. +The Bessel function solution of SIII (2.13) is given by +σn(x; µ, ε) = 2x d +dx {ln τn(x; µ, ε)} + 1 +2εx2 + µ2 − n2 + 2n, +(3.10a) +for the parameters +a = n, +b = µ. +(3.10b) +Lemma 3.7. The determinant τn(x; µ, ε) given by (3.9) satisfies the equation +x2 +� +τn +d2τn +dx2 − +�dτn +dx +�2� ++ xτn +dτn +dx = τn+1τn−1, +(3.11) +or equivalently +� +x d +dx +�2 +ln τn = τn+1τn−1 +τ 2n +. +(3.12) +Proof. See Okamoto [46, Theorem 2]. +4 +B¨acklund transformations +We note that deg-PV (1.1) has the symmetries +S1 : +w1(z; α1, β1, γ1) = w(−z; α, β, γ), +(α1, β1, γ1) = (α, β, −γ), +(4.1) +S2 : +w2(z; α2, β2, γ2) = 1/w(z; α, β, γ), +(α2, β2, γ2) = (−β, −α, −γ), +(4.2) +where w(z; α, β, γ) is a solution of (1.1). +5 + +Theorem 4.1. Suppose that w = w(z; α, β, γ) satisfies (1.1) with parameters +α = 1 +2a2, +β = − 1 +2b2, +γ = c. +Then wj = w(z; αj, βj, γj) given by +W1 : +w1 = +{zw′ + (w − 1)(aw − b)} {zw′ + (w − 1)(aw + b)} +z2(w′)2 + 2azw(w − 1)w′ + 2cz2w(w − 1) + (w − 1)2(a2w2 − b2), +(4.3a) +W2 : +w2 = +{zw′ − (w − 1)(aw − b)} {zw′ − (w − 1)(aw + b)} +z2(w′)2 − 2azw(w − 1)w′ + 2cz2w(w − 1) + (w − 1)2(a2w2 − b2), +(4.3b) +W3 : +w3 = z2(w′)2 + 2bz(w − 1)w′ + 2cz2w2(w − 1) − (w − 1)2(a2w2 − b2) +{zw′ − (w − 1)(aw − b)} {zw′ + (w − 1)(aw + b)} +, +(4.3c) +W4 : +w4 = z2(w′)2 − 2bz(w − 1)w′ + 2cz2w2(w − 1) − (w − 1)2(a2w2 − b2) +{zw′ − (w − 1)(aw − b)} {zw′ + (w − 1)(aw + b)} +, +(4.3d) +satisfy (1.1) with parameters +α1 = 1 +2(a + 1)2, +β1 = − 1 +2b2, +γ1 = c, +α2 = 1 +2(a − 1)2, +β2 = − 1 +2b2, +γ2 = c, +α3 = 1 +2a2, +β3 = − 1 +2(b + 1)2, +γ3 = c, +α4 = 1 +2a2, +β4 = − 1 +2(b − 1)2, +γ4 = c, +respectively. +Proof. See Adler [2]; also Filipuk and Van Assche [18]. +5 +Classical solutions of deg-PV +To discuss classical solutions of deg-PV (1.1), it is convenient to make the transformation +w(z) = u(x), +z = 1 +2x2, +(5.1) +in (1.1), which gives +d2u +dx2 = +� 1 +2u + +1 +u − 1 +� �du +dx +�2 +− 1 +x +du +dx + 4(u − 1)2(αu2 + β) +x2u ++ 2γu. +(5.2) +We could fix the parameter γ in (5.2), by rescaling x if necessary, but it is more convenient not to do so. +Instead classical solutions will be classified for γ = ±1. From Corollary 2.3 and (5.1), we have that if +σ(x; a, b, ε) is a solution of SIII (2.13), then +u(x; a, b, ε) = σ′(x; a, b, ε) + εx +σ′(x; a, b, ε) − εx, +(5.3) +is a solution of (5.2) with γ = ε. +Theorem 5.1. Supppose that u = u(x; α, β, γ) satisfies (5.2) with parameters +α = 1 +2a2, +β = − 1 +2b2, +γ = c. +Then uj = u(x; αj, βj, γj) given by +U1 : +u1 = +{xu′ + 2(u − 1)(au − b)} {xu′ + 2(u − 1)(au + b)} +x2(u′)2 + 4axu(u − 1)u′ + 4cu(u − 1)x2 + 4(u − 1)2(a2u2 − b2), +(5.4a) +U2 : +u2 = +{xu′ − 2(u − 1)(au − b)} {xu′ − 2(u − 1)(au + b)} +x2(u′)2 − 4axu(u − 1)u′ + 4cu(u − 1)x2 + 4(u − 1)2(a2u2 − b2), +(5.4b) +U3 : +u3 = x2(u′)2 + 4bx(u − 1)u′ + 4cx2u2(u − 1) − 4(u − 1)2(a2u2 − b2) +{xu′ − 2(u − 1)(au − b)} {xu′ + 2(u − 1)(au + b)} +, +(5.4c) +U4 : +u4 = x2(u′)2 − 4bx(u − 1)u′ + 4cx2u2(u − 1) − 4(u − 1)2(a2u2 − b2) +{xu′ − 2(u − 1)(au − b)} {xu′ + 2(u − 1)(au + b)} +, +(5.4d) +6 + +satisfy (5.2) with parameters +α1 = 1 +2(a + 1)2, +β1 = − 1 +2b2, +γ1 = c, +α2 = 1 +2(a − 1)2, +β2 = − 1 +2b2, +γ2 = c, +α3 = 1 +2a2, +β3 = − 1 +2(b + 1)2, +γ3 = c, +α4 = 1 +2a2, +β4 = − 1 +2(b − 1)2, +γ4 = c, +respectively. +Proof. This is easily proved by applying (5.1) to B¨acklund transformations in Theorem 4.1. +5.1 +Algebraic solutions +Algebraic solutions of (1.1) are equivalent to rational solutions of (5.2) and so we discuss rational +solutions of (5.2), which are classified in the following Theorem. +Theorem 5.2. Necessary and sufficient conditions for the existence of rational solutions of (5.2) are +either +(α, β, γ) = +� 1 +2(n + 1 +2), − 1 +2µ2, 1 +� +, +(5.5) +or +(α, β, γ) = +� 1 +2µ2, − 1 +2(n + 1 +2), −1 +� +, +(5.6) +where n ∈ Z and µ is an arbitrary constant. +Proof. For details see Gromak, Laine and Shimomura [22, §38]; see also [39, 40]. +We remark that the solutions of (5.2) satisfying (5.5) are related to those satisfying (5.6) through +the analog of the symmetry (4.2). Consequently we shall be concerned only with rational solutions of +(5.2) for the parameters given by (5.5). +Theorem 5.3. The rational solution of (5.2) for the parameters (5.5) is given by +un(x; µ) = 1 − +xS2 +n(x; µ) +Sn+1(x; µ)Sn−1(x; µ), +n ≥ 0, +(5.7) +where Sn(x; µ) is the Umemura polynomial (3.2). +Proof. Substituting the rational solution of SIII (2.13) given by (3.3) into (5.3) and then using the +reccurence relation (3.1) gives the result. +Remark 5.4. The Umemura polynomial Sn(x; µ) satisfies the difference equation +Sn+1(x; µ)Sn−1(x; µ) = xS2 +n(x; µ) + µSn(x; µ + 1) Sn(x; µ − 1). +(5.8) +Hence from (5.7) there are two alternative representations of the rational solution +un(x; µ) = +µSn(x; µ + 1) Sn(x; µ − 1) +µSn(x; µ + 1) Sn(x; µ − 1) + xS2n(x; µ), +un(x; µ) = µSn(x; µ + 1) Sn(x; µ − 1) +Sn+1(x; µ)Sn−1(x; µ) +. +5.2 +Bessel function solutions +Theorem 5.5. Necessary and sufficient conditions for the existence of Bessel function solutions of (5.2) +are either +(α, β, γ) = +� 1 +2n2, − 1 +2µ2, ε +� +, +(5.9) +or +(α, β, γ) = +� 1 +2µ2, − 1 +2n2, −ε +� +, +(5.10) +with ε = ±1, and where n ∈ Z+ and µ is an arbitrary constant. +7 + +Proof. From (2.5) and (2.9), the parameters in PIII (2.1) and deg-PV (5.2) are given by +(A, B) = +� +2(a − b), 2ε(a + b + 1) +� +, +(α, β, γ) = ( 1 +2a2, − 1 +2b2, ε), +respectively, for parameters a, b and ε. The result then follows from Theorem 3.5. +Theorem 5.6. The Bessel function solution of (5.2) for the parameters +(α, β, γ) = +� 1 +2n2, − 1 +2µ2, ε +� +, +is given by +un(x; µ, ε) = 1 + +εx2τ 2 +n(x; µ, ε) +τn+1(x; µ, ε) τn−1(x; µ, ε), +n ≥ 1, +(5.11) +where +τn(x; µ, ε) = det +�� +x d +dx +�j+k +ϕµ(x; ε) +�n−1 +j,k=0 +, +(5.12) +and τ0(x; µ, ε) = 1, with +ϕµ(x; ε) = +� +c1Jµ(x) + c2Yµ(x), +if +ε = 1, +c1Iµ(x) + c2Kµ(x), +if +ε = −1, +(5.13) +c1 and c2 arbitrary constants, and Jµ(x), Yµ(x), Iµ(x) and Kµ(x) Bessel functions. +Proof. Substituting the Bessel function solution of SIII (2.13) given by (3.10) into (5.3) and then using +(3.11) gives the result. +Corollary 5.7. The Bessel function solution of (5.2) for the parameters +(α, β, γ) = +� 1 +2n2, − 1 +2µ2, 2ε +� +, +is given by +wn(z; µ, ε) = 1 + +εzT 2 +n (z; µ, ε) +Tn+1(z; µ, ε) Tn−1(z; µ, ε), +n ≥ 1, +(5.14) +where +Tn(z; µ, ε) = det +�� +z d +dz +�j+k +ψµ(z; ε) +�n−1 +j,k=0 +, +(5.15) +and T0(z; µ, ε) = 1, with +ϕµ(z; ε) = +� +c1Jµ(2√z) + c2Yµ(2√z), +if +ε = 1, +c1Iµ(2√z) + c2Kµ(2√z), +if +ε = −1, +(5.16) +c1 and c2 arbitrary constants, and Jµ(x), Yµ(x), Iµ(x) and Kµ(x) Bessel functions. +In the next Lemma, it is shown that the first solution u1(x; µ, ε), the “seed solution”, satisfies a +first-order, second-degree equation. +Lemma 5.8. The solution of (5.2) for the parameters +(α, β, γ) = +� 1 +2, − 1 +2µ2, ε +� +, +is +u1(x; µ, ε) = +ϕµ+1(x; ε) [xϕµ+1(x; ε) − 2εµϕµ(x; ε)] +xϕ2 +µ+1(x; ε) − 2εµϕµ+1(x; ε)ϕµ(x; ε) + εxϕ2µ(x; ε), +(5.17) +where +ϕµ(x; ε) = +� +c1Jµ(x) + c2Yµ(x), +if +ε = 1, +c1Iµ(x) + c2Kµ(x), +if +ε = −1, +with c1 and c2 constants, satisfies the first-order, second-degree equation +x2 +�du +dx +�2 +− 4xu(u − 1)du +dx + 4εx2u(u − 1) + 4(u − 1)2(u2 − µ2) = 0. +(5.18) +8 + +Proof. Define +Φµ(x; ε) = ϕµ+1(x; ε) +ϕµ(x; ε) , +then from (5.17) +u1(x; µ, ε) = 1 − +x +εxΦ2µ − 2µΦµ + x, +(5.19) +and Φµ(x; ε) satisfies the Riccati equation +xdΦµ +dx = εxΦ2 +µ − (2µ + 1)Φµ + x. +(5.20) +Next we assume that u1(x; µ, ε) satisfies a first-order, second-degree equation of the form +x2 +�du +dx +�2 ++ x +� +f2(x, µ, ε)u2 + f1(x, µ, ε)u + f0(x, µ, ε) +� du +dx + +4 +� +j=0 +gj(x, µ, ε)uj = 0, +(5.21) +where {fj(x, µ, ε)}2 +j=0 and {gj(x, µ, ε)}4 +j=0 are to be determined. Then substituting (5.19) into (5.21), +using the fact that Φµ(x; ε) satisfies (5.20) and equating coefficients of powers of Φµ yields +f2 = −4, +f1 = 4, +f0 = 0, +g4 = 4, +g3 = −8, +g2 = 4εx2 − 4µ2 + 4, +g1 = −4εx2 + 8µ2, +g0 = −4µ2. +Hence we obtain equation (5.18), as required. +This demonstrates that special function solutions of (5.2), and hence also deg-PV (1.1) , are different +from special function solutions of PII–PVI where the “seed solution” satisfies a Riccati equation, a first- +order, first-degree equation. +6 +Applications +6.1 +Complex sine-Gordon equation +Consider the two-dimensional complex sine-Gordon equation +∇2ψ + (∇ψ)2ψ +1 − |ψ|2 + ψ +� +1 − |ψ|2� += 0, +(6.1) +where ∇ψ = (ψx, ψy). Making the transformation +ψ(x, y) = cos(ϕ(x, y)) exp{iη(x, y)}, +ψ(x, y) = cos(ϕ(x, y)) exp{−iη(x, y)}, +in the complex sine-Gordon equation (6.1) yields +∇2ϕ + cos ϕ +sin3 ϕ(∇η)2 − 1 +2 sin(2ϕ) = 0, +sin(2ϕ) ∇2η = 4∇ϕ •∇η, +which is the Pohlmeyer-Lund-Regge model [33, 34, 50]. +The complex sine-Gordon equation (6.1) has a separable solution in polar coordinates given by +ψ(r, θ) = Rn(r) einθ, where Rn(r) satisfies +d2Rn +dr2 ++ 1 +r +dRn +dr ++ +Rn +1 − R2n +��dRn +dr +�2 +− n2 +r2 +� ++ Rn +� +1 − R2 +n +� += 0, +(6.2) +We remark that this equation also arises in extended quantum systems [4, 5, 6], in relativity [20] and +in coefficients in the three-term recurrence relation for orthogonal polynomials with respect to the +weight w(θ) = et cos θ on the unit circle, see [56, equation (3.13)]. The orthogonal polynomials for this +weight on the unit circle are related to unitary random matrices [49]. +Equation (6.2) can be shown to possess the Painlev´e property, though is not in the list of 50 equa- +tions given in [25, Chapter 14]. Equation (6.2) can be transformed to the fifth Painlev´e equation (1.2) +in two different ways. +9 + +(i) If Rn(r) satisfies (6.2) then making the transformation +Rn(r) = 1 + un(z) +1 − un(z), +r = 1 +2z, +(6.3) +yields +d2un +dz2 = +� 1 +2un ++ +1 +un − 1 +� �dun +dz +�2 +− 1 +z +dun +dz + n2(un − 1)2(u2 +n − 1) +8z2un +− un(un + 1) +2(un − 1) , +(6.4) +which is PV (1.2) with α = 1 +8n2, β = − 1 +8n2, γ = 0 and δ = − 1 +2. +(ii) If Rn(r) satisfies (6.2) then making the transformation +Rn(r) = +1 +� +1 − vn(x) +, +r = √x, +(6.5) +yields +d2vn +dx2 = +� 1 +2vn ++ +1 +vn − 1 +� �dvn +dx +�2 +− 1 +x +dvn +dx − n2(vn − 1)2 +2x2vn ++ vn +2x, +(6.6) +which is deg-PV (1.1) with α = 0, β = − 1 +2n2 and γ = 1 +2 so is equivalent to PIII (2.1), as mentioned +above. +This shows that solutions of equations (6.4) and (6.6) are related by +vn(x) = +4un(z) +1 + u2n(z), +x = 1 +4z2. +The function Rn(r) satisfies the ordinary differential equation (6.2), the differential-difference equa- +tions +dRn +dr ++ n +r Rn − +� +1 − R2 +n +� +Rn−1 = 0, +(6.7a) +dRn−1 +dr +− n − 1 +r +Rn−1 + +� +1 − R2 +n−1 +� +Rn = 0, +(6.7b) +since solving (6.7a) for Rn−1(r) and substituting in (6.7b) yields equation (6.2). Also eliminating the +derivatives in (6.7), after letting n → n + 1 in (6.7b), yields the difference equation +Rn+1 + Rn−1 = 2n +r +Rn +1 − R2n +, +(6.8) +which is known as the discrete Painlev´e II equation [41, 49]. +If n = 1 then equations (6.7) have the solution +R0(r) = 1, +R1(r) = C1I1(r) − C2K1(r) +C1I0(r) + C2K0(r), +where I0(r), K0(r), I1(r) and K1(r) are the imaginary Bessel functions and C1 and C2 are arbitrary +constants. For solutions which are bounded at r = 0 then necesssarily C2 = 0 and so +R0(r) = 1, +R1(r) = I1(r) +I0(r). +(6.9) +Hence one can use the difference equation (6.8) to determine Rn(r), for n ≥ 2, which yields +R2(r) = −rR2 +1(r) + 2R1(r) − r +r [R2 +1(r) − 1] +, +R3(r) = R3 +1(r) − rR2 +1(r) − 2R1(r) + r +R1(r) [rR2 +1(r) + R1(r) − r] , +R4(r) = +r(r2 + 5)R4 +1(r) + 4R3 +1(r) − 2r(r2 + 3)R2 +1(r) + r3 +r [(r2 − 1)R4 +1(r) + 4rR3 +1(r) − 2(r2 + 2)R2 +1(r) − 4rR1(r) + r2]. +These results suggest that (6.2) should be solvable in terms of PIII (2.1), which is illustrated in the +following theorem. +10 + +Theorem 6.1. If Rn(r) satisfies (6.2) then wn(r) = Rn+1(r)/Rn(r) satisfies +d2wn +dr2 += 1 +wn +�dwn +dr +�2 +− 1 +r +dwn +dr +− 2n +r w2 +n + 2(n + 1) +r ++ w3 +n − 1 +wn +, +(6.10) +which is PIII (2.1) with parameters α = −2n and β = 2(n + 1). +Proof. See Hisakado [23] and Tracy & Widom [52]; see also [56, §3.1]. +We note that since the parameters in (6.10) satisfy −α + β = 4n + 2, with n ∈ Z+, then the equation +has solutions expressible in terms of the modified Bessel functions I0(r) and I1(r) (as well as K0(r) and +K1(r), but these are not needed here). +Theorem 6.2. Let τn(r; ν) be the n × n determinant +τn(r; ν) = det +�� +r d +dr +�j+k +Iν(r) +�n−1 +j,k=0 +, +(6.11) +with Iν(r) the modified Bessel function, then +wn(r; ν) = τn+1(r; ν + 1) τn(r; ν) +τn+1(r; ν) τn(r; ν + 1) ≡ d +dz +� +ln τn+1(z; ν) +τn(z; ν + 1) +� +− n + ν +z +, +n ≥ 0, +(6.12) +satisfies PIII (2.1) with α = 2(ν − n) and β = 2(ν + n + 1). +Proof. See, for example, [19, 38]. +Theorem 6.3. Equation (6.2) has the solution +Rn(r) = τn(r; 1) +τn(r; 0), +(6.13) +where τn(r; ν) is the determinant given by (6.11). +Proof. The proof is straightforward using induction. From (6.9) we have +R1(r) = I1(r) +I0(r) = τ1(r; 1) +τ1(r; 0), +so (6.13) is true if n = 1. Assuming (6.13) holds then from Theorems 6.1 and 6.2 +Rn+1(r) = wn(r; 0)Rn(r) = τn+1(r; 1) τn(r; 0) +τn+1(r; 0) τn(r; 1) × τn(r; 1) +τn(r; 0) = τn+1(r; 1) +τn+1(r; 0), +as required, and so the result follows by induction. +Corollary 6.4. Equations (6.4) and (6.6) have the Bessel function solutions +un(z) = τn( 1 +2z; 1) + τn( 1 +2z; 0) +τn( 1 +2z; 1) − τn( 1 +2z; 0), +vn(x) = 1 − τ 2 +n(√x; 0) +τ 2n(√x; 1), +respectively, with τn(r; ν) the determinant given by (6.11). +Lemma 6.5. The formal asymptotic behaviour of the vortex solution Rn(r) is given by +Rn(r) = +rn +2n n! +� +1 − +r2 +4(n + 1) + O +� +r4�� +, +as +r → 0, +(6.14) +Rn(r) = 1 − n +2r − n2 +8r2 − n(n2 + 1) +16r3 ++ O(r−4), +as +r → ∞. +(6.15) +Proof. These are determined from (6.8) and (6.9). +11 + +6.2 +Generalised Charlier polynomials +The Charlier polynomials Cn(k; z) are a family of orthogonal polynomials introduced in 1905 by Char- +lier [7] given by +Cn(k; z) = 2F0 (−n, −k; ; −1/z) = (−1)nn!L(−1−k) +n +(−1/z) , +z > 0, +(6.16) +where 2F0(a, b; ; z) is the hypergeometric function and L(α) +n (z) is the associated Laguerre polynomial, +see, for example, [48, §18.19]. The Charlier polynomials are orthogonal on the lattice N with respect to +the Poisson distribution +ω(k) = zk +k! , +z > 0, +(6.17) +and satisfy the orthogonality condition +∞ +� +k=0 +Cm(k; z)Cn(k; z)zk +k! = n! ez +zn δm,n. +Smet and Van Assche [51] generalized the Charlier weight (6.17) with one additional parameter +through the weight function +ω(k; ν) = +Γ(ν + 1) zk +Γ(ν + k + 1) Γ(k + 1), +z > 0, +with ν a parameter such that ν > −1. This gives the discrete weight +ω(k; ν) = +zk +(ν + 1)k k!, +z > 0, +(6.18) +where (ν + 1)k = Γ(ν + 1 + k)/Γ(ν + 1) is the Pochhammer symbol, on the lattice N. Discrete orthogonal +polynomials are characterized by the discrete Pearson equation +∆ +� +σ(k)ω(k) +� += τ(k)ω(k), +(6.19) +where ∆ is the forward difference operator +∆f(k) = f(k + 1) − f(k). +The weight (6.18) satisfies the discrete Pearson equation (6.19) with +σ(k) = k(k + ν), +τ(k) = −k2 − νk + z, +and so the generalised Charlier polynomials are semi-classical orthogonal polynomials since τ(k) is a +polynomial with deg(τ) > 1. The special case β = 0 was first considered by Hounkonnou, Hounga and +Ronveaux [24] and later studied by Van Assche and Foupouagnigni [57]. +For the generalised Charlier weight (6.18), the orthonormal polynomials pn(k; z) satisfy the orthog- +onality condition +∞ +� +k=0 +pm(k; z)pn(k; z) +zk +(ν + 1)k k! = δm,n, +and the three-term recurrence relation +kpn(k; z) = an+1(z)pn+1(k; z) + bn(z)pn(k; z) + an(z)pn−1(k; z), +(6.20) +with p−1(k; z) = 0 and p0(k; z) = 1. Our interest is in the coefficients an(z) and bn(z) in the recurrence +relation (6.20). +Smet and Van Assche [51, Theorem 2.1] proved the following theorem for recurrence coefficients +associated with the generalised Charlier weight (6.18). +12 + +Theorem 6.6. The recurrence coefficients an(z) and bn(z) for orthonormal polynomials associated with +the generalised Charlier weight (6.18) on the lattice N satisfy the discrete system +(a2 +n+1 − z)(a2 +n − z) = z(bn − n)(bn − n + ν), +bn + bn−1 − n + ν + 1 = nz/a2 +n, +(6.21) +with initial conditions +a2 +0 = 0, +b0 = +√z Iν+1(2√z) +Iν(2√z) += z d +dz +� +ln Iν(2√z) +� +− ν +2 , +(6.22) +with Iν(k) the modified Bessel function. +Remark 6.7. The discrete system such as (6.21) for recurrence coefficients is sometimes known as the +Laguerre-Freud equations, cf. [3, 24, 35]. +The recurrence coefficients an(z) and bn(z) also satisfy the Toda lattice, cf. [56, Theorem 3.8] +z d +dz a2 +n = a2 +n(bn − bn−1), +(6.23a) +z d +dz bn = a2 +n+1 − a2 +n. +(6.23b) +Letting a2 +n(z) = xn(z) and bn(z) = yn(z) in (6.21) and (6.23) yields +(xn+1 − z)(xn − z) = t(yn − n)(yn − n + ν), +z dxn +dt += xn(yn − yn−1), +yn + yn−1 − n + ν + 1 = nz +xn +, +z dyn +dz = xn+1 − xn. +Eliminating xn+1 and yn−1 in these equations yields the differential system +z dxn +dz = xn(2yn + ν − n + 1) − nz, +(6.24a) +z dyn +dz = −xn + z + (yn − n)(yn − n + ν)z +xn − z +. +(6.24b) +Solving (6.24a) for yn gives +yn = +z +2xn +dxn +dz + nz +2xn ++ n − ν − 1 +2 +, +and substituting this into (6.24b) yields +d2xn +dz2 = 1 +2 +� 1 +xn ++ +1 +xn − z +� +− +xn +z(xn − z) +dxn +dz − 2x2 +n +z2 + 4xn + n2 − ν2 + 1 +2z +− n2 +2xn ++ +1 − ν2 +2(xn − z). +(6.25) +Making the transformation +xn(z) = +z +1 − wn(z). +(6.26) +in (6.25) yields +d2wn +dz2 += +� 1 +2wn ++ +1 +wn − 1 +��dwn +dz +�2 +− 1 +z +dwn +dz ++ (wn − 1)2(n2w2 +n − ν2) +2wnz2 +− 2wn +z , +(6.27) +which is deg-PV (1.1) with parameters α = 1 +2n2, β = − 1 +2ν2 and γ = −2. +Solving (6.24b) for xn gives +xn = − 1 +2z dyn +dz + z + 1 +2Xn, +(6.28) +where +X2 +n = z2 +�dyn +dz +�2 ++ 4z(yn − n)(yn − n + ν). +(6.29) +13 + +From (6.29) we get +dXn +dz += z2 +Xn +d2yn +dz2 +dyn +dz + z +Xn +�dyn +dz +�2 ++ 2z(2yn − 2n + ν) +Xn +dyn +dz + 2(yn − n)(yn − n + ν) +Xn +(6.30) +Substituting (6.28) into (6.24a), then using (6.30), solving for Xn, and substituting into (6.29) yields +the second-order, second-degree equation +� +2z d2yn +dz2 + dyn +dz + 8yn − 8n + 4ν +�2 += (4yn − 2n + 2ν + 1)2 +z +� +z +�dyn +dz +�2 ++ 4(yn − n)(yn − n + ν) +� +. (6.31) +Making the transformation +yn(z) = 1 +2vn(x) + 1 +2n − 1 +2ν − 1 +4, +x = 2√z, +in (6.31) yields +�d2vn +dx2 + 4vn − 4n − 2 +�2 += 4v2 +n +x2 +��dvn +dx +�2 ++ 4v2 +n − 4(2n + 1)vn + (2n + 1)2 − 4ν2 +� +. +(6.32) +Equation (A.5) in [14] is +�d2v +dx2 − av − b +�2 += 4v2 +x2 +��dv +dx +�2 +− av2 − 2bv − c +� +, +(6.33) +with a, b and c parameters, an equation derived by Chazy [8], and is the primed version of equation +SD-III in [15]. Hence equation (6.32) is the special case of equation (6.33) with +a = −4, +b = 4n + 2, +c = 4ν2 − (2n + 1)2. +Cosgrove [14] showed that equation (6.33) is solvable in terms of solutions of PIII (2.1). Consequently, +the solution of (6.32) is given by +vn(x) = x +2q +� dq +dx + q2 + 1 +� +, +where q(x) satisfies PIII (2.1) for the parameters A = 2ν − 2n − 2 and B = 2ν + 2n. +Theorem 6.8. The recurrence relations an(z) and bn(z) are given by +a2 +n(z) = xn(z) = Tn+1(z; ν)Tn−1(z; ν) +T 2 +n (z; ν) +, +(6.34a) +bn(z) = yn(z) = z d +dz +� +ln Tn+1(z; ν) +Tn(z; ν) +� +− ν +2 , +(6.34b) +where +Tn(z; ν) = det +�� +z d +dz +�j+k +Iν +� +2√z +� +�n−1 +j,k=0 +, +with T0(z; ν) = 1, and Iν(x) is the modified Bessel function. +Proof. The expression (6.34a) for a2 +n(z) follows immediately by substituting (5.14) in (6.26). To prove +the result (6.34b) for bn(z) we use induction and the factor that from equation (6.23b), a2 +n(z) = xn(z) +and bn(z) = yn(z) are related by +z dxn +dt += xn(yn − yn−1), +and initially +y0(z) = z d +dz +� +ln T1(z; ν) +� +} − ν +2 . +14 + +Hence +y1(z) = z d +dz +� +ln x1(z) +� ++ y0(z) += z d +dz +� +ln T2(z; ν)T0(z; ν) +T 2 +1 (z; ν) +� ++ z d +dz {ln T1(z; ν)} − ν +2 += z d +dz +� +ln T2(z; ν) +T1(z; ν) +� +− ν +2 , +so (6.34b) is true for n = 1. Now suppose that (6.34b) is true, then +yn+1(z) = z d +dz +� +ln xn(z) +� ++ yn(z) += z d +dz +� +ln Tn+2(z; ν)Tn(z; ν) +T 2 +n+1(z; ν) +� ++ z d +dz +� +ln Tn+1(z; ν) +Tn(z; ν) +� +− ν +2 += z d +dz +� +ln Tn+2(z; ν) +Tn+1(z; ν) +� +− ν +2 , +as required, and so the result follows by induction. We remark that equation (6.23a) is identically +satisfied by a2 +n(z) and bn(z) given by (6.34). +In a recent paper, Fern´andez-Irisarri and Ma˜nas [17, §2] discuss the generalised Charlier weight +(6.18), in particular properties of the coefficients in the recurrence relation. The relationship between +the notations in [17] and those here are xn(z) = γn(η) and yn(z) = βn(η). Fern´andez-Irisarri and Ma˜nas +[17] relate xn(z) and yn(z) to Okamoto’s Hamiltonian for PIII′ [46] and derive two ordinary differential +equations for xn(z). +1. Equation (45) in [17, Theorem 4] is the third order equation +δz +�xn +z +� +δ2 +z(ln xn) + 2xn +� ++ n2z +xn +� += 2xn, +δz(f) = z df +dz , +i.e. +d3xn +dz3 = +1 +zx2n +� +z dxn +dz − xn +� � +2xn +d2xn +dz2 − +�dxn +dz +�2 ++ n2 +� +− 4xn +z2 +dxn +dz + 2xn(xn + z) +z3 +, +(6.35) +and the state that this equation “should have the Painlev´e property”. Equation (6.35) can be +integrate to give equation (6.25), with ν2 as the constant of integration. Since equation (6.25) is +equivalent to deg-PV (5.2) then equation (6.35) does have the Painlev´e property. +2. Equation (60) in [17, Theorem 5] is the second order equation +� +1 − xn +z +� � +δz +�δz(xn) + nz +xn +� ++ 2xn +� ++ 2{xn − z + (n − b)n} += − 1 +2 +�δz(xn) + nz +xn +�2 ++ (n + 1) +�δz(xn) + nz +xn +� ++ (n − b − 1)(3n − b + 1), +which is equation (6.25) with +ν2 = 2(b − n)2 + n2 − 2n − 1. +7 +Discussion +In this paper the classical solutions of deg-PV (5.2) have been classified. Ohyama and Okumura [43, +Theorem 2.1] give a list of classical solutions of PI to PV and state that “deg-P5 with α = 1 +2a2, β = − 1 +8, +γ = −2 has the algebraic solution w(z) = 1 + 2√z/a”2 and “deg-P5 with β = 0 has the Riccati type +2As noted in [1], there is typo in [43] who say β = −8 rather than β = − 1 +8. +15 + +solutions”. The results in this paper show that there are more classical solutions of deg-PV (1.1). The +algebraic solution is equivalent to the “seed solution” obtained by setting n = 0 in (5.7), i.e. +u0(x; µ) = +µ +x + µ, +and there is a more general hierarchy of “Riccati type solutions” which are described in Theorem 5.6. +All solutions of PII–PVI that are expressible in terms of special functions satisfy a first-order equa- +tion of the form +�du +dx +�n += +n−1 +� +j=0 +Fj(u, x) +�du +dx +�j +, +(7.1) +where Fj(u, x) is polynomial in u with coefficients that are rational functions of x. It can be shown +that the Bessel function solutions of PIII (2.1) satisfy a first-order equation of the form (7.1) for n odd, +whereas the Bessel function solutions of deg-PV (5.2) satisfy a first-order equation of the form (7.1) for +n even. +The relationship between PIII (2.1) and deg-PV (1.1) is similar to that between the second Painlev´e +equation (PII) +d2q +dx2 = 2q3 + xq, +(7.2) +with α a parameter, and Painlev´e XXXIV equation (P34) +d2p +dx2 = 1 +2p +� dp +dx +�2 ++ 2p2 − xp − (α + 1 +2)2 +2p +, +(7.3) +which is equivalent to equation XXXIV of Chapter 14 in [25], in that both pairs of equations arise from +a Hamiltonian. The Hamiltonian associated with PII (7.2) and P34 (7.3) is +HII(q, p, z; α) = 1 +2p2 − (q2 + 1 +2z)p − (α + 1 +2)q +(7.4) +and so +dq +dz = p − q2 − 1 +2z, +dp +dz = 2qp + α + 1 +2, +(7.5) +see [28, 44]. It is known that PII (7.2) and P34 (7.3) have special function solutions in terms of Airy +functions, cf. 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Phys., 10(suppl. 2) (2003) 231–237. +18 + diff --git a/D9AzT4oBgHgl3EQfwv5m/content/tmp_files/load_file.txt b/D9AzT4oBgHgl3EQfwv5m/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f584ada80b330bb4bae257f6bb1290424a544f8c --- /dev/null +++ b/D9AzT4oBgHgl3EQfwv5m/content/tmp_files/load_file.txt @@ -0,0 +1,1150 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf,len=1149 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='01727v1 [nlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='SI] 4 Jan 2023 Classical Solutions of the Degenerate Fifth Painlev´e Equation Peter A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Clarkson School of Mathematics, Statistics and Actuarial Science, University of Kent, Canterbury, CT2 7FS, UK Email: P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='Clarkson@kent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='uk January 5, 2023 Abstract In this paper classical solutions of the degenerate fifth Painlev´e equation are classified, which include hierarchies of algebraic solutions and solutions expressible in terms of Bessel functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Solu- tions of the degenerate fifth Painlev´e equation are known to expressible in terms of the third Painlev´e equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Two applications of these classical solutions are discussed, deriving exact solutions of the complex sine-Gordon equation and of the coefficients in the three-term recurrence relation associated with generalised Charlier polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' 1 Introduction In this paper we are concerned with solutions of the equation d2w dz2 = � 1 2w + 1 w − 1 � �dw dz �2 − 1 z dw dz + (w − 1)2(αw2 + β) z2w + γw z , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1) with α, β and γ constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1) is the special case of the fifth Painlev´e equation (PV) d2w dz2 = � 1 2w + 1 w − 1 � �dw dz �2 − 1 z dw dz + (w − 1)2(αw2 + β) z2w + γw z + δw(w + 1) w − 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2) with α, β, γ and δ constants, when δ = 0 and is known as the degenerate fifth Painlev´e equation (deg- PV), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' The six Painlev´e equations (PI–PVI), were discovered by Painlev´e, Gambier and their colleagues whilst studying second order ordinary differential equations of the form d2w dz2 = F � z, w, dw dz � , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='3) where F is rational in dw/dz and w and analytic in z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' The Painlev´e equations can be thought of as nonlinear analogues of the classical special functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' The general solutions of the Painlev´e equations are transcendental in the sense that they cannot be expressed in terms of known elementary functions and so require the introduction of a new transcendental function to describe their solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' However, it is well known that PII–PVI possess rational solutions, algebraic solutions and solutions expressed in terms of the classical special functions — Airy, Bessel, parabolic cylinder, Kummer and hypergeometric functions, respectively — for special values of the parameters, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' [11, 22] and the references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' These hierarchies are usually generated from “seed solutions” using the associated B¨acklund transformations and frequently can be expressed in the form of determinants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' These solutions of the Painlev´e equations are often called “classical solutions”, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' [53, 54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' It is well known that solutions of deg-PV (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1) are related to solutions of the third Painlev´e equation d2q dx2 = 1 q � dq dx �2 − 1 x dq dx + Aq2 + B x + Cq3 + D q , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='4) 1 with A, B, C and D constants, a result originally due to Gromak [21];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' see also [22, §34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' The purpose of this paper is to give a classification and description of the classical solutions of deg-PV (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1) directly, rather than indirectly through (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' In §2, the relationship between deg-PV (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1) and the third Painlev´e equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='4) is discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' In §3, classical solutions of the third Painlev´e equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='4) are reviewed, the rational solutions in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1 and the Bessel function solutions in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' In §4, B¨acklund transformations of deg-PV (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1) are given, which can be used to derive a hierarchy of solutions from a “seed solution”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' In §5, classical solutions of deg-PV (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1) are classified, the algebraic solutions in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1 and the Bessel function solutions in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' In §6, two applications of classical solutions of deg-PV (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1) are given to derive exact solutions of the complex sine-Gordon equation, which is equivalent to the Pohlmeyer-Lund-Regge model, and to derive explicit representations of the coefficients in the three-term recurrence relation satisfied by generalised Charlier polynomials, which are discrete orthogonal polynonials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' 2 The relationship between deg-PV and PIII In the generic case when CD ̸= 0 in the third Painlev´e equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='4), we set C = 1 and D = −1, without loss of generality (by rescaling the variables if necessary), and so consider the equation d2q dx2 = 1 q � dq dx �2 − 1 x dq dx + Aq2 + B x + q3 − 1 q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1) In the sequel, we shall refer to this equation as PIII since it is the generic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Consider the Hamiltonian associated with PIII (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1) given by HIII(q, p, x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' a, b, ε) = q2p2 − xq2p − (2a + 2b + 1)qp + εxp + 2bxq, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2) with a and b parameters and ε = ±1, see [28, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Then p(x) and q(x) satisfy the Hamiltonian system x dq dx = ∂HIII ∂p = 2q2p − xq2 − (2a + 2b + 1)q + εx, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='3a) x dp dx = −∂HIII ∂q = −2qp2 + 2xqp + (2a + 2b + 1)p − 2bx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='3b) Solving (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='3a) for p(x) gives p(x) = 1 2q � x dq dx + xq2 + (2a + 2b + 1)q − εx � , and then substituting this in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='3b) gives d2q dx2 = 1 q � dq dx �2 − 1 x dq dx + 2(a − b)q2 x + 2ε(a + b + 1) x + q3 − 1 q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='4) which is PIII (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1), with parameters A = 2(a − b), B = 2ε(a + b + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='5) Solving (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='3a) for q(x) gives q(x) = 1 2p(x − p) � x dp dx − (2a + 2b + 1) + 2bx � , and then substituting this in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='3a) gives d2p dx2 = 1 2 �1 p + 1 p − x � � dp dx �2 − p x(p − x) dp dx + 2εp − 2b2 p − 4a2 − 1 2(p − x) + 1 − 4(a2 − b2) − 4εp2 2x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='6) Then making the transformation p(x) = 2√z w(z) w(z) − 1 , x = 2√z, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='7) 2 in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='6) gives d2w dz2 = � 1 2w + 1 w − 1 � �dw dz �2 − 1 z dw dz + (w − 1)2(a2w2 − b2) 2z2w + εw z , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='8) which is deg-PV (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1) with parameters α = 1 2a2, β = − 1 2b2, γ = ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='9) Hence we have the following result;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' see also [22, Theorem 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' If q(x) is a solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='4) then w(z) = xq′(x) + xq2(x) + (2a + 2b + 1)q(x) − εx xq′(x) − xq2(x) + (2a + 2b + 1)q(x) − εx, z = 1 2x2, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='10) with ′ ≡ d/dx is a solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='8), provided that x dq dx − xq2 + (2a + 2b + 1)q − εx ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Conversely, if w(z) is a solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='8), then q(x) = 1 2√z w � z dw dz + (w − 1)(aw + b) � , x = √ 2z, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='11) is a solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Solving (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='3a) for p(x), substituting in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='7) and solving for w(z) gives (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Also solving (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='3b) for q(x) and substituting (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='7) gives (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' An alternative method of deriving solutions of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='8) involves the second-order, second-degree equa- tion satisfied associated with the Hamiltonian (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2), due to Jimbo and Miwa [28] and Okamoto [46], which is often called the “σ-equation”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' If HIII(q, p, x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' a, b, ε) is given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2), then σ(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' a, b, ε) = HIII(q, p, x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' a, b, ε) + qp − 1 2εx2 + (a + b)2, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='12) where q(x) and p(x) satisfy the system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='3), satisfies the second-order, second-degree equation (SIII) � xd2σ dx2 − dσ dx �2 + 2 ��dσ dx �2 − x2 � � xdσ dx − 2σ � − 8ε(a2 − b2)xdσ dx = 8(a2 + b2)x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='13) Conversely, if σ(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' a, b, ε) satisfies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='13) then the solution of the Hamiltonian system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='3) is given by q(x) = εxσ′′(x) − ε(2a + 2b + 1)σ′(x) − 2(a − b)x x2 − [σ′(x)]2 , p(x) = 1 2εσ′(x) + 1 2x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='14) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' See Jimbo and Miwa [28] and Okamoto [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Consequently solutions of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='8) can be expressed in terms of solutions of SIII (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' If σ(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' a, b, ε) is a solution of SIII (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='13), then w(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' a, b, ε) = σ′(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' a, b, ε) + εx σ′(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' a, b, ε) − εx, z = 1 2x2, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='15) is a solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' This immediately follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='7) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' 3 3 Classical solutions of PIII and SIII 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1 Rational solutions of PIII and SIII Rational solutions of PIII (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1) are classified in the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1) has a rational solution if and only if ε1A + ε2B = 4n, with n ∈ Z and ε2 1 = 1, ε2 2 = 1, independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' For details see Lukashevich [32];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' see also [39, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Umemura [55]1 derived special polynomials associated with rational solutions of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1), which we now define;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' see also [9, 29, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' The Umemura polynomial Sn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ) is given by the recursion relation Sn+1Sn−1 = −x � Sn d2Sn dx2 − �dSn dx �2� − Sn dSn dx + (x + µ)S2 n, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1) where S−1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ) = S0(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ) = 1, with µ an arbitrary parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' The Umemura polynomial Sn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ) has the Wronskian representation Sn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ) = cnW (ϕ1, ϕ3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' , ϕ2n−1) , cn = n � k=0 (2k + 1)n−k, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2a) where ϕm(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ) = L(µ−2m+1) 2m−1 (−x), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2b) with L(α) k (x) the Laguerre polynomial, for details see Kajiwara and Masuda [30];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' see also [9, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' The rational function solution of SIII (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='13) is given by σn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ, ε) = 2x d dx {ln Sn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ)} − 1 2x2 − 2µx − 1 4, n ≥ 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='3a) with Sn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ) the Umemura polynomial, for the parameters a = n + 1 2, b = µ, ε = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='3b) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' See Clarkson [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2 Special function solutions of PIII and SIII Special function solutions of PIII (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1), which are expressed in terms of Bessel functions and are classi- fied in the following Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1) has solutions expressible in terms of the Riccati equation x dq dx = ε1xq2 + (Aε1 − 1)q + ε2x, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='4) if and only if ε1A + ε2B = 4n + 2, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='5) with n ∈ Z and ε2 1 = 1, ε2 2 = 1, independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Further, the Riccati equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='4) has the solution q(x) = −ε1 d dx ln ψν(x), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='6) 1The original manuscript was written by Umemura in 1996 for the proceedings of the conference “Theory of nonlinear special functions: the Painlev´e transcendents” in Montreal, which were not published;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' for further details see [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' 4 where ψν(x) satisfies xd2ψν dx2 + (1 − 2ε1ν)dψν dx + ε1ε2xψν = 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='7) which has solution ψν(x) = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f3 xν {C1Jν(x) + C2Yν(x)} , if ε1 = 1, ε2 = 1, x−ν {C1Jν(x) + C2Yν(x)} , if ε1 = −1, ε2 = −1, xν {C1Iν(x) + C2Kν(x)} , if ε1 = 1, ε2 = −1, x−ν {C1Iν(x) + C2Kν(x)} , if ε1 = −1, ε2 = 1, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='8) with C1, C2 arbitrary constants, and Jν(x), Yν(x), Iν(x), Kν(x) Bessel functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' For details see Okamoto [46];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' see also [11, 22, 36, 39, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Determinantal representations of special function solutions of PIII (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1) were given by Okamoto [46];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' see also [19, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Suppose τn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ, ε) is the determinant given by τn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ, ε) = det �� x d dx �j+k ϕµ(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ε) �n−1 j,k=0 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='9a) where ϕµ(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ε) = � c1Jµ(x) + c2Yµ(x), if ε = 1, c1Iµ(x) + c2Kµ(x), if ε = −1, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='9b) with c1, c2 arbitrary constants, and Jµ(z), Yµ(z), Iµ(z), Kµ(z) Bessel functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' The Bessel function solution of SIII (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='13) is given by σn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ, ε) = 2x d dx {ln τn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ, ε)} + 1 2εx2 + µ2 − n2 + 2n, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='10a) for the parameters a = n, b = µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='10b) Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' The determinant τn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ, ε) given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='9) satisfies the equation x2 � τn d2τn dx2 − �dτn dx �2� + xτn dτn dx = τn+1τn−1, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='11) or equivalently � x d dx �2 ln τn = τn+1τn−1 τ 2n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='12) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' See Okamoto [46, Theorem 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' 4 B¨acklund transformations We note that deg-PV (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1) has the symmetries S1 : w1(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' α1, β1, γ1) = w(−z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' α, β, γ), (α1, β1, γ1) = (α, β, −γ), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1) S2 : w2(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' α2, β2, γ2) = 1/w(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' α, β, γ), (α2, β2, γ2) = (−β, −α, −γ), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2) where w(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' α, β, γ) is a solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' 5 Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Suppose that w = w(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' α, β, γ) satisfies (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1) with parameters α = 1 2a2, β = − 1 2b2, γ = c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Then wj = w(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' αj, βj, γj) given by W1 : w1 = {zw′ + (w − 1)(aw − b)} {zw′ + (w − 1)(aw + b)} z2(w′)2 + 2azw(w − 1)w′ + 2cz2w(w − 1) + (w − 1)2(a2w2 − b2), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='3a) W2 : w2 = {zw′ − (w − 1)(aw − b)} {zw′ − (w − 1)(aw + b)} z2(w′)2 − 2azw(w − 1)w′ + 2cz2w(w − 1) + (w − 1)2(a2w2 − b2), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='3b) W3 : w3 = z2(w′)2 + 2bz(w − 1)w′ + 2cz2w2(w − 1) − (w − 1)2(a2w2 − b2) {zw′ − (w − 1)(aw − b)} {zw′ + (w − 1)(aw + b)} , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='3c) W4 : w4 = z2(w′)2 − 2bz(w − 1)w′ + 2cz2w2(w − 1) − (w − 1)2(a2w2 − b2) {zw′ − (w − 1)(aw − b)} {zw′ + (w − 1)(aw + b)} , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='3d) satisfy (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1) with parameters α1 = 1 2(a + 1)2, β1 = − 1 2b2, γ1 = c, α2 = 1 2(a − 1)2, β2 = − 1 2b2, γ2 = c, α3 = 1 2a2, β3 = − 1 2(b + 1)2, γ3 = c, α4 = 1 2a2, β4 = − 1 2(b − 1)2, γ4 = c, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' See Adler [2];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' also Filipuk and Van Assche [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' 5 Classical solutions of deg-PV To discuss classical solutions of deg-PV (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1), it is convenient to make the transformation w(z) = u(x), z = 1 2x2, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1) in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1), which gives d2u dx2 = � 1 2u + 1 u − 1 � �du dx �2 − 1 x du dx + 4(u − 1)2(αu2 + β) x2u + 2γu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2) We could fix the parameter γ in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2), by rescaling x if necessary, but it is more convenient not to do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Instead classical solutions will be classified for γ = ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' From Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='3 and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1), we have that if σ(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' a, b, ε) is a solution of SIII (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='13), then u(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' a, b, ε) = σ′(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' a, b, ε) + εx σ′(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' a, b, ε) − εx, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='3) is a solution of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2) with γ = ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Supppose that u = u(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' α, β, γ) satisfies (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2) with parameters α = 1 2a2, β = − 1 2b2, γ = c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Then uj = u(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' αj, βj, γj) given by U1 : u1 = {xu′ + 2(u − 1)(au − b)} {xu′ + 2(u − 1)(au + b)} x2(u′)2 + 4axu(u − 1)u′ + 4cu(u − 1)x2 + 4(u − 1)2(a2u2 − b2), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='4a) U2 : u2 = {xu′ − 2(u − 1)(au − b)} {xu′ − 2(u − 1)(au + b)} x2(u′)2 − 4axu(u − 1)u′ + 4cu(u − 1)x2 + 4(u − 1)2(a2u2 − b2), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='4b) U3 : u3 = x2(u′)2 + 4bx(u − 1)u′ + 4cx2u2(u − 1) − 4(u − 1)2(a2u2 − b2) {xu′ − 2(u − 1)(au − b)} {xu′ + 2(u − 1)(au + b)} , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='4c) U4 : u4 = x2(u′)2 − 4bx(u − 1)u′ + 4cx2u2(u − 1) − 4(u − 1)2(a2u2 − b2) {xu′ − 2(u − 1)(au − b)} {xu′ + 2(u − 1)(au + b)} , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='4d) 6 satisfy (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2) with parameters α1 = 1 2(a + 1)2, β1 = − 1 2b2, γ1 = c, α2 = 1 2(a − 1)2, β2 = − 1 2b2, γ2 = c, α3 = 1 2a2, β3 = − 1 2(b + 1)2, γ3 = c, α4 = 1 2a2, β4 = − 1 2(b − 1)2, γ4 = c, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' This is easily proved by applying (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1) to B¨acklund transformations in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1 Algebraic solutions Algebraic solutions of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1) are equivalent to rational solutions of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2) and so we discuss rational solutions of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2), which are classified in the following Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Necessary and sufficient conditions for the existence of rational solutions of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2) are either (α, β, γ) = � 1 2(n + 1 2), − 1 2µ2, 1 � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='5) or (α, β, γ) = � 1 2µ2, − 1 2(n + 1 2), −1 � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='6) where n ∈ Z and µ is an arbitrary constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' For details see Gromak, Laine and Shimomura [22, §38];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' see also [39, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' We remark that the solutions of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2) satisfying (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='5) are related to those satisfying (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='6) through the analog of the symmetry (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Consequently we shall be concerned only with rational solutions of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2) for the parameters given by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' The rational solution of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2) for the parameters (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='5) is given by un(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ) = 1 − xS2 n(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ) Sn+1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ)Sn−1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ), n ≥ 0, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='7) where Sn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ) is the Umemura polynomial (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Substituting the rational solution of SIII (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='13) given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='3) into (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='3) and then using the reccurence relation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1) gives the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' The Umemura polynomial Sn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ) satisfies the difference equation Sn+1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ)Sn−1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ) = xS2 n(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ) + µSn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ + 1) Sn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='8) Hence from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='7) there are two alternative representations of the rational solution un(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ) = µSn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ + 1) Sn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ − 1) µSn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ + 1) Sn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ − 1) + xS2n(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ), un(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ) = µSn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ + 1) Sn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ − 1) Sn+1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ)Sn−1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2 Bessel function solutions Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Necessary and sufficient conditions for the existence of Bessel function solutions of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2) are either (α, β, γ) = � 1 2n2, − 1 2µ2, ε � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='9) or (α, β, γ) = � 1 2µ2, − 1 2n2, −ε � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='10) with ε = ±1, and where n ∈ Z+ and µ is an arbitrary constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' 7 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' From (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='5) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='9), the parameters in PIII (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1) and deg-PV (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2) are given by (A, B) = � 2(a − b), 2ε(a + b + 1) � , (α, β, γ) = ( 1 2a2, − 1 2b2, ε), respectively, for parameters a, b and ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' The result then follows from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' The Bessel function solution of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2) for the parameters (α, β, γ) = � 1 2n2, − 1 2µ2, ε � , is given by un(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ, ε) = 1 + εx2τ 2 n(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ, ε) τn+1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ, ε) τn−1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ, ε), n ≥ 1, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='11) where τn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ, ε) = det �� x d dx �j+k ϕµ(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ε) �n−1 j,k=0 , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='12) and τ0(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ, ε) = 1, with ϕµ(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ε) = � c1Jµ(x) + c2Yµ(x), if ε = 1, c1Iµ(x) + c2Kµ(x), if ε = −1, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='13) c1 and c2 arbitrary constants, and Jµ(x), Yµ(x), Iµ(x) and Kµ(x) Bessel functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Substituting the Bessel function solution of SIII (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='13) given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='10) into (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='3) and then using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='11) gives the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' The Bessel function solution of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2) for the parameters (α, β, γ) = � 1 2n2, − 1 2µ2, 2ε � , is given by wn(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ, ε) = 1 + εzT 2 n (z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ, ε) Tn+1(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ, ε) Tn−1(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ, ε), n ≥ 1, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='14) where Tn(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ, ε) = det �� z d dz �j+k ψµ(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ε) �n−1 j,k=0 , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='15) and T0(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ, ε) = 1, with ϕµ(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ε) = � c1Jµ(2√z) + c2Yµ(2√z), if ε = 1, c1Iµ(2√z) + c2Kµ(2√z), if ε = −1, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='16) c1 and c2 arbitrary constants, and Jµ(x), Yµ(x), Iµ(x) and Kµ(x) Bessel functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' In the next Lemma, it is shown that the first solution u1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ, ε), the “seed solution”, satisfies a first-order, second-degree equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' The solution of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2) for the parameters (α, β, γ) = � 1 2, − 1 2µ2, ε � , is u1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ, ε) = ϕµ+1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ε) [xϕµ+1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ε) − 2εµϕµ(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ε)] xϕ2 µ+1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ε) − 2εµϕµ+1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ε)ϕµ(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ε) + εxϕ2µ(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ε), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='17) where ϕµ(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ε) = � c1Jµ(x) + c2Yµ(x), if ε = 1, c1Iµ(x) + c2Kµ(x), if ε = −1, with c1 and c2 constants, satisfies the first-order, second-degree equation x2 �du dx �2 − 4xu(u − 1)du dx + 4εx2u(u − 1) + 4(u − 1)2(u2 − µ2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='18) 8 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Define Φµ(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ε) = ϕµ+1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ε) ϕµ(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ε) , then from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='17) u1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ, ε) = 1 − x εxΦ2µ − 2µΦµ + x, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='19) and Φµ(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ε) satisfies the Riccati equation xdΦµ dx = εxΦ2 µ − (2µ + 1)Φµ + x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='20) Next we assume that u1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ, ε) satisfies a first-order, second-degree equation of the form x2 �du dx �2 + x � f2(x, µ, ε)u2 + f1(x, µ, ε)u + f0(x, µ, ε) � du dx + 4 � j=0 gj(x, µ, ε)uj = 0, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='21) where {fj(x, µ, ε)}2 j=0 and {gj(x, µ, ε)}4 j=0 are to be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Then substituting (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='19) into (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='21), using the fact that Φµ(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ε) satisfies (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='20) and equating coefficients of powers of Φµ yields f2 = −4, f1 = 4, f0 = 0, g4 = 4, g3 = −8, g2 = 4εx2 − 4µ2 + 4, g1 = −4εx2 + 8µ2, g0 = −4µ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Hence we obtain equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='18), as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' This demonstrates that special function solutions of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2), and hence also deg-PV (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1) , are different from special function solutions of PII–PVI where the “seed solution” satisfies a Riccati equation, a first- order, first-degree equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' 6 Applications 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1 Complex sine-Gordon equation Consider the two-dimensional complex sine-Gordon equation ∇2ψ + (∇ψ)2ψ 1 − |ψ|2 + ψ � 1 − |ψ|2� = 0, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1) where ∇ψ = (ψx, ψy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Making the transformation ψ(x, y) = cos(ϕ(x, y)) exp{iη(x, y)}, ψ(x, y) = cos(ϕ(x, y)) exp{−iη(x, y)}, in the complex sine-Gordon equation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1) yields ∇2ϕ + cos ϕ sin3 ϕ(∇η)2 − 1 2 sin(2ϕ) = 0, sin(2ϕ) ∇2η = 4∇ϕ •∇η, which is the Pohlmeyer-Lund-Regge model [33, 34, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' The complex sine-Gordon equation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1) has a separable solution in polar coordinates given by ψ(r, θ) = Rn(r) einθ, where Rn(r) satisfies d2Rn dr2 + 1 r dRn dr + Rn 1 − R2n ��dRn dr �2 − n2 r2 � + Rn � 1 − R2 n � = 0, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2) We remark that this equation also arises in extended quantum systems [4, 5, 6], in relativity [20] and in coefficients in the three-term recurrence relation for orthogonal polynomials with respect to the weight w(θ) = et cos θ on the unit circle, see [56, equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='13)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' The orthogonal polynomials for this weight on the unit circle are related to unitary random matrices [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Equation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2) can be shown to possess the Painlev´e property, though is not in the list of 50 equa- tions given in [25, Chapter 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Equation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2) can be transformed to the fifth Painlev´e equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2) in two different ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' 9 (i) If Rn(r) satisfies (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2) then making the transformation Rn(r) = 1 + un(z) 1 − un(z), r = 1 2z, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='3) yields d2un dz2 = � 1 2un + 1 un − 1 � �dun dz �2 − 1 z dun dz + n2(un − 1)2(u2 n − 1) 8z2un − un(un + 1) 2(un − 1) , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='4) which is PV (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2) with α = 1 8n2, β = − 1 8n2, γ = 0 and δ = − 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' (ii) If Rn(r) satisfies (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2) then making the transformation Rn(r) = 1 � 1 − vn(x) , r = √x, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='5) yields d2vn dx2 = � 1 2vn + 1 vn − 1 � �dvn dx �2 − 1 x dvn dx − n2(vn − 1)2 2x2vn + vn 2x, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='6) which is deg-PV (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1) with α = 0, β = − 1 2n2 and γ = 1 2 so is equivalent to PIII (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1), as mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' This shows that solutions of equations (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='4) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='6) are related by vn(x) = 4un(z) 1 + u2n(z), x = 1 4z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' The function Rn(r) satisfies the ordinary differential equation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2), the differential-difference equa- tions dRn dr + n r Rn − � 1 − R2 n � Rn−1 = 0, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='7a) dRn−1 dr − n − 1 r Rn−1 + � 1 − R2 n−1 � Rn = 0, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='7b) since solving (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='7a) for Rn−1(r) and substituting in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='7b) yields equation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Also eliminating the derivatives in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='7), after letting n → n + 1 in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='7b), yields the difference equation Rn+1 + Rn−1 = 2n r Rn 1 − R2n , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='8) which is known as the discrete Painlev´e II equation [41, 49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' If n = 1 then equations (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='7) have the solution R0(r) = 1, R1(r) = C1I1(r) − C2K1(r) C1I0(r) + C2K0(r), where I0(r), K0(r), I1(r) and K1(r) are the imaginary Bessel functions and C1 and C2 are arbitrary constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' For solutions which are bounded at r = 0 then necesssarily C2 = 0 and so R0(r) = 1, R1(r) = I1(r) I0(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='9) Hence one can use the difference equation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='8) to determine Rn(r), for n ≥ 2, which yields R2(r) = −rR2 1(r) + 2R1(r) − r r [R2 1(r) − 1] , R3(r) = R3 1(r) − rR2 1(r) − 2R1(r) + r R1(r) [rR2 1(r) + R1(r) − r] , R4(r) = r(r2 + 5)R4 1(r) + 4R3 1(r) − 2r(r2 + 3)R2 1(r) + r3 r [(r2 − 1)R4 1(r) + 4rR3 1(r) − 2(r2 + 2)R2 1(r) − 4rR1(r) + r2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' These results suggest that (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2) should be solvable in terms of PIII (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1), which is illustrated in the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' 10 Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' If Rn(r) satisfies (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2) then wn(r) = Rn+1(r)/Rn(r) satisfies d2wn dr2 = 1 wn �dwn dr �2 − 1 r dwn dr − 2n r w2 n + 2(n + 1) r + w3 n − 1 wn , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='10) which is PIII (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1) with parameters α = −2n and β = 2(n + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' See Hisakado [23] and Tracy & Widom [52];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' see also [56, §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' We note that since the parameters in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='10) satisfy −α + β = 4n + 2, with n ∈ Z+, then the equation has solutions expressible in terms of the modified Bessel functions I0(r) and I1(r) (as well as K0(r) and K1(r), but these are not needed here).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Let τn(r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ν) be the n × n determinant τn(r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ν) = det �� r d dr �j+k Iν(r) �n−1 j,k=0 , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='11) with Iν(r) the modified Bessel function, then wn(r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ν) = τn+1(r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ν + 1) τn(r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ν) τn+1(r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ν) τn(r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ν + 1) ≡ d dz � ln τn+1(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ν) τn(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ν + 1) � − n + ν z , n ≥ 0, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='12) satisfies PIII (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1) with α = 2(ν − n) and β = 2(ν + n + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' See, for example, [19, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Equation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2) has the solution Rn(r) = τn(r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' 1) τn(r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' 0), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='13) where τn(r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ν) is the determinant given by (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' The proof is straightforward using induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' From (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='9) we have R1(r) = I1(r) I0(r) = τ1(r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' 1) τ1(r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' 0), so (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='13) is true if n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Assuming (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='13) holds then from Theorems 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2 Rn+1(r) = wn(r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' 0)Rn(r) = τn+1(r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' 1) τn(r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' 0) τn+1(r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' 0) τn(r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' 1) × τn(r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' 1) τn(r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' 0) = τn+1(r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' 1) τn+1(r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' 0), as required, and so the result follows by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Equations (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='4) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='6) have the Bessel function solutions un(z) = τn( 1 2z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' 1) + τn( 1 2z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' 0) τn( 1 2z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' 1) − τn( 1 2z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' 0), vn(x) = 1 − τ 2 n(√x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' 0) τ 2n(√x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' 1), respectively, with τn(r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ν) the determinant given by (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' The formal asymptotic behaviour of the vortex solution Rn(r) is given by Rn(r) = rn 2n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' � 1 − r2 4(n + 1) + O � r4�� , as r → 0, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='14) Rn(r) = 1 − n 2r − n2 8r2 − n(n2 + 1) 16r3 + O(r−4), as r → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='15) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' These are determined from (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='8) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' 11 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2 Generalised Charlier polynomials The Charlier polynomials Cn(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' z) are a family of orthogonal polynomials introduced in 1905 by Char- lier [7] given by Cn(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' z) = 2F0 (−n, −k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' −1/z) = (−1)nn!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='L(−1−k) n (−1/z) , z > 0, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='16) where 2F0(a, b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' z) is the hypergeometric function and L(α) n (z) is the associated Laguerre polynomial, see, for example, [48, §18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' The Charlier polynomials are orthogonal on the lattice N with respect to the Poisson distribution ω(k) = zk k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' , z > 0, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='17) and satisfy the orthogonality condition ∞ � k=0 Cm(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' z)Cn(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' z)zk k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' = n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ez zn δm,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Smet and Van Assche [51] generalized the Charlier weight (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='17) with one additional parameter through the weight function ω(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ν) = Γ(ν + 1) zk Γ(ν + k + 1) Γ(k + 1), z > 0, with ν a parameter such that ν > −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' This gives the discrete weight ω(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ν) = zk (ν + 1)k k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=', z > 0, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='18) where (ν + 1)k = Γ(ν + 1 + k)/Γ(ν + 1) is the Pochhammer symbol, on the lattice N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Discrete orthogonal polynomials are characterized by the discrete Pearson equation ∆ � σ(k)ω(k) � = τ(k)ω(k), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='19) where ∆ is the forward difference operator ∆f(k) = f(k + 1) − f(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' The weight (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='18) satisfies the discrete Pearson equation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='19) with σ(k) = k(k + ν), τ(k) = −k2 − νk + z, and so the generalised Charlier polynomials are semi-classical orthogonal polynomials since τ(k) is a polynomial with deg(τ) > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' The special case β = 0 was first considered by Hounkonnou, Hounga and Ronveaux [24] and later studied by Van Assche and Foupouagnigni [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' For the generalised Charlier weight (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='18), the orthonormal polynomials pn(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' z) satisfy the orthog- onality condition ∞ � k=0 pm(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' z)pn(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' z) zk (ν + 1)k k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' = δm,n, and the three-term recurrence relation kpn(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' z) = an+1(z)pn+1(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' z) + bn(z)pn(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' z) + an(z)pn−1(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' z), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='20) with p−1(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' z) = 0 and p0(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' z) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Our interest is in the coefficients an(z) and bn(z) in the recurrence relation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Smet and Van Assche [51, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1] proved the following theorem for recurrence coefficients associated with the generalised Charlier weight (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' 12 Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' The recurrence coefficients an(z) and bn(z) for orthonormal polynomials associated with the generalised Charlier weight (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='18) on the lattice N satisfy the discrete system (a2 n+1 − z)(a2 n − z) = z(bn − n)(bn − n + ν), bn + bn−1 − n + ν + 1 = nz/a2 n, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='21) with initial conditions a2 0 = 0, b0 = √z Iν+1(2√z) Iν(2√z) = z d dz � ln Iν(2√z) � − ν 2 , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='22) with Iν(k) the modified Bessel function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' The discrete system such as (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='21) for recurrence coefficients is sometimes known as the Laguerre-Freud equations, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' [3, 24, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' The recurrence coefficients an(z) and bn(z) also satisfy the Toda lattice, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' [56, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='8] z d dz a2 n = a2 n(bn − bn−1), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='23a) z d dz bn = a2 n+1 − a2 n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='23b) Letting a2 n(z) = xn(z) and bn(z) = yn(z) in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='21) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='23) yields (xn+1 − z)(xn − z) = t(yn − n)(yn − n + ν), z dxn dt = xn(yn − yn−1), yn + yn−1 − n + ν + 1 = nz xn , z dyn dz = xn+1 − xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Eliminating xn+1 and yn−1 in these equations yields the differential system z dxn dz = xn(2yn + ν − n + 1) − nz, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='24a) z dyn dz = −xn + z + (yn − n)(yn − n + ν)z xn − z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='24b) Solving (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='24a) for yn gives yn = z 2xn dxn dz + nz 2xn + n − ν − 1 2 , and substituting this into (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='24b) yields d2xn dz2 = 1 2 � 1 xn + 1 xn − z � − xn z(xn − z) dxn dz − 2x2 n z2 + 4xn + n2 − ν2 + 1 2z − n2 2xn + 1 − ν2 2(xn − z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='25) Making the transformation xn(z) = z 1 − wn(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='26) in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='25) yields d2wn dz2 = � 1 2wn + 1 wn − 1 ��dwn dz �2 − 1 z dwn dz + (wn − 1)2(n2w2 n − ν2) 2wnz2 − 2wn z , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='27) which is deg-PV (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1) with parameters α = 1 2n2, β = − 1 2ν2 and γ = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Solving (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='24b) for xn gives xn = − 1 2z dyn dz + z + 1 2Xn, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='28) where X2 n = z2 �dyn dz �2 + 4z(yn − n)(yn − n + ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='29) 13 From (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='29) we get dXn dz = z2 Xn d2yn dz2 dyn dz + z Xn �dyn dz �2 + 2z(2yn − 2n + ν) Xn dyn dz + 2(yn − n)(yn − n + ν) Xn (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='30) Substituting (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='28) into (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='24a), then using (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='30), solving for Xn, and substituting into (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='29) yields the second-order, second-degree equation � 2z d2yn dz2 + dyn dz + 8yn − 8n + 4ν �2 = (4yn − 2n + 2ν + 1)2 z � z �dyn dz �2 + 4(yn − n)(yn − n + ν) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='31) Making the transformation yn(z) = 1 2vn(x) + 1 2n − 1 2ν − 1 4, x = 2√z, in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='31) yields �d2vn dx2 + 4vn − 4n − 2 �2 = 4v2 n x2 ��dvn dx �2 + 4v2 n − 4(2n + 1)vn + (2n + 1)2 − 4ν2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='32) Equation (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='5) in [14] is �d2v dx2 − av − b �2 = 4v2 x2 ��dv dx �2 − av2 − 2bv − c � , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='33) with a, b and c parameters, an equation derived by Chazy [8], and is the primed version of equation SD-III in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Hence equation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='32) is the special case of equation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='33) with a = −4, b = 4n + 2, c = 4ν2 − (2n + 1)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Cosgrove [14] showed that equation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='33) is solvable in terms of solutions of PIII (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Consequently, the solution of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='32) is given by vn(x) = x 2q � dq dx + q2 + 1 � , where q(x) satisfies PIII (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1) for the parameters A = 2ν − 2n − 2 and B = 2ν + 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' The recurrence relations an(z) and bn(z) are given by a2 n(z) = xn(z) = Tn+1(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ν)Tn−1(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ν) T 2 n (z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ν) , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='34a) bn(z) = yn(z) = z d dz � ln Tn+1(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ν) Tn(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ν) � − ν 2 , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='34b) where Tn(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ν) = det �� z d dz �j+k Iν � 2√z � �n−1 j,k=0 , with T0(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ν) = 1, and Iν(x) is the modified Bessel function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' The expression (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='34a) for a2 n(z) follows immediately by substituting (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='14) in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' To prove the result (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='34b) for bn(z) we use induction and the factor that from equation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='23b), a2 n(z) = xn(z) and bn(z) = yn(z) are related by z dxn dt = xn(yn − yn−1), and initially y0(z) = z d dz � ln T1(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ν) � } − ν 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' 14 Hence y1(z) = z d dz � ln x1(z) � + y0(z) = z d dz � ln T2(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ν)T0(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ν) T 2 1 (z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ν) � + z d dz {ln T1(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ν)} − ν 2 = z d dz � ln T2(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ν) T1(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ν) � − ν 2 , so (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='34b) is true for n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Now suppose that (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='34b) is true, then yn+1(z) = z d dz � ln xn(z) � + yn(z) = z d dz � ln Tn+2(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ν)Tn(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ν) T 2 n+1(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ν) � + z d dz � ln Tn+1(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ν) Tn(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ν) � − ν 2 = z d dz � ln Tn+2(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ν) Tn+1(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ν) � − ν 2 , as required, and so the result follows by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' We remark that equation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='23a) is identically satisfied by a2 n(z) and bn(z) given by (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' In a recent paper, Fern´andez-Irisarri and Ma˜nas [17, §2] discuss the generalised Charlier weight (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='18), in particular properties of the coefficients in the recurrence relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' The relationship between the notations in [17] and those here are xn(z) = γn(η) and yn(z) = βn(η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Fern´andez-Irisarri and Ma˜nas [17] relate xn(z) and yn(z) to Okamoto’s Hamiltonian for PIII′ [46] and derive two ordinary differential equations for xn(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Equation (45) in [17, Theorem 4] is the third order equation δz �xn z � δ2 z(ln xn) + 2xn � + n2z xn � = 2xn, δz(f) = z df dz , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' d3xn dz3 = 1 zx2n � z dxn dz − xn � � 2xn d2xn dz2 − �dxn dz �2 + n2 � − 4xn z2 dxn dz + 2xn(xn + z) z3 , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='35) and the state that this equation “should have the Painlev´e property”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Equation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='35) can be integrate to give equation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='25), with ν2 as the constant of integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Since equation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='25) is equivalent to deg-PV (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2) then equation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='35) does have the Painlev´e property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Equation (60) in [17, Theorem 5] is the second order equation � 1 − xn z � � δz �δz(xn) + nz xn � + 2xn � + 2{xn − z + (n − b)n} = − 1 2 �δz(xn) + nz xn �2 + (n + 1) �δz(xn) + nz xn � + (n − b − 1)(3n − b + 1), which is equation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='25) with ν2 = 2(b − n)2 + n2 − 2n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' 7 Discussion In this paper the classical solutions of deg-PV (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2) have been classified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Ohyama and Okumura [43, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1] give a list of classical solutions of PI to PV and state that “deg-P5 with α = 1 2a2, β = − 1 8, γ = −2 has the algebraic solution w(z) = 1 + 2√z/a”2 and “deg-P5 with β = 0 has the Riccati type 2As noted in [1], there is typo in [43] who say β = −8 rather than β = − 1 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' 15 solutions”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' The results in this paper show that there are more classical solutions of deg-PV (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' The algebraic solution is equivalent to the “seed solution” obtained by setting n = 0 in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='7), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' u0(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ) = µ x + µ, and there is a more general hierarchy of “Riccati type solutions” which are described in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' All solutions of PII–PVI that are expressible in terms of special functions satisfy a first-order equa- tion of the form �du dx �n = n−1 � j=0 Fj(u, x) �du dx �j , (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1) where Fj(u, x) is polynomial in u with coefficients that are rational functions of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' It can be shown that the Bessel function solutions of PIII (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1) satisfy a first-order equation of the form (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1) for n odd, whereas the Bessel function solutions of deg-PV (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2) satisfy a first-order equation of the form (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1) for n even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' The relationship between PIII (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1) and deg-PV (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1) is similar to that between the second Painlev´e equation (PII) d2q dx2 = 2q3 + xq, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2) with α a parameter, and Painlev´e XXXIV equation (P34) d2p dx2 = 1 2p � dp dx �2 + 2p2 − xp − (α + 1 2)2 2p , (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='3) which is equivalent to equation XXXIV of Chapter 14 in [25], in that both pairs of equations arise from a Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' The Hamiltonian associated with PII (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2) and P34 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='3) is HII(q, p, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' α) = 1 2p2 − (q2 + 1 2z)p − (α + 1 2)q (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='4) and so dq dz = p − q2 − 1 2z, dp dz = 2qp + α + 1 2, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='5) see [28, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' It is known that PII (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2) and P34 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='3) have special function solutions in terms of Airy functions, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' It can be shown that the Airy function solutions of PII (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2) satisfy first-order equation of the form (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1) for n odd, whereas the Airy function solutions of P34 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='3) satisfy a first-order equation of the form (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1) for n even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Acknowledgements I thank Clare Dunning and Steffen Krusch for helpful comments and illuminating discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' References [1] P.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' 2) (2003) 231–237.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' 18' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} diff --git a/D9E2T4oBgHgl3EQfSgcI/content/tmp_files/2301.03792v1.pdf.txt b/D9E2T4oBgHgl3EQfSgcI/content/tmp_files/2301.03792v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..d83e0100e809a18435e04a34322dfceb3223b1d4 --- /dev/null +++ b/D9E2T4oBgHgl3EQfSgcI/content/tmp_files/2301.03792v1.pdf.txt @@ -0,0 +1,775 @@ +arXiv:2301.03792v1 [math.GT] 10 Jan 2023 +A G-FAMILY OF SINGQUANDLES AND INVARIANTS OF DICHROMATIC +SINGULAR LINKS +MOHD IBRAHIM SHEIKH, MOHAMED ELHAMDADI, AND DANISH ALI +ABSTRACT. We introduce and investigate dichromatic singular links. We also construct G-Family +of singquandles and use them to define counting invariants for unoriented dichromatic singular links. +We provide some examples to show that these invariants distinguish some dichromatic singular links. +CONTENTS +1. +Introduction +1 +2. +Singular links, Singquandles and Dichromatic Links +2 +3. +Dichromatic Singular Links +5 +4. +G-Family of Singquandles (Disingquandles) +6 +5. +Computable Invariants for Unoriented Dichromatic Singular Links +10 +Acknowledgement +14 +References +15 +Mathematics Subject Classifications (2020): 57M25, 57M27. +Key words and Phrases: Knot; Link; Singular knot; Singular link; Dichromatic link; Dichromatic +singular link; Quandle; Singquandle; Disingquandle; Disingquandle counting invariant. +1. INTRODUCTION +A knot is a simple closed curve in three dimensional space R3 and a disjoint union of two or +more knots forms a link with two or more components [8]. Knots and links are categorised in many +ways. One way is to use the crossing type as a tool to define a knot or link type. Classical, virtual +and singular knots and links serve as examples as they are all recognised by the type of crossing +they contain. The other way to define link types is by labelling the components of a classical link. +Dichromatic links are defined by using this technique as their components are either labelled by +“1” or “2” [1, 2, 10, 11, 14]. A singular link is a link with at least one singular crossing. In this +paper we use such labelling technique for singular links and define a new type of links which we +call dichromatic singular links. +A quandle is an algebraic structure satisfying some axioms that result from the Reidemeister +moves for oriented classical knots and links. If furthermore all right multiplications by fixed ele- +ments of the quandle are involutions then such structures are called involutory quandles or Kei’s +They are used to investigate unoriented knots and links. Quandles were independently introduced +by Joyce and Matveev [13, 16]. Since then they have been used to construct invariants of knots +and links [4, 6, 17]. Quandles have been also used to define new algebraic systems by taking a +1 + +family of quandles at a time. Such systems are called G-Family of quandles and this notion was +introduced in 2013 by Ishii, Iwakiri, Jang and Oshiro [12]. A G-Family of quandles were used to +define invariants for handlebody-knots. Also in [15] Lee and Sheikh used Z2-Family of quandles +to construct algebraic invariants for oriented dichromatic links. +In this paper, we introduce the notions of G-Family of singquandles and dichromatic singular +links. A dichromatic singular link is an n component singular link with each of its component +labelled as “1” or “2”. A singquandle is an algebraic system whose axioms are motivated by +Reidemeister moves of unoriented singular knots. By taking a family of such algebaraic systems +(Singquandles), we define a new algebraic system which we call G-Family of singquandles or +disingquandle. The axioms of the latter are motivated by generalized Reidemeister moves of un- +oriented dichromatic singular links. We discuss various examples and some properties of G-Family +of singquandles, and also show that a G-Family of singquandles X enables us to distinguish unori- +ented dichromatic singular links by computing their sets of all X-colorings and proving that these +sets are different when their arcs are colored by the elements of X. +This paper is organized as follows. Section 2 reviews some preliminaries about singular links, +singquandles as well as dichromatic links and their generalized Reidemeister moves. In Section 3 +we introduce the notion of dichromatic singular links with some typical examples of unoriented +dichromatic singular links. Section 4 introduces the notion of G-Family of singquandles (dis- +ingquandles) with some typical examples of G-Family of singquandles. Section 5 discusses how +G-Family of singquandles is related to unoriented dichromatic singular links and develop com- +putable invariants for unoriented dichromatic singular links. We discuss some examples which +show how the invariants distinguish unoriented dichromatic singular links, and especially how +they detect the change of component labelings. +2. SINGULAR LINKS, SINGQUANDLES AND DICHROMATIC LINKS +In this section we review some preliminaries about singular links, singquandles and dichromatic +links. Most of the terminologies of this section can be found in [5, 9, 15]. We begin with the +definition of a singular link. +Definition 2.1. A singular link in S3 is the image of a smooth immersion of n circles in S3 that has +finitely many double points, called singular points. +A singular link in R3 is represented by a singular link diagram in the plane R2, which is a +classical link diagram with one or more singularities. A singularity is a rigid vertex where a link is +glued to itself. Figure 1 gives two examples of singular links. +FIGURE 1. Singular Links +2 + +Two singular links L� and L� are isotopy equivalent if one can be obtained from the other by a +finite sequence generalized Reidemeister moves for singular links as shown in the following figure. +Let D� and D� be two singular link diagrams in R2 representing L� and L�, respectively. Then +L� and L� are equivalent if and only if D� and D� can be transformed into each other by a finite +sequence of classical and singular Reidemeister moves shown in Figure 2. +FIGURE 2. Classical and Singular Reidemeister Moves +Definition 2.2. [5] Let (X, ∗) be an involutive quandle. Let R1 and R2 be two maps from X × X +to X. The quadruple (X, ∗, R1, R2) is called a singquandle if the following axioms are satisfied +(2.2.1) +x = R1(y, R2(x, y)) = R2(R2(x, y), R1(x, y)), +(2.2.2) +y = R2(R2(x, y), x) = R1(R2(x, y), R1(x, y)), +(2.2.3) +R(x, y) = (R1(y, R2(x, y)), R2(R2(x, y), x)), +(2.2.4) +(y ∗ z) ∗ R2(x, z) = (y ∗ x) ∗ R1(x, z), +(2.2.5) +R1(x, y) = R2(y ∗ x, x), +(2.2.6) +R2(x, y) = R1(y ∗ x, x) ∗ R2(y ∗ x, x), +(2.2.7) +R1(x ∗ y, z) ∗ y = R1(x, z ∗ y), +3 + +(2.2.8) +R2(x ∗ y, z) = R2(x, z ∗ y) ∗ y. +We remind the reader that the singquandle axioms come from the generalized Reidemeister +moves for unoriented singular knots. Singquandles were introduced as a ramification of quandles +with the purpose of studying singular links, see for example [4,5,17]. +The following are few typical examples of singquandles. +• For an involutive quandle (X, ∗) with x ∗ y = 2y − x and X = Zn, the quadruple +(X, ∗, R1, R2) forms a singquandle if and only if the following conditions are satisfied: +(1) R2(x, y) = R1(x, y) + y − x, +(2) R1(x, y) = R1(2x − y, x) + y − x, +(3) R1(x, 2y − z) = 2y − R1(2y − x, z), +(4) R2(2y − x, z) = 2y − R2(x, 2x − z). +• For an involutive quandle (X, ∗) where X is a group G and x ∗ y = yx−1y, the quadruple +(X, ∗, R1, R2) forms a singquandle if and only if the following conditions are satisfied: +(1) R2(x, z)z−1yz−1R2(x, z) = R1(x, z)x−1yx−1R1(x, z), +(2) R1(x, y) = R2(xy−1x, x), +(3) R2(x, y) = R2(xy−1x, x)[R1(xy−1x, x)]−1R2(xy−1x, x), +(4) y[R1(yx−1y, z)]−1y = R1(x, yz−1y), +(5) R1(yx−1y, z) = y[R2(x, yz−1y)]−1y. +Definition 2.3. For a positive integer n ≥ 1. A dichromatic link is a smooth imbedding of n circles +in R3 such that each component is labeled as “1” or “2”. +In R2 every dichromatic link is represented by a dichromatic link diagram which is a classical link +diagram with each component labelled either “1” or “2”. For example, see Figure 3. +1 +1 +2 +2 +FIGURE 3. Dichromatic Links +Two dichromatic links L� and L� are isotopy equivalent if one can be obtained from the other +by a finite sequence of generalized Reidemeister moves for the dichromatic links as shown in +the figure 4. Let D� and D� be two dichromatic link diagrams in R2 representing L� and L�, +respectively. Then L� and L� are equivalent if and only D� and D� can be transformed into each +other by a finite sequence of generalized Reidemeister moves shown in the following Figure 4. +4 + +i +i +i +j +j +i +i +k +j +i +k +j +FIGURE 4. Generalized Reidemeister Moves for Dichromatic Links +3. DICHROMATIC SINGULAR LINKS +This section is devoted to dichromatic singular links which is a generalization of singular links. +To generate a dichromatic singular link we label a singular link’s components with “1” or “2”. +Thus We have the following definition. +Definition 3.1. A singular link L in R3 whose each component is colored (labelled) by either “1” +or “2” is called a dichromatic singular link. +A dichromatic singular link L in R3 is represented by a dichromatic singular link diagram D in +R2 in which each component is labelled “1” or “2”. Figure 5 shows two examples of unoriented +dichromatic singular link diagrams. +1 +2 +1 +2 +FIGURE 5. Dichromatic Singular Links +Two dichromatic singular links L� and L� in R3 are ambient isotopic if there exists a self home- +omorphism h : R3 → R3 that takes one link to the other and preserves the singularities as well +as the labels “1”, “2” such that h(L�) = L�. Thus two singular dichromatic links L� and L� are +equivalent if one can be obtained from the other by a finite sequence of generalized dichromatic +singular Reidemeister moves preserving the label of each component as shown in the Figure 6. Let +D� and D� be two dichromatic singular link diagrams in R2 representing L� and L�, respectively. +Then L� and L� are equivalent if and only if D� and D� can be transformed into each other by +a finite sequence of generalized dichromatic singular Reidemeister moves shown in the following +Figure 6 where i, j, k ∈ {1, 2}. +A dichromatic singular link with n components is called as an n-component dichromatic singular +link. Thus an n-component dichromatic singular link in R3 can be defined as L = K1 ∪ · · · ∪ Kn. +5 + +i +k +k +j +i +i +j +j +i +j +i +k +k +j +i +j +i +i +i +j +j +i +i +k +j +i +k +j +FIGURE 6. Regular Dichromatic Reidemeister Moves RI, RII and RIII on the +top and Dichromatic Singular Reidemeister Moves RIV a, RIV b and RV in the +middle and on the bottom. +Taking n = 2, we obtain 2-component dichromatic singular links. Some 2-component unoriented +dichromatic singular link diagrams (see p 814 of [18]) are shown in Figure 12. +Proposition 3.2. Let L� and L� be two unoriented dichromatic singular links in R3 and let D� +and D� be two unoriented dichromatic singular link diagrams in R2 representing L� and L�, +respectively. Then L� and L� are equivalent if and only if D� and D� are transformed into each +other by a finite sequence of generalized Reidemeister moves for unoriented dichromatic singular +links which preserve the singularities and the label of each component as shown in the Fig. 6 +where i, j, k ∈ {1, 2} and ambient isotopies of R2. +4. G-FAMILY OF SINGQUANDLES (DISINGQUANDLES) +Before introducing the notion of G-Family of Singquandles, we first recall the definition of +G-family of quandles from [12]. +Definition 4.1. Given a group G and a set X, a G-family of quandles, denoted by (G, X), is +a choice of quandle operation ∗g on the set X for each element g ∈ G such that the following +axioms are satisfied +(1) For all g ∈ G and for all x ∈ X, x ∗g x = x, +(2) For all g, h ∈ G and for all x, y ∈ X, (x ∗g y) ∗h y = x ∗gh y, +(3) For all x, y ∈ X, x ∗e x = x, where e is the identity element of G, +6 + +(4) For all x, y, z ∈ X, (x ∗g y) ∗h z = (x ∗h z) ∗h−1gh (y ∗h z) +The following are two examples of G-families of quandles. +• For any group G and any set X, defining x ∗g y = x for all x, y ∈ X and all g ∈ G. This +gives a G-family of quandles called the trivial G-family of quandles. +• Let (X, ∗) be a quandle of cyclic type [19] with cardinality n. Let Rx denotes the right +multiplication by x, thus by definition Rx +(n−1) is the identity map. Then define x ∗i y = +Ry +i(x) then it is shown in Proposition 2.3 of [12] that (Z, X) is a Z-family of quandles and +also Z(n−1)-family of quandles. +A G-family of quandles (G, X) induces a quandle operation on the set G × X by +(g, x) ∗ (h, y) = (h−1gx, x ∗h y). +The notion of G-family of quandles was introduced by Ishii, Iwakiri, Jang and Oshiro in 2013 +in [12] in order to produce invariants of handlebody knots. They defined coloring invariants and +cocycle invariants of handlebody knots. They used these invariants to detect chirality of some han- +dlebody knots. Later in 2015, Ishii independently studied the notion of G-family of quandles in +connection with the multiple conjugation quandle and showed that the later one can be obtained +from the first one. In 2017 and 2018 Ishii, Nelson and Ishii, Iwakiri, Kamada, Kim, Matsuzaki, Os- +hiro respectively, used this work and introduced the notions of partially multiplicative biquandles +and multiple conjugation biquandle. In 2021 Lee and Sheikh jointly used G-family of quandles +to construct algebraic invariants for oriented dichromatic links [15]. We introduce the following +definition. +Definition 4.2. Let X be a set equipped with two binary operations ∗1 and ∗2 such that both +(X, ∗1), (X, ∗2) are involutive quandles. Let R1, R2 be two maps from X × X to X such that the +quadruples (X, ∗1, R1, R2) and (X, ∗2, R1, R2) are singquandles. Then the quintuple (X, ∗1, ∗2, R1, R2) +is called a disingquandle or Z2-family of singquandles if the following axioms are satisfied +(4.2.1) +(x ∗1 y) ∗2 z = (x ∗2 z) ∗1 (y ∗2 z), +(4.2.2) +(x ∗2 y) ∗1 z = (x ∗1 z) ∗2 (y ∗1 z), +(4.2.3) +(y ∗1 z) ∗2 R2(x, z) = (y ∗2 x) ∗1 R1(x, z), +(4.2.4) +(y ∗2 z) ∗1 R2(x, z) = (y ∗1 x) ∗2 R1(x, z), +(4.2.5) +R2(x, y) = R1(y ∗1 x, x) ∗2 R2(y ∗1 x, x), +(4.2.6) +R2(x, y) = R1(y ∗2 x, x) ∗1 R2(y ∗2 x, x), +The above axioms of a disingquandle come from the generalized dichromatic singular Reide- +meister moves shown in Figure 6 when we take the coloring rule shown in Figure 7 under consid- +eration. +7 + +i +i +j +i/j +j/i +j +i +y +x +y +x +y +x +x +x +y +x +* +j x +y* +R ( ) +x +1 +2 +y, +R ( ) +x y, +FIGURE 7. Coloring by a disingquandle +The following lemma is motivated by the above construction. +Lemma 4.3. The set of colorings of a dichromatic singular link by a disingquandle does not change +by the dichromatic singular Reidemeister moves shown in Figure 6. +Proof. As in the case of classical and singular knot theories, there is one to one correspondence +between colorings before and after each of the dichromatic singular Reidemeister moves. The +invariance follows directly from the equations 4.2.1, 4.2.2, 4.2.3, 4.2.4, 4.2.5 and 4.2.6 given in +Definition 4.2. +□ +Example 4.4. Let (X, ∗1, R1, R2) and (X, ∗2, R1, R2) be two singquandles such that such that +x ∗1 y = x = x ∗2 y and R1(x, y) = R2(x, y), then (X, ∗1, ∗2, R1, R2) forms a disingquandle. +Example 4.5. Let (X, ∗1, R1, R2) and (X, ∗2, R1, R2) be two singquandles. If for all x, y ∈ X +we have x ∗1 y = x ∗2 y then (X, ∗1, ∗2, R1, R2) forms a disingquandle. +Now Example 4.5 combined with Proposition 3.6 in [5] gives the following example. +Example 4.6. Let Λ = Z[t, B]/(t2 − 1, B(1 + t), t − (1 − B)2) and X be an Λ-module. Define +x ∗1 y = tx + (1 − t)y, R1(x, y) = (1 − t − b)x + (t + b)y and R2(x, y) = (1 − B)x + By, then +by setting ∗2 = ∗1, then one obtains that (X, ∗1, ∗2, R1, R2) forms a disingquandle. +Example 4.7. Let X be a module over Λ = Z[t]. Define x∗1y = x∗2y = tx+(1−t)y, R1(x, y) = +(1 − t − B)x + (t + B)y and R2(x, y) = (1 − B)x + By. Setting t = −1 and X = Z7, then the +quintuple (X, ∗1, ∗2, R1, R2) forms a disingquandle if B = 4 or if B = 5. +This example can be generalized to Zp, where p is a prime as follows. +Example 4.8. Let p be an odd prime and let B ∈ Zp. Consider Zp with x ∗1 y = x ∗2 y = +−x + 2y, R1(x, y) = (2 − B)x + (−1 + B)y and R2(x, y) = (1 − B)x + By. Let ζ be a +primitive root of unity in Zp so that ζ +p−1 +2 += −1. By choosing 1 − B = ζ +p−1 +2 +we obtain that +(Zp, ∗1, ∗2, R1, R2) forms a disingquandle. +Example 4.9. Let X = G be a multiplicative group with the involutive quandle operations +x ∗1 y = x ∗2 y = yx−1y (core quandle on G), then a direct computation gives the fact that +the quintuple (X, ∗1, ∗2, R1, R2) forms a disingquandle if and only if R1 and R2 satisfies the +following equations: +(5.1) +R2(x, z)z−1yz−1R2(x, z) = R1(x, z)x−1yx−1R1(x, z), +(5.2) +R2(x, y) = R2(xy−1x, x)[R1(xy−1x, x)]−1R2(xy−1x, x), +8 + +A straightforward computation gives the following solution +R1(x, y) = x and R2(x, y) = y, for all x, y, z ∈ G. +Now assume that G is an abelian group without 2-torsion, so that x ∗ y = −x + 2y, then +(X, ∗1, ∗2, R1, R2) forms a disingquandle if and only if R2(x, y) = R1(x, y) + y − x, where R1 +satisfies the identity R1(x, y) = R1(−x + 2y, x) + y − x. For example for any integer m, the map +R1(x, y) = mx + (2m + 1)y give a solution. Thus we have a family of solutions parametrized by +the integer m: +R1(x, y) = mx + (2m + 1)y, R2(x, y) = (m − 1)x + 2(m + 1)y. +Definition 4.10. A map f : X → Y is called a homomorphism of disingquandle (X, ∗1, ∗2, R1, R2) +and (Y, ∗′ +1, ∗′ +2, R′ +1, R′ +2) if the following conditions are satisfied for all x, y, z ∈ X +(i) f(x ∗1 y) = f(x) ∗′ +1 f(y), +(ii) f(x ∗2 y) = f(x) ∗′ +2 f(y), +(iii) f(R1(x, y)) = R′ +1(f(x), f(y)), +(iv) f(R2(x, y)) = R′ +1(f(x), f(y)). +If a homomorphism of disingquandle is bijective, then it is called an isomorphism of disingquan- +dle. We say that two Z2-families of singquandles are isomorphic if there exists an ismorphism of +disingquandle between them. +Definition 4.11. Let (X, ∗1, ∗2, R1, R2) be a disingquandle. A subset Y ⊂ X is called a sub- +disingquandle if (Y, ∗1, ∗2, R1, R2) is itself a disingquandle. +Example 4.12. We use Example 4.9 to get the following 2 examples: +• Let X = Z9 be the dihedral quandle with x ∗ y = −x + 2y, R1(x, y) = mx + (2m + +1)y, R2(x, y) = (m−1)x+ 2(m+ 1)y.. Then (Y, ∗1, ∗2, R1, R2) is itself a disingquandle +with Y = Z3. +• Let X = Z25 be the dihedral quandle with x ∗ y = −x + 2y, R1(x, y) = mx + (2m + +1)y, R2(x, y) = (m−1)x+ 2(m+ 1)y.. Then (Y, ∗1, ∗2, R1, R2) is itself a disingquandle +with Y = Z5. +Given a homomorphism of disingquandles, we obtain the following lemma. +Lemma 4.13. The image Im(f) of any homomorphism of disingquandle f defined from (X, ∗1, ∗2, R1, R2) +to (Y, ∗′ +1, ∗′ +2, R′ +1, R′ +2) is always a sub-disingquandle. +Proof. Given that f : (X, ∗1, ∗2, R1, R2) → (Y, ∗′ +1, ∗′ +2, R′ +1, R′ +2) is a homomorphism. Then the +equations (i), (ii), (iii) and (iv) of Definition 4.10 imply that Im(f) is closed under ∗1, ∗2, R1 and +R2. Then the axioms of disingquandle are satisfied in Y . Hence they are automatically satisfied in +Im(f). This ends the proof of the lemma. +□ +Now we introduce the notion of fundamental disingquandle of an unoriented dichromatic sin- +gular link and provide an illustrative example. Let D be a diagram of an unoriented dichromatic +singular link L in R2. We define the fundamental disingquandle of D, denoted by DSQ(D), as +the set of equivalence classes of disingquandle words W-DSQ(D) under the equivalence relation +generated by the axioms of disingquandle and the crossing relations shown in Figure 7, where W- +DSQ(D) are defined by taking a set of generators X = {x1, x2, x3, ....., xn} which corresponds +bijectively with the semi arcs in D, recursively by the following two rules: +9 + +(1) X ⊂ W-DSQ(D), +(2) If x, y ∈ W-DSQ(D), then +x ∗1 y, x ∗2 y, R1(x, y), R2(x, y) ∈ W-DSQ(D). +Example 4.14. Consider the following unoriented dichromatic singular link L. +1 +2 +x +z +u +v +y +FIGURE 8. Fundamental Disingquandle of Unoriented Dichromatic Singular Links +The fundamental disingquandle of L is given by +DSQ(L) = ⟨x, y, z, u, v| z = x ∗2 y; u = y ∗1 z; v = z ∗2 u; x = R1(u, v); y = R2(u, v)⟩. +This presentation can be simplified to the following presentation of DSQ(L) +⟨x, y| x = R1(y∗1(x∗2y), (x∗2y)∗2(y∗1(x∗2y))); y = R2(y∗1(x∗2y), (x∗2y)∗2(y∗1(x∗2y)))⟩. +5. COMPUTABLE INVARIANTS FOR UNORIENTED DICHROMATIC SINGULAR LINKS +Let D be an unoriented dichromatic singular link diagram and let A(D) denote the set of all +arcs of D. Let (X, ∗1, ∗2, R1, R2) be a disingquandle. A disingquandle coloring of D by X, or +simply disingquandle X-coloring of D, is a map C : A(D) → X such that at every classical and +singular crossing, the relations depicted in Figure 7 hold. The disingquandle element C(s) is called +a color of the arc s and the pair (D, C) is called the X-colored unoriented dichromatic singular +link diagram by C. The set of all disingquandle X-colorings of D is denoted by Coldsq +X (D). Then +we have the following: +Lemma 5.1. Let D and D′ be two unoriented dichromatic singular link diagramss in R2 that +can be transformed into each other by unoriented generalized dichromatic singular Reidemeis- +ter moves as shown in the Figure 6. Then for any finite disingquandle X, there is a one-to-one +correspondence between Coldsq +X (D) and Coldsq +X (D′). +10 + +Proof. It suffices to prove the assertion for the case that D′ is obtained from D by a single an +unoriented generalized dichromatic singular Reidemeister move. Let E be an open disk in R2 +where the unoriented generalized dichromatic singular Reidemeister move under consideration is +applied. Then D∩(R2−E) = D′∩(R2−E). Now let C be a disingquandle X-coloring of D. Since +(X, ∗1, R1, R2) and (X, ∗2, R1, R2) are both singquandles by the disingquandle definition 4.2, it +is obviously seen from the Figure 6 that the restriction of C to D ∩(R2 −E)(= D′ ∩(R2 −E)) can +be extended to a unique disingquandle X-coloring of D′ for unoriented generalized dichromatic +singular Reidemeister moves RI, RII and RIII. Also, using the disingquandle axioms 4.2.1 to +4.2.6, it is easily seen from the Figure 6 that the restriction of C to D ∩ (R2 − E)(= D′ ∩ (R2 − +E)) can be extended to a unique disingquandle X-coloring of D′ for an unoriented generalized +dichromatic Reidemeister moves RIV a, RIV b and RV . This completes the proof. +□ +In an X-colored unoriented dichromatic singular link diagram (D, C), we think of elements +of a disingquandle X as labels for the arcs in D with different operations at crossings as shown +in Figure 7. Then it is seen from Lemma 5.1 that the disingquandle axioms of Definition 4.2 +are transcriptions of a generating set of unoriented generalized Reidemeister moves for unoriented +dichromatic singular links which are sufficient to generate any other unoriented generalized dichro- +matic singular Reidemeister moves. That is, the axioms 4.2.1 and 4.2.2 come from the unoriented +generalized dichromatic singular Reidemeister move RIV a, the axioms 2.2.3 and 2.2.4 come from +the unoriented generalized dichromatic singular Reidemeister move RIV b and the axioms 2.2.5 +and 2.2.6 come from the unoriented generalized dichromatic singular Reidemeister move RV as +seen in Figure 6. +Theorem 5.2. Let L be an unoriented dichromatic singular link in R3 and let D be a diagram of +L. Then for any finite disingquandle X, the cardinality ♯Coldsq +X (L) is an invariant of L. +Proof. Let D′ be any other unoriented dichromatic singular link diagram of L obtained from D +by applying a finite number of unoriented generalized dichromatic singular Reidemeister moves. +Then it is direct from Lemma 5.1 that ♯Coldsq +X (D′) = ♯Coldsq +X (D). This completes the proof. +□ +If X is a finite disingquandle, we call the cardinality ♯Coldsq +X (D) the disingquandle X-coloring +number or the disingquandle counting invariant of L, and denote it by Zdsq +X (L), i.e., Zdsq +X (L) = +♯Coldsq +X (D). +Theorem 5.3. Let L be an unoriented dichromatic singular link and let X be a disingquandle. +Then there is a one-to-one correspondence between Coldsq +X (L) and Hom(DSQ(L), X). Conse- +quently, Zdsq +X (L) = ♯Hom(DSQ(L), X). +Proof. Since the disingquandle X-colorings of L generate the fundamental disingquandle DSQ(L) +of a link L which is generated by its arc labels. Also each arc of L is assigned an element of X, +for a disingquandle X-coloring of L, so we can associate each coloring a map f : DSQ(L) → X +where if an arc is labelled a in the fundamental disingquandle and is assigned the color x ∈ X, +then f(a) = x. This completes the proof. +□ +Now we give an example. +Example 5.4. Now, we give an explicit example of three unoriented dichromatic singular links L1, +L2 and L3 and show that the coloring invariant distinguishes them from each other. Consider the +singquandle (X, ∗, R1, R2), where X = Z6, x∗1 y = x∗2 y = −x+2y = x∗y, R1(x, y) = x+3, +11 + +and R2(x, y) = 3x2 + 3x + y + 3 (see page 9 of [7]). By checking directly that the equations of +Definition 4.2 hold we obtain that the quintuple (X, ∗1, ∗2, R1, R2) form a disingquandle. Now +coloring the two top arcs of link L1 by x and y as in the figure 9 below gives that the coloring +equations are: +x = R1(R1(x, y), R2(x, y)) +and +y = R2(R1(x, y), R2(x, y)). +One then gets the system, +� +x = 3 + 3 + x, +y = 3 + 3(3 + x) + 3(3 + x)2 + (3 + 3x + 3x2). +R ( ) +x +1 +y, +2 +R ( ) +x y, +x +y +1 +2 +FIGURE 9. Unoriented Dichromatic Singular Link(L1) +Any pair (x, y) gives a solution to this system over Z6 and thus the set Coldsq +X (L1) is equal to Z2 +6. +Now coloring the link L2 as in the figure 10 below gives that the coloring equations are: +R1(R1(x, y), x ∗ R1(x, y)) = R2(x, y) ∗ y +and +R2(R1(x, y), x ∗ R1(x, y)) = y. +One then obtain that the solution is given by y = 3x2 + 4x + 3, thus the Coldsq +X (L2) is +R ( ) +x +1 +y, +R ( ) +x +1 +y, +2 +R ( ) +x y, +x +x +y +* +1 +2 +FIGURE 10. Unoriented Dichromatic Singular Link(L2) +12 + +{(0, 3), (1, 4), (2, 5), (3, 0), (4, 1), (5, 2)}. +Now we consider the link L3 (dichromatic singular Whitehead) as in the following figure 11. +R ( ) +x +1 +y, +R ( ) +x +1 +y, +2 +R ( ) +x y, +x +y +y +y +* +2 +R ( +u x u) +, +* +x u +* +u := +1 +R ( +u x u) +, +* +2 +R ( ) +x y, +* +y +1 +2 +FIGURE 11. Unoriented Dichromatic Singular Link(L3) +The coloring equations are: +R2(y ∗ R1(x, y), x ∗ (y ∗ R1(x, y))) = R1(x, y), +and +R1(y ∗ R1(x, y), x ∗ (y ∗ R1(x, y))) = R2(x, y) ∗ y. +The system of these two equations reduces to +� +0 = 3y2 + y + 2x, +0 = 3x2 + x + 2y, +and thus we obtain that 2(y − x) = 0 giving x = y or y = x + 3. +Then Coldsq +X (L3) = {(x, x), x ∈ X} ∪ {(x, x + 3), x ∈ X}. +Thus the three links L1, L2 and L3 are pairwise distinct. +Example 5.5. Let 12 +1, 32 +1, 42 +1, 52 +1, 52 +2, 52 +3, 62 +1, 62 +2, 62 +3, 62 +4, 62 +5, 62 +6, 62 +7, 62 +8, 62 +9, 62 +10, 62 +11, and 62 +12 be the eigh- +teen unoriented dichromatic singular links in Figure 12 and let X be the disingquandle in Example +5.4. By similar calculations as in the example, we obtain the following table: +L +#Coldsq +X (L) +62 +2 +0 +62 +6 +2 +42 +1, 62 +12 +18 +12 +1, 32 +1, 52 +1, 52 +2, 52 +3, 62 +1, 62 +3, 62 +4, 62 +5, 62 +7, 62 +8, 62 +9, 62 +10, 62 +11 +6 +This table shows that the disingquandle counting invariant Zdsq +X (L) distinguishes some of these +eighteen unoriented dichromatic singular links. +13 + +11 +31 +41 +51 +1 +2 +1 +2 +1 +2 +1 +2 +1 +2 +1 +2 +1 +2 +1 +2 +1 +2 +1 +2 +1 +2 +1 +2 +1 +2 +1 +2 +1 +2 +1 +2 +1 +2 +1 +2 +2 +2 +2 +2 +52 +53 +61 +62 +2 +2 +2 +2 +63 +64 +65 +66 +2 +2 +2 +2 +67 +68 +69 +610 +2 +2 +611 +612 +2 +2 +2 +2 +FIGURE 12. Table of Unoriented Dichromatic Singular Links +ACKNOWLEDGEMENT +Mohamed Elhamdadi was partially supported by Simons Foundation collaboration grant 712462. +14 + +REFERENCES +[1] K. Bataineh, On skein theory of dichromatic links and invariants of finite type, Journal of Knot Theory and Its +Ramifications. 26(13) (2017) 1750092. +[2] K. Bataineh and I. Saidi, Involutory quandles and dichromatic links, Symmetry. 12(1) (2020) 111. +[3] Madeline Brown and Sam Nelson, G-family polynomials, J. Knot Theory Ramifications 30 (2021), no. 9, Paper +No. 2150070, 15, doi: 10.1142/S021821652150070X. MR4358333 +[4] Jose Ceniceros, Indu R. Churchill, and Mohamed Elhamdadi, Singquandle shadows and singular knot invariants, +Canad. Math. Bull. 65 (2022), no. 3, 770–787, doi: 10.4153/S0008439521000837. MR4472501 +[5] Indu R. U. Churchill, Mohamed Elhamdadi, Mustafa Hajij, and Sam Nelson, Singular knots and involutive +quandles, J. 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MR3127819 +DEPARTMENT OF MATHEMATICS, GRADUATE SCHOOL OF NATURAL SCIENCES PUSAN NATIONAL UNIVER- +SITY, BUSAN 46241, REPUBLIC OF KOREA +Email address: ibrahimsheikh@pusan.ac.kr +UNIVERSITY OF SOUTH FLORIDA, TAMPA, FLORIDA, USA +Email address: emohamed@usf.edu +DEPARTMENT OF MATHEMATICS, DALIAN UNIVERSITY OF TECHNOLOGY, CHINA +Email address: danishali@mail.dlut.edu.cn +15 + diff --git a/D9E2T4oBgHgl3EQfSgcI/content/tmp_files/load_file.txt b/D9E2T4oBgHgl3EQfSgcI/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6daabde5e5086353a7b4600a6096b8a0af7251b7 --- /dev/null +++ b/D9E2T4oBgHgl3EQfSgcI/content/tmp_files/load_file.txt @@ -0,0 +1,486 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf,len=485 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='03792v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='GT] 10 Jan 2023 A G-FAMILY OF SINGQUANDLES AND INVARIANTS OF DICHROMATIC SINGULAR LINKS MOHD IBRAHIM SHEIKH, MOHAMED ELHAMDADI, AND DANISH ALI ABSTRACT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' We introduce and investigate dichromatic singular links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' We also construct G-Family of singquandles and use them to define counting invariants for unoriented dichromatic singular links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' We provide some examples to show that these invariants distinguish some dichromatic singular links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' CONTENTS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Introduction 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Singular links, Singquandles and Dichromatic Links 2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Dichromatic Singular Links 5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' G-Family of Singquandles (Disingquandles) 6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Computable Invariants for Unoriented Dichromatic Singular Links 10 Acknowledgement 14 References 15 Mathematics Subject Classifications (2020): 57M25, 57M27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Key words and Phrases: Knot;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Link;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Singular knot;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Singular link;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Dichromatic link;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Dichromatic singular link;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Quandle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Singquandle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Disingquandle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Disingquandle counting invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' INTRODUCTION A knot is a simple closed curve in three dimensional space R3 and a disjoint union of two or more knots forms a link with two or more components [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Knots and links are categorised in many ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' One way is to use the crossing type as a tool to define a knot or link type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Classical, virtual and singular knots and links serve as examples as they are all recognised by the type of crossing they contain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' The other way to define link types is by labelling the components of a classical link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Dichromatic links are defined by using this technique as their components are either labelled by “1” or “2” [1, 2, 10, 11, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' A singular link is a link with at least one singular crossing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' In this paper we use such labelling technique for singular links and define a new type of links which we call dichromatic singular links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' A quandle is an algebraic structure satisfying some axioms that result from the Reidemeister moves for oriented classical knots and links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' If furthermore all right multiplications by fixed ele- ments of the quandle are involutions then such structures are called involutory quandles or Kei’s They are used to investigate unoriented knots and links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Quandles were independently introduced by Joyce and Matveev [13, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Since then they have been used to construct invariants of knots and links [4, 6, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Quandles have been also used to define new algebraic systems by taking a 1 family of quandles at a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Such systems are called G-Family of quandles and this notion was introduced in 2013 by Ishii, Iwakiri, Jang and Oshiro [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' A G-Family of quandles were used to define invariants for handlebody-knots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Also in [15] Lee and Sheikh used Z2-Family of quandles to construct algebraic invariants for oriented dichromatic links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' In this paper, we introduce the notions of G-Family of singquandles and dichromatic singular links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' A dichromatic singular link is an n component singular link with each of its component labelled as “1” or “2”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' A singquandle is an algebraic system whose axioms are motivated by Reidemeister moves of unoriented singular knots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' By taking a family of such algebaraic systems (Singquandles), we define a new algebraic system which we call G-Family of singquandles or disingquandle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' The axioms of the latter are motivated by generalized Reidemeister moves of un- oriented dichromatic singular links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' We discuss various examples and some properties of G-Family of singquandles, and also show that a G-Family of singquandles X enables us to distinguish unori- ented dichromatic singular links by computing their sets of all X-colorings and proving that these sets are different when their arcs are colored by the elements of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Section 2 reviews some preliminaries about singular links, singquandles as well as dichromatic links and their generalized Reidemeister moves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' In Section 3 we introduce the notion of dichromatic singular links with some typical examples of unoriented dichromatic singular links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Section 4 introduces the notion of G-Family of singquandles (dis- ingquandles) with some typical examples of G-Family of singquandles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Section 5 discusses how G-Family of singquandles is related to unoriented dichromatic singular links and develop com- putable invariants for unoriented dichromatic singular links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' We discuss some examples which show how the invariants distinguish unoriented dichromatic singular links, and especially how they detect the change of component labelings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' SINGULAR LINKS, SINGQUANDLES AND DICHROMATIC LINKS In this section we review some preliminaries about singular links, singquandles and dichromatic links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Most of the terminologies of this section can be found in [5, 9, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' We begin with the definition of a singular link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' A singular link in S3 is the image of a smooth immersion of n circles in S3 that has finitely many double points, called singular points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' A singular link in R3 is represented by a singular link diagram in the plane R2, which is a classical link diagram with one or more singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' A singularity is a rigid vertex where a link is glued to itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Figure 1 gives two examples of singular links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' FIGURE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Singular Links 2 Two singular links L� and L� are isotopy equivalent if one can be obtained from the other by a finite sequence generalized Reidemeister moves for singular links as shown in the following figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Let D� and D� be two singular link diagrams in R2 representing L� and L�, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Then L� and L� are equivalent if and only if D� and D� can be transformed into each other by a finite sequence of classical and singular Reidemeister moves shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' FIGURE 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Classical and Singular Reidemeister Moves Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' [5] Let (X, ∗) be an involutive quandle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Let R1 and R2 be two maps from X × X to X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' The quadruple (X, ∗, R1, R2) is called a singquandle if the following axioms are satisfied (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='1) x = R1(y, R2(x, y)) = R2(R2(x, y), R1(x, y)), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='2) y = R2(R2(x, y), x) = R1(R2(x, y), R1(x, y)), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='3) R(x, y) = (R1(y, R2(x, y)), R2(R2(x, y), x)), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='4) (y ∗ z) ∗ R2(x, z) = (y ∗ x) ∗ R1(x, z), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='5) R1(x, y) = R2(y ∗ x, x), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='6) R2(x, y) = R1(y ∗ x, x) ∗ R2(y ∗ x, x), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='7) R1(x ∗ y, z) ∗ y = R1(x, z ∗ y), 3 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='8) R2(x ∗ y, z) = R2(x, z ∗ y) ∗ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' We remind the reader that the singquandle axioms come from the generalized Reidemeister moves for unoriented singular knots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Singquandles were introduced as a ramification of quandles with the purpose of studying singular links, see for example [4,5,17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' The following are few typical examples of singquandles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' For an involutive quandle (X, ∗) with x ∗ y = 2y − x and X = Zn, the quadruple (X, ∗, R1, R2) forms a singquandle if and only if the following conditions are satisfied: (1) R2(x, y) = R1(x, y) + y − x, (2) R1(x, y) = R1(2x − y, x) + y − x, (3) R1(x, 2y − z) = 2y − R1(2y − x, z), (4) R2(2y − x, z) = 2y − R2(x, 2x − z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' For an involutive quandle (X, ∗) where X is a group G and x ∗ y = yx−1y, the quadruple (X, ∗, R1, R2) forms a singquandle if and only if the following conditions are satisfied: (1) R2(x, z)z−1yz−1R2(x, z) = R1(x, z)x−1yx−1R1(x, z), (2) R1(x, y) = R2(xy−1x, x), (3) R2(x, y) = R2(xy−1x, x)[R1(xy−1x, x)]−1R2(xy−1x, x), (4) y[R1(yx−1y, z)]−1y = R1(x, yz−1y), (5) R1(yx−1y, z) = y[R2(x, yz−1y)]−1y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' For a positive integer n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' A dichromatic link is a smooth imbedding of n circles in R3 such that each component is labeled as “1” or “2”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' In R2 every dichromatic link is represented by a dichromatic link diagram which is a classical link diagram with each component labelled either “1” or “2”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' For example, see Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' 1 1 2 2 FIGURE 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Dichromatic Links Two dichromatic links L� and L� are isotopy equivalent if one can be obtained from the other by a finite sequence of generalized Reidemeister moves for the dichromatic links as shown in the figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Let D� and D� be two dichromatic link diagrams in R2 representing L� and L�, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Then L� and L� are equivalent if and only D� and D� can be transformed into each other by a finite sequence of generalized Reidemeister moves shown in the following Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' 4 i i i j j i i k j i k j FIGURE 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Generalized Reidemeister Moves for Dichromatic Links 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' DICHROMATIC SINGULAR LINKS This section is devoted to dichromatic singular links which is a generalization of singular links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' To generate a dichromatic singular link we label a singular link’s components with “1” or “2”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Thus We have the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' A singular link L in R3 whose each component is colored (labelled) by either “1” or “2” is called a dichromatic singular link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' A dichromatic singular link L in R3 is represented by a dichromatic singular link diagram D in R2 in which each component is labelled “1” or “2”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Figure 5 shows two examples of unoriented dichromatic singular link diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' 1 2 1 2 FIGURE 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Dichromatic Singular Links Two dichromatic singular links L� and L� in R3 are ambient isotopic if there exists a self home- omorphism h : R3 → R3 that takes one link to the other and preserves the singularities as well as the labels “1”, “2” such that h(L�) = L�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Thus two singular dichromatic links L� and L� are equivalent if one can be obtained from the other by a finite sequence of generalized dichromatic singular Reidemeister moves preserving the label of each component as shown in the Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Let D� and D� be two dichromatic singular link diagrams in R2 representing L� and L�, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Then L� and L� are equivalent if and only if D� and D� can be transformed into each other by a finite sequence of generalized dichromatic singular Reidemeister moves shown in the following Figure 6 where i, j, k ∈ {1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' A dichromatic singular link with n components is called as an n-component dichromatic singular link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Thus an n-component dichromatic singular link in R3 can be defined as L = K1 ∪ · · · ∪ Kn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' 5 i k k j i i j j i j i k k j i j i i i j j i i k j i k j FIGURE 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Regular Dichromatic Reidemeister Moves RI, RII and RIII on the top and Dichromatic Singular Reidemeister Moves RIV a, RIV b and RV in the middle and on the bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Taking n = 2, we obtain 2-component dichromatic singular links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Some 2-component unoriented dichromatic singular link diagrams (see p 814 of [18]) are shown in Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Let L� and L� be two unoriented dichromatic singular links in R3 and let D� and D� be two unoriented dichromatic singular link diagrams in R2 representing L� and L�, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Then L� and L� are equivalent if and only if D� and D� are transformed into each other by a finite sequence of generalized Reidemeister moves for unoriented dichromatic singular links which preserve the singularities and the label of each component as shown in the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' 6 where i, j, k ∈ {1, 2} and ambient isotopies of R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' G-FAMILY OF SINGQUANDLES (DISINGQUANDLES) Before introducing the notion of G-Family of Singquandles, we first recall the definition of G-family of quandles from [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Given a group G and a set X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' a G-family of quandles,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' denoted by (G,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' X),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' is a choice of quandle operation ∗g on the set X for each element g ∈ G such that the following axioms are satisfied (1) For all g ∈ G and for all x ∈ X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' x ∗g x = x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' (2) For all g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' h ∈ G and for all x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' y ∈ X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' (x ∗g y) ∗h y = x ∗gh y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' (3) For all x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' y ∈ X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' x ∗e x = x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' where e is the identity element of G,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' 6 (4) For all x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' z ∈ X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' (x ∗g y) ∗h z = (x ∗h z) ∗h−1gh (y ∗h z) The following are two examples of G-families of quandles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' For any group G and any set X, defining x ∗g y = x for all x, y ∈ X and all g ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' This gives a G-family of quandles called the trivial G-family of quandles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Let (X, ∗) be a quandle of cyclic type [19] with cardinality n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Let Rx denotes the right multiplication by x, thus by definition Rx (n−1) is the identity map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Then define x ∗i y = Ry i(x) then it is shown in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='3 of [12] that (Z, X) is a Z-family of quandles and also Z(n−1)-family of quandles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' A G-family of quandles (G, X) induces a quandle operation on the set G × X by (g, x) ∗ (h, y) = (h−1gx, x ∗h y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' The notion of G-family of quandles was introduced by Ishii, Iwakiri, Jang and Oshiro in 2013 in [12] in order to produce invariants of handlebody knots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' They defined coloring invariants and cocycle invariants of handlebody knots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' They used these invariants to detect chirality of some han- dlebody knots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Later in 2015, Ishii independently studied the notion of G-family of quandles in connection with the multiple conjugation quandle and showed that the later one can be obtained from the first one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' In 2017 and 2018 Ishii, Nelson and Ishii, Iwakiri, Kamada, Kim, Matsuzaki, Os- hiro respectively, used this work and introduced the notions of partially multiplicative biquandles and multiple conjugation biquandle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' In 2021 Lee and Sheikh jointly used G-family of quandles to construct algebraic invariants for oriented dichromatic links [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' We introduce the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Let X be a set equipped with two binary operations ∗1 and ∗2 such that both (X, ∗1), (X, ∗2) are involutive quandles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Let R1, R2 be two maps from X × X to X such that the quadruples (X, ∗1, R1, R2) and (X, ∗2, R1, R2) are singquandles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Then the quintuple (X, ∗1, ∗2, R1, R2) is called a disingquandle or Z2-family of singquandles if the following axioms are satisfied (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='1) (x ∗1 y) ∗2 z = (x ∗2 z) ∗1 (y ∗2 z), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='2) (x ∗2 y) ∗1 z = (x ∗1 z) ∗2 (y ∗1 z), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='3) (y ∗1 z) ∗2 R2(x, z) = (y ∗2 x) ∗1 R1(x, z), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='4) (y ∗2 z) ∗1 R2(x, z) = (y ∗1 x) ∗2 R1(x, z), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='5) R2(x, y) = R1(y ∗1 x, x) ∗2 R2(y ∗1 x, x), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='6) R2(x, y) = R1(y ∗2 x, x) ∗1 R2(y ∗2 x, x), The above axioms of a disingquandle come from the generalized dichromatic singular Reide- meister moves shown in Figure 6 when we take the coloring rule shown in Figure 7 under consid- eration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' 7 i i j i/j j/i j i y x y x y x x x y x j x y* R ( ) x 1 2 y, R ( ) x y, FIGURE 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Coloring by a disingquandle The following lemma is motivated by the above construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' The set of colorings of a dichromatic singular link by a disingquandle does not change by the dichromatic singular Reidemeister moves shown in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' As in the case of classical and singular knot theories, there is one to one correspondence between colorings before and after each of the dichromatic singular Reidemeister moves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' The invariance follows directly from the equations 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='1, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='2, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='3, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='4, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='5 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='6 given in Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' □ Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Let (X, ∗1, R1, R2) and (X, ∗2, R1, R2) be two singquandles such that such that x ∗1 y = x = x ∗2 y and R1(x, y) = R2(x, y), then (X, ∗1, ∗2, R1, R2) forms a disingquandle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Let (X, ∗1, R1, R2) and (X, ∗2, R1, R2) be two singquandles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' If for all x, y ∈ X we have x ∗1 y = x ∗2 y then (X, ∗1, ∗2, R1, R2) forms a disingquandle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Now Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='5 combined with Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='6 in [5] gives the following example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Let Λ = Z[t, B]/(t2 − 1, B(1 + t), t − (1 − B)2) and X be an Λ-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Define x ∗1 y = tx + (1 − t)y, R1(x, y) = (1 − t − b)x + (t + b)y and R2(x, y) = (1 − B)x + By, then by setting ∗2 = ∗1, then one obtains that (X, ∗1, ∗2, R1, R2) forms a disingquandle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Let X be a module over Λ = Z[t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Define x∗1y = x∗2y = tx+(1−t)y, R1(x, y) = (1 − t − B)x + (t + B)y and R2(x, y) = (1 − B)x + By.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Setting t = −1 and X = Z7, then the quintuple (X, ∗1, ∗2, R1, R2) forms a disingquandle if B = 4 or if B = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' This example can be generalized to Zp, where p is a prime as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Let p be an odd prime and let B ∈ Zp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Consider Zp with x ∗1 y = x ∗2 y = −x + 2y, R1(x, y) = (2 − B)x + (−1 + B)y and R2(x, y) = (1 − B)x + By.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Let ζ be a primitive root of unity in Zp so that ζ p−1 2 = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' By choosing 1 − B = ζ p−1 2 we obtain that (Zp, ∗1, ∗2, R1, R2) forms a disingquandle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Let X = G be a multiplicative group with the involutive quandle operations x ∗1 y = x ∗2 y = yx−1y (core quandle on G), then a direct computation gives the fact that the quintuple (X, ∗1, ∗2, R1, R2) forms a disingquandle if and only if R1 and R2 satisfies the following equations: (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='1) R2(x, z)z−1yz−1R2(x, z) = R1(x, z)x−1yx−1R1(x, z), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='2) R2(x, y) = R2(xy−1x, x)[R1(xy−1x, x)]−1R2(xy−1x, x), 8 A straightforward computation gives the following solution R1(x, y) = x and R2(x, y) = y, for all x, y, z ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Now assume that G is an abelian group without 2-torsion, so that x ∗ y = −x + 2y, then (X, ∗1, ∗2, R1, R2) forms a disingquandle if and only if R2(x, y) = R1(x, y) + y − x, where R1 satisfies the identity R1(x, y) = R1(−x + 2y, x) + y − x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' For example for any integer m, the map R1(x, y) = mx + (2m + 1)y give a solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Thus we have a family of solutions parametrized by the integer m: R1(x, y) = mx + (2m + 1)y, R2(x, y) = (m − 1)x + 2(m + 1)y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' A map f : X → Y is called a homomorphism of disingquandle (X, ∗1, ∗2, R1, R2) and (Y, ∗′ 1, ∗′ 2, R′ 1, R′ 2) if the following conditions are satisfied for all x, y, z ∈ X (i) f(x ∗1 y) = f(x) ∗′ 1 f(y), (ii) f(x ∗2 y) = f(x) ∗′ 2 f(y), (iii) f(R1(x, y)) = R′ 1(f(x), f(y)), (iv) f(R2(x, y)) = R′ 1(f(x), f(y)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' If a homomorphism of disingquandle is bijective, then it is called an isomorphism of disingquan- dle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' We say that two Z2-families of singquandles are isomorphic if there exists an ismorphism of disingquandle between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Let (X, ∗1, ∗2, R1, R2) be a disingquandle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' A subset Y ⊂ X is called a sub- disingquandle if (Y, ∗1, ∗2, R1, R2) is itself a disingquandle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' We use Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='9 to get the following 2 examples: Let X = Z9 be the dihedral quandle with x ∗ y = −x + 2y, R1(x, y) = mx + (2m + 1)y, R2(x, y) = (m−1)x+ 2(m+ 1)y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='. Then (Y, ∗1, ∗2, R1, R2) is itself a disingquandle with Y = Z3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Let X = Z25 be the dihedral quandle with x ∗ y = −x + 2y, R1(x, y) = mx + (2m + 1)y, R2(x, y) = (m−1)x+ 2(m+ 1)y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='. Then (Y, ∗1, ∗2, R1, R2) is itself a disingquandle with Y = Z5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Given a homomorphism of disingquandles, we obtain the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' The image Im(f) of any homomorphism of disingquandle f defined from (X, ∗1, ∗2, R1, R2) to (Y, ∗′ 1, ∗′ 2, R′ 1, R′ 2) is always a sub-disingquandle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Given that f : (X, ∗1, ∗2, R1, R2) → (Y, ∗′ 1, ∗′ 2, R′ 1, R′ 2) is a homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Then the equations (i), (ii), (iii) and (iv) of Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='10 imply that Im(f) is closed under ∗1, ∗2, R1 and R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Then the axioms of disingquandle are satisfied in Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Hence they are automatically satisfied in Im(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' This ends the proof of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' □ Now we introduce the notion of fundamental disingquandle of an unoriented dichromatic sin- gular link and provide an illustrative example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Let D be a diagram of an unoriented dichromatic singular link L in R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' We define the fundamental disingquandle of D, denoted by DSQ(D), as the set of equivalence classes of disingquandle words W-DSQ(D) under the equivalence relation generated by the axioms of disingquandle and the crossing relations shown in Figure 7, where W- DSQ(D) are defined by taking a set of generators X = {x1, x2, x3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=', xn} which corresponds bijectively with the semi arcs in D, recursively by the following two rules: 9 (1) X ⊂ W-DSQ(D), (2) If x, y ∈ W-DSQ(D), then x ∗1 y, x ∗2 y, R1(x, y), R2(x, y) ∈ W-DSQ(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Consider the following unoriented dichromatic singular link L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' 1 2 x z u v y FIGURE 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Fundamental Disingquandle of Unoriented Dichromatic Singular Links The fundamental disingquandle of L is given by DSQ(L) = ⟨x, y, z, u, v| z = x ∗2 y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' u = y ∗1 z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' v = z ∗2 u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' x = R1(u, v);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' y = R2(u, v)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' This presentation can be simplified to the following presentation of DSQ(L) ⟨x, y| x = R1(y∗1(x∗2y), (x∗2y)∗2(y∗1(x∗2y)));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' y = R2(y∗1(x∗2y), (x∗2y)∗2(y∗1(x∗2y)))⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' COMPUTABLE INVARIANTS FOR UNORIENTED DICHROMATIC SINGULAR LINKS Let D be an unoriented dichromatic singular link diagram and let A(D) denote the set of all arcs of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Let (X, ∗1, ∗2, R1, R2) be a disingquandle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' A disingquandle coloring of D by X, or simply disingquandle X-coloring of D, is a map C : A(D) → X such that at every classical and singular crossing, the relations depicted in Figure 7 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' The disingquandle element C(s) is called a color of the arc s and the pair (D, C) is called the X-colored unoriented dichromatic singular link diagram by C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' The set of all disingquandle X-colorings of D is denoted by Coldsq X (D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Then we have the following: Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Let D and D′ be two unoriented dichromatic singular link diagramss in R2 that can be transformed into each other by unoriented generalized dichromatic singular Reidemeis- ter moves as shown in the Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Then for any finite disingquandle X, there is a one-to-one correspondence between Coldsq X (D) and Coldsq X (D′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' 10 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' It suffices to prove the assertion for the case that D′ is obtained from D by a single an unoriented generalized dichromatic singular Reidemeister move.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Let E be an open disk in R2 where the unoriented generalized dichromatic singular Reidemeister move under consideration is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Then D∩(R2−E) = D′∩(R2−E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Now let C be a disingquandle X-coloring of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Since (X, ∗1, R1, R2) and (X, ∗2, R1, R2) are both singquandles by the disingquandle definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='2, it is obviously seen from the Figure 6 that the restriction of C to D ∩(R2 −E)(= D′ ∩(R2 −E)) can be extended to a unique disingquandle X-coloring of D′ for unoriented generalized dichromatic singular Reidemeister moves RI, RII and RIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Also, using the disingquandle axioms 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='1 to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='6, it is easily seen from the Figure 6 that the restriction of C to D ∩ (R2 − E)(= D′ ∩ (R2 − E)) can be extended to a unique disingquandle X-coloring of D′ for an unoriented generalized dichromatic Reidemeister moves RIV a, RIV b and RV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' □ In an X-colored unoriented dichromatic singular link diagram (D, C), we think of elements of a disingquandle X as labels for the arcs in D with different operations at crossings as shown in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Then it is seen from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='1 that the disingquandle axioms of Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='2 are transcriptions of a generating set of unoriented generalized Reidemeister moves for unoriented dichromatic singular links which are sufficient to generate any other unoriented generalized dichro- matic singular Reidemeister moves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' That is, the axioms 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='2 come from the unoriented generalized dichromatic singular Reidemeister move RIV a, the axioms 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='3 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='4 come from the unoriented generalized dichromatic singular Reidemeister move RIV b and the axioms 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='5 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='6 come from the unoriented generalized dichromatic singular Reidemeister move RV as seen in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Let L be an unoriented dichromatic singular link in R3 and let D be a diagram of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Then for any finite disingquandle X, the cardinality ♯Coldsq X (L) is an invariant of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Let D′ be any other unoriented dichromatic singular link diagram of L obtained from D by applying a finite number of unoriented generalized dichromatic singular Reidemeister moves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Then it is direct from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='1 that ♯Coldsq X (D′) = ♯Coldsq X (D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' □ If X is a finite disingquandle, we call the cardinality ♯Coldsq X (D) the disingquandle X-coloring number or the disingquandle counting invariant of L, and denote it by Zdsq X (L), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=', Zdsq X (L) = ♯Coldsq X (D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Let L be an unoriented dichromatic singular link and let X be a disingquandle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Then there is a one-to-one correspondence between Coldsq X (L) and Hom(DSQ(L), X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Conse- quently, Zdsq X (L) = ♯Hom(DSQ(L), X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Since the disingquandle X-colorings of L generate the fundamental disingquandle DSQ(L) of a link L which is generated by its arc labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Also each arc of L is assigned an element of X, for a disingquandle X-coloring of L, so we can associate each coloring a map f : DSQ(L) → X where if an arc is labelled a in the fundamental disingquandle and is assigned the color x ∈ X, then f(a) = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' □ Now we give an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Now, we give an explicit example of three unoriented dichromatic singular links L1, L2 and L3 and show that the coloring invariant distinguishes them from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Consider the singquandle (X, ∗, R1, R2), where X = Z6, x∗1 y = x∗2 y = −x+2y = x∗y, R1(x, y) = x+3, 11 and R2(x, y) = 3x2 + 3x + y + 3 (see page 9 of [7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' By checking directly that the equations of Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='2 hold we obtain that the quintuple (X, ∗1, ∗2, R1, R2) form a disingquandle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Now coloring the two top arcs of link L1 by x and y as in the figure 9 below gives that the coloring equations are: x = R1(R1(x, y), R2(x, y)) and y = R2(R1(x, y), R2(x, y)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' One then gets the system, � x = 3 + 3 + x, y = 3 + 3(3 + x) + 3(3 + x)2 + (3 + 3x + 3x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' R ( ) x 1 y, 2 R ( ) x y, x y 1 2 FIGURE 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Unoriented Dichromatic Singular Link(L1) Any pair (x, y) gives a solution to this system over Z6 and thus the set Coldsq X (L1) is equal to Z2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Now coloring the link L2 as in the figure 10 below gives that the coloring equations are: R1(R1(x, y), x ∗ R1(x, y)) = R2(x, y) ∗ y and R2(R1(x, y), x ∗ R1(x, y)) = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' One then obtain that the solution is given by y = 3x2 + 4x + 3, thus the Coldsq X (L2) is R ( ) x 1 y, R ( ) x 1 y, 2 R ( ) x y, x x y 1 2 FIGURE 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Unoriented Dichromatic Singular Link(L2) 12 {(0, 3), (1, 4), (2, 5), (3, 0), (4, 1), (5, 2)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Now we consider the link L3 (dichromatic singular Whitehead) as in the following figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' R ( ) x 1 y, R ( ) x 1 y, 2 R ( ) x y, x y y y 2 R ( u x u) , x u u := 1 R ( u x u) , 2 R ( ) x y, y 1 2 FIGURE 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Unoriented Dichromatic Singular Link(L3) The coloring equations are: R2(y ∗ R1(x, y), x ∗ (y ∗ R1(x, y))) = R1(x, y), and R1(y ∗ R1(x, y), x ∗ (y ∗ R1(x, y))) = R2(x, y) ∗ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' The system of these two equations reduces to � 0 = 3y2 + y + 2x, 0 = 3x2 + x + 2y, and thus we obtain that 2(y − x) = 0 giving x = y or y = x + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Then Coldsq X (L3) = {(x, x), x ∈ X} ∪ {(x, x + 3), x ∈ X}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Thus the three links L1, L2 and L3 are pairwise distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Let 12 1, 32 1, 42 1, 52 1, 52 2, 52 3, 62 1, 62 2, 62 3, 62 4, 62 5, 62 6, 62 7, 62 8, 62 9, 62 10, 62 11, and 62 12 be the eigh- teen unoriented dichromatic singular links in Figure 12 and let X be the disingquandle in Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' By similar calculations as in the example, we obtain the following table: L #Coldsq X (L) 62 2 0 62 6 2 42 1, 62 12 18 12 1, 32 1, 52 1, 52 2, 52 3, 62 1, 62 3, 62 4, 62 5, 62 7, 62 8, 62 9, 62 10, 62 11 6 This table shows that the disingquandle counting invariant Zdsq X (L) distinguishes some of these eighteen unoriented dichromatic singular links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' 13 11 31 41 51 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 2 2 2 2 52 53 61 62 2 2 2 2 63 64 65 66 2 2 2 2 67 68 69 610 2 2 611 612 2 2 2 2 FIGURE 12.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='topol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='049.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' MR3431017 [19] Hiroshi Tamaru, Two-point homogeneous quandles with prime cardinality, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' Japan 65 (2013), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' 4, 1117–1134, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='2969/jmsj/06541117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content=' MR3127819 DEPARTMENT OF MATHEMATICS, GRADUATE SCHOOL OF NATURAL SCIENCES PUSAN NATIONAL UNIVER- SITY, BUSAN 46241, REPUBLIC OF KOREA Email address: ibrahimsheikh@pusan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='kr UNIVERSITY OF SOUTH FLORIDA, TAMPA, FLORIDA, USA Email address: emohamed@usf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='edu DEPARTMENT OF MATHEMATICS, DALIAN UNIVERSITY OF TECHNOLOGY, CHINA Email address: danishali@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='dlut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} +page_content='cn 15' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'} diff --git a/DNFRT4oBgHgl3EQfxDge/content/tmp_files/2301.13640v1.pdf.txt b/DNFRT4oBgHgl3EQfxDge/content/tmp_files/2301.13640v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..cadc46241365d931e41586a14b0db56768fba7fc --- /dev/null +++ b/DNFRT4oBgHgl3EQfxDge/content/tmp_files/2301.13640v1.pdf.txt @@ -0,0 +1,894 @@ +Vacuum enhanced charging of a quantum battery +Tiago F. F. Santos,1 Yohan Vianna de Almeida,1 and Marcelo F. Santos1, ∗ +1Instituto de F´ısica, Universidade Federal do Rio de Janeiro, +CP68528, Rio de Janeiro, Rio de Janeiro 21941-972, Brazil +(Dated: February 1, 2023) +Quantum batteries are quantum systems that store energy which can then be used for quantum +tasks. One relevant question about such systems concerns the differences and eventual advantages +over their classical counterparts, whether in the efficiency of the energy transference, input power, +total stored energy or other relevant physical quantities. Here, we show how a purely quantum +effect related to the vacuum of the electromagnetic field can enhance the charging of a quantum +battery. In particular, we demonstrate how an anti-Jaynes Cummings interaction derived from an +off-resonant Raman configuration can be used to increase the stored energy of an effective two-level +atom when compared to its classically driven counterpart, eventually achieving full charging of the +battery with zero entropic cost. +The quest for advanced quantum technologies or the ir- +reversible role of measurements in quantum dynamics are +examples of subjects that have stimulated the study of +thermodynamics in the microscopic world. An important +recent topic of investigation involves the role played by +quantum resources in the storage and use of energy by +quantized systems [1–19]. For example, coherence and +entanglement have been proven useful to speed up or +to super-extend the charging of quantum batteries [20– +27]. Experimental results have also shown advances to- +wards the production of microscopic quantum thermal +machines and quantum batteries [28–33]. Most results re- +garding quantum properties influencing the performance +of quantum batteries, however, focus on increasing the +power of the process rather than enhancing the charg- +ing capacity. That is because the latter usually requires +entropy producing mechanisms [7–12, 18] that have dele- +terious effects in properties such as coherence and entan- +glement. +In this work we investigate how the quantized nature +of part of an entropy preserving charging circuit can in- +fluence the charging of a quantum battery. The circuit +comprises a classical power source (p.s.) and an auxiliary +frequency changer (f.c.). We compare the variation of the +internal energy stored in the battery and the efficiency of +the work extraction from the p.s., both for a classical and +quantum version of the f.c. component. In both cases, +the overall dynamics is unitary and, therefore, comes at +zero entropic cost. +In the classical scenario, both p.s. +and f.c. are connected to the battery for a fixed amount +of time, τc (“c” for classical), unitarily charging its ini- +tially thermal state: ρB(τc) = Uc(τc)ρT +BU −1 +c +(τc), where +Uc(τc) is derived from the coupling Hamiltonian Hc = +HB0 + Vp.s.(t) + Vf.c.(t), Vj(t) is the potential created +by the circuit component j and HB0 is the free Hamilto- +nian of the battery. Thermal states are free resources in +thermodynamics [34–36] and, therefore, ideal to establish +the classical benchmark to be challenged by the quantum +version. The charging is measured by the variation ∆U +of internal energy of the battery, where U = Tr[ρBHB0]. +In the quantized version, Vf.c.(t) is replaced by the inter- +action Hamiltonian HB−f.c.(t) and the initial state must +include the f.c. system which is also in a thermal state: +ρ(0) = ρT +B ⊗ ρT +f.c.. The variation of energy of the bat- +tery is now given by ∆U = Tr{[ρB(τq) − ρT +B]HB0} ( “q” +for quantum) where ρB(τq) = Trf.c.Uq(τq)ρ(0)U −1(τq) +and Uq is the time evolution operator obtained from +Hq = HB0 + Hf.c.0 + HB−f.c(t) + Vp.s.(t). Note that, +in both cases we assume isolation from the environment +and the charging does not produce any entropy. For com- +pleteness, we later add dissipative non-unitary terms to +the dynamics to verify how our results are affected by +the heat exchanged with surrounding reservoirs. +We investigate the classical protocol in a particular +setup where the battery is an oscillating two-level sys- +tem of frequency ωeg, the p.s. +generates an oscillat- +ing potential of frequency ωL > ωeg and the f.c.. gen- +erates another potential of frequency ωq = ωL − ωeg. +This situation is commonly found in many different +quantum optical experiments [37–43], where the battery +consists of two non-degenerate ground states {|g⟩, |e⟩} +(ωeg ≡ ωe − ωg > 0) of a real or artificial atom and +two modes of the electromagnetic field play the role of +power supply and f.c.. +The couplings are intermedi- +ated by a third atomic level |m⟩ working as an ancilla +as depicted in Fig. +(1a). +Level |m⟩ should only con- +tribute virtually to the transference of energy and has +to be adiabatically eliminated from the dynamics. This +is achieved when each of p.s. +and f.c. +couples off- +resonantly one of the lower levels of the battery to |m⟩ in +a Raman configuration, where HB0 = ℏ � +j=g,e,m ωjσjj +(σjk ≡ |j⟩⟨k|), Vf.c.(t) = ℏΩq(σemeiωqt + σmee−iωqt) +and Vp.s.(t) = ℏΩL(σgmeiωLt + σmge−iωLt). +If ∆ = +ωmg − ωL = ωme − ωq ≫ ΩL, Ωq, the corresponding +time evolution Uc(t) induces Rabi oscillations between +levels |g⟩ and |e⟩ that are equivalent to directly cou- +pling them through one effective classical field of cou- +pling strength ¯Ω = +ΩLΩq +∆ +[44]. +The optimal charg- +ing of the battery is then obtained for a full Rabi flip +that swaps the populations pT +g,e in the original ther- +arXiv:2301.13640v1 [quant-ph] 31 Jan 2023 + +2 +mal state ρT +B = � +j=g,e pT +j σjj, where pT +j = +e +ℏωj +KBT +ZB +and +ZB = � +j e +ℏωj +KBT . +In this case, ∆Uc = ℏωeg[pT +g − pT +e ]. +Note that this is the most that a unitary transforma- +tion can charge an initially thermalized two-level bat- +tery and corresponds to the ergotropy Ec of the resulting +state, ρB(τc) = pT +e σgg + pT +g σee. +Ergotropy is defined +as Eρ(τ) = � +k,j rkEj(|⟨rk|Ej⟩|2 − δkj), where Ej are the +eigenenergies of H0 in increasing magnitude, i.e., Ei ≥ Ej +for i > j, and rk are the eigenvalues of ρ(τ) in decreasing +order, i.e., ri ≤ rj for i > j [45]. +|m〉 +|e〉 +|g〉 +Δ +𝛀L +gq, 𝛀q +𝚪mg +𝚪gm +𝚪em +𝚪me +|e〉 +|g〉 +N = 1 +|e, 0〉 |e, 1〉 +|g, 0〉 +N = 2 +N = k +|e, 2〉 +|e, k〉 +|g, 1〉 +|g, k-1〉 +N = 1 +N = 2 +N = k +(a) +(b) +FIG. 1. (a) off-resonant Raman configuration: the battery is +a two-level atom ({|g⟩, |e⟩}); the p.s. is a laser of frequency +ωL (coupling ΩL), the f.c. is another harmonic oscillator of +frequency ωq and couplings Ωq (classical) and gq (quantum). +Level |m⟩ is an ancilla that intermediates both couplings. +Each channel can also exchange heat with the surrounding +reservoirs.the battery. (b) Selective scheme to charge the bat- +tery: in each step N, a selective Rabi flip transfers energy +from |g, N − 1⟩ to |e, N⟩. +If, now, the classical f.c. is replaced by a quantized +field, we need to add its free energy Hf.c.0 = ℏωqˆb†ˆb +to the Hamiltonian, where ˆb† creates an excitation, +and replace Vf.c.(t) by the interaction term HB−f.c. = +ℏgq(σemˆb†+σmeˆb). Once again, for ∆ ≫ ΩL, gq, we elim- +inate level |m⟩ and, as shown in [46–48], the dynamics of +the Battery-f.c. system becomes approximately given by +the effective Hamiltonian (ℏ = 1) +Heff = −g2 +qN +∆ σgg − g2 +qˆb†ˆb +∆ +σee + ΩLgq +∆ +(σgeˆb + σegˆb†), +(1) +Note that Heff also includes a small correction to the +energy difference between levels |g⟩ and |e⟩, given by +ℏ∆N +eg = ℏ +Ω2 +L−g2 +qN +∆ +. This term, of the same order of Heff, +does not affect the conditions for eliminating |m⟩ and can +be physically implemented by applying a d.c. Stark shift +to the atom. +There are a few aspects of Heff useful for us: first, +the a.c. Stark shift correction to level |e⟩ depends on the +number of excitations of the f.c. and |e, 0⟩ is an eigen- +state of Heff with eigenvalue 0; second, the Rabi oscil- +lations occur in the joint Hilbert space of atom and f.c., +splitting it into doublets {|g, n⟩, |e, n + 1⟩}. This corre- +sponds to the anti-Jaynes-Cummings (anti-JC) configu- +ration where the p.s. excites both the battery and the +f.c. at the same time. Third, each doublet oscillates at its +own Rabi frequency given by Ωn = +� +∆2n/4 + G2n, where +∆n = +r2Ω2 +L(n+1−N) +∆ +, Gn = +rΩ2 +L +√n+1 +∆ +and r ≡ +gq +ΩL , i.e. +each doublet is detuned from resonance by an amount +∆n proportional to the number of excitations of the f.c.. +Such Hamiltonians were predicted and implemented +in trapped ions, cavity QED and superconducting cir- +cuits, and for r ≫ 1, they operate in a selective regime +where ∆n ≫ Gn and the Rabi oscillation in all the dou- +blets is highly detuned except if n = N − 1. +In this +case, {|g, N −1⟩, |e, N⟩} oscillates resonantly (∆N−1 = 0, +ΩN−1 = rΩ2 +L +√ +N +∆ +). Therefore, by properly choosing ∆N +eg +the battery population exchange is conditioned on the +number of excitations of the f.c. field as shown in [46, 47]. +For example, for N += 1, after an interaction time +τq = +π∆ +2rΩ2 +L , the population in the {|g, 0⟩, |e, 1⟩} subspace +swaps while all other states only gain number dependent +phases. That takes the initial state ρ(0) = ρT +B ⊗ ρT +f.c. to +ρ(τq) = pT +e pT +0 |e, 0⟩⟨e, 0| + pT +g pT +0 |e, 1⟩⟨e, 1| + pT +e pT +1 |g, 0⟩⟨g, 0| ++ pT +g pT +1 |g, 1⟩⟨g, 1| + ( +� +n>1 +pT +n|n⟩⟨n|) ⊗ ρT +B. +(2) +Here, ρT +f.c. = � +n pT +nσnn, pT +n = e− +nℏωq +KBT (1 − e− +ℏωq +KBT ). A +simple algebraic manipulation shows that this swap in- +creases the charge of the battery by ∆Uq = (pT +0 pT +g − +pT +e pT +1 )ℏωeg. In this case, there is an advantage over ∆Uc +if pT +e +pT +g > 1−pT +0 +1−pT +1 . We can better understand this condition +at low temperatures. When KBT ≪ ℏωq, ℏωm, the prob- +abilities pT +n are negligible for n > 1 and so is pT +m and we +can approximate 1 − pT +1 ≈ pT +0 and pT +e ≈ 1 − pT +g , mean- +ing that ∆Uq > ∆Uc if pT +e pT +0 +pT +g pT +1 ≈ e +ℏωeg(ξ−1) +KBT +> 1, where +ξ = +ωq +ωeg . This happens whenever ξ > 1, i.e. whenever +the battery’s gap is smaller than one excitation of field ˆb. +In principle, the larger the value of ξ, the more accentu- +ated the enhancement due to the vacuum of field ˆb. This +is a purely quantum effect due solely to the vacuum of +the f.c. component. +Note, however, that the quantum protocol allows for +the relaxation of the ξ > 1 condition and an even more +enhanced charging, which is a much more powerful result, +due to the selectivity of Heff. In fact, similar Rabi flips +can be sequentially applied, each one tuned to resonance +by adjusting ∆N +eg in consecutive subspaces (N = 2, 3, ...) +as pictorially shown in Fig. (1b). In principle, this se- +quence must be infinite to maximize the charging of the +battery but, in practice, pT +n tends rapidly to zero unless +T is very high, and only a few cycles are required to ap- +proach maximum charging. After the sequence, the final + +3 +state reads ρ(� +j τqj) ≈ [pT +e (1 − pT +0 )σgg + (pT +g + pT +e pT +0 )σee +and the variation of internal energy is ∆Uq = ∆Uc + +pT +e pT +0 ℏωeg ≥ ∆Uc. This shows an advantage for any pos- +itive temperature and independent of ξ. More than that, +in the limit of ℏωq ≫ KBT, pT +0 → 1 and the quantized +protocol fully charges the battery, independent of its ini- +tial state. This is a purely quantum effect due to the +vacuum of the f.c. and consists in the main result of this +paper. Not that similar charging can be obtained with +open system entropy producing dynamics, such as opti- +cal pumping. Here, we match it in an entropy preserving +protocol. +This sequence of cycles, however, can be cumbersome +and, in practice, escape from the isentropic condition of +no heat exchanged with external reservoirs. Furthermore, +the classical protocol is much faster, only requiring one +Rabi flip. One may wonder, then, if the quantized ad- +vantage still holds under equivalent restrictions. To an- +alyze this, we compute, from now on, single shot scenar- +ios designed with a sole detuning adjustment. The en- +ergy variation is obtained by solving the Von-Neumann +equation with Heff. The separation of Heff in doublets +makes it easy to derive the time evolution of the eigen- +states of HB0 + Hf.c.0. The anti-JC dynamics is similar +to the JC and it is simple to show that an initial state +|Ψ(0)⟩ = |g, n⟩ evolves to |Ψ(t)⟩ = e−i∆nt/2[(cos Ωnt + +i∆n +2Ωn sin Ωnt)|g, n⟩ − iGn +Ωn sin Ωnt|e, n + 1⟩]. A similar ex- +pression can be found for the initial state |e, n+1⟩. There- +fore, after evolving for τq, the state of the battery changes +to ρB(τq) = Trf.c.[e−iHeff τq/ℏ(ρT +B ⊗ ρT +f.c.)eiHeff τq/ℏ] = +� pjσjj where pg = pT +g − S(τq), pe = pT +e + S(τq) and +pm = pT +m (due to the elimination of level |m⟩). +Here, +S(τq) = �∞ +n=0 An[pT +g pT +n(0) − pT +e pT +n+1] sin2 � +Ωnτq +2 +� +, An = +1 +1+ r2(n+1−N0)2 +4(n+1) +and r = gq +ΩL (see Sup. Mat. for full deriva- +tion). In this case, ∆Uq = ℏωegS(τq) and the battery’s +ergotropy reads Eq = ℏωeg[pT +e −pT +g +2S(τq)] = 2∆Uq−Ec. +The quantized version will be advantageous whenever +∆Uq > Ec. +A quick inspection of S(τq) shows that, for single shots +(ss), it is the non-selective regime of r ≪ 1 that optimizes +the charging of the atom. In this case, all the doublets +evolve almost resonantly, each of them contributing to +enhance the charge. Because they oscillate at different +Rabi frequencies, it is impossible to choose a τq,ss that +simultaneously maximizes the energy transfer in all of +them. The optimal interaction time, which depends on +T, has to be numerically extracted by maximizing S(t) +and, because higher excited states oscillate faster, it gets +shorter for higher temperatures. In Fig. (2) we plot the +relative gain Kq ≡ ∆Uq +ss−∆Uc +∆Uc += ∆Uq +ss +∆Uc − 1 induced by the +single shot quantized protocol as a function of ξ and for +two temperatures. Note that, similar to the single shot +selective case, Kq increases with T and requires ξ > 1 to +represent positive gain over the classical counterpart. +We also plot in the same figure the efficiency of the +work extraction, defined as η ≡ +Eq +WL , where WL is the +work injected by the power supply. +The first law of +thermodynamics says that WL = ∆Uq + ∆Ufc where +∆Ufc = ℏωqS(τq) is the energy variation of the f.c.. +Therefore, the efficiency assumes the very simple formula +η = +1 +1+ξ +1+2Kq +1+Kq . For a fixed value of ξ, the best efficiency, +η = +2 +1+ξ, is achieved when Kq ≫ 1. On the other hand, +because ξ > 1 is a necessary condition for the advan- +tage of the single shot quantum protocol and because Kq +increases for larger values of ξ, it is clear that the best +gains are achieved at lower efficiencies. This should be +expected since ξ ≫ 1 means that most of the energy in- +jected by the power supply is actually going to the f.c.. +Note that for each temperature, there is an ideal value of +ξ if one wishes for the best gain at a given efficiency. +FIG. 2. Relative gain Kq = +∆Uq−∆Uc +∆Uc +(blue, straight) and +efficiency η = +Eq +WL (red, curved) as a function of parameter +ξ = +ωq +ωeg for different values of the adimensional temperature +¯T = KBT +ℏωm (≈ 0.1 for solid and ≈ 0.4 for dashed lines). +∆ +2π = 1 +MHz, gq = +∆ +600, ΩL = ∆ +20. +So far, we have considered the isentropic injection of +energy by the external source. +However, neither the +battery nor the f.c. +are ever fully isolated from their +environment and there will always be heat exchanged +with the external reservoir. From the battery’s perspec- +tive, if both |g⟩ → |m⟩ and |e⟩ → |m⟩ transitions are +dipole coupled, levels |g⟩ and |e⟩ must be of the same +parity and, therefore, cannot be dipole coupled them- +selves. That means that the time scale for direct energy +exchange between them is usually much slower than any +other time scale of the problem and, in general, the cor- +responding heat channel can be ignored. +Considering +the standard weak coupling to thermal reservoirs, the +overall dynamics of the system is, then, governed by a +master equation of the form ˙ρ = − i +ℏ[Hq, ρ] + L(ρ) [49], +where L(ρ) = � +s Γs[2LsρL† +s − {L† +sLs, ρ}], with s = +gm, mg, em, me, +, −. The rates of the non-unitary parts +are given by Γjm = γ0j(¯nj + 1), Γmj = γ0j¯nj, Γ− = + +0 +20 +40 +60 +80 +100 +35 6 +0.6 +Kq +n +30 E +0.5 +25 +0.4 +20 +0.3 +15 +0.2 +10 E +0.1 +0.0 +0 +20 +40 +60 +80 +100 +54 +γ0q(¯nq + 1), and Γ+ = γ0q¯nq. +Here, the γ0’s indicate +the spontaneous decay rates and ¯n’s the average number +of photons of the thermal reservoir at frequencies ωmj +and ωq. The respective jump operators are Ljk = σjk, +L− = ˆb and L+ = ˆb†. +0.0 +0.5 +1.0 +1.5 +2.0 +T +0 +10 +20 +30 +40 +50 +60 +Kq +0 = 0 +0 = 0.01 +gq +L +0 = 0.1 +gq +L +0 = +gq +L +0 = 10 +gq +L +FIG. 3. Relative gain as a function of the adimensional tem- +perature ¯T ≡ kBT +ℏωm for different values of spontaneous decay +rates γ0 and for ξ = 99, +∆ +2π = 1 MHz, gq = +∆ +600, ΩL = ∆ +20. The +solid curve is obtained from the unitary evolution with Heff. +The dotted curves are numerical solutions of the open system +dynamics (master equation) with full Hamiltonian Hq. +The couplings to the thermal reservoirs establish at +least four typical regimes to the problem, depending on +their strength. The first one, already addressed, corre- +sponds to γ0’s much smaller than the effective coupling +gqΩL +∆ +and kBT ≪ ℏωeg, ℏωq. This is well approximated +by the isentropic dynamics considered so far. However, +we saw that the higher the temperature, the more advan- +tageous the quantum protocol is. This may not hold true +when we take into consideration the heat exchanges with +the reservoir. As the spontaneous decay rates increase, +a combination of effects begin to affect the charging of +the battery and may even create optimal temperatures +for better quantum gain. +In Fig. (3) we present Kq as a function of the adimen- +sional temperature ¯T ≡ kBT +ℏωm for different values of γ0. ¯T +is relevant to the problem because it regulates the pop- +ulation of level |m⟩. Although each reservoir has its own +spontaneous decay rate, they all produce similar effects +on both Kq and η, therefore we have considered a single +γ0 for all of them. The result was obtained by solving +the full dynamics of the open quantum system and choos- +ing the best τq,ss for each temperature. In these plots, +ωm +2π = 1012Hz, ∆ = 2πMHz = 600g = 20ΩL, ξ = 99, +r = 1/30. As previously discussed, for γ0 ≪ +gqΩL +∆ +we +reach the unitary regime calculated with Hamiltonian (1) +(solid curve), except for very high temperatures ( ¯T ∼ 2) +when the population of level |m⟩ becomes too significant +and start to affect the protocol as a whole. As we in- +crease γ0, effects such as decoherence of the f.c. +field +and the augmented relaxation rates Γj begin to limit the +quantum advantage. These effects become particularly +relevant when Γ’s rates approach the effective battery- +f.c. coupling gqΩL/∆. Note, however, that even for such +values of dissipation, the quantum protocol can still pro- +duce gains 30 times larger than its classical counterparts +for ξ = 99. Finally, a fourth effect takes place for higher +values of γ0 and at much higher temperatures: when Γ’s +become of the order of ∆ the heat exchange eventually +brings the transitions back into resonance in which case +level |m⟩ cannot be adiabatically eliminated anymore and +the charging scheme breaks down. +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +2.00 +T +0.0000 +0.0025 +0.0050 +0.0075 +0.0100 +0.0125 +0.0150 +0.0175 +0.0200 +0 = 0 +0 = 0.01 +gq +L +0 = 0.1 +gq +L +0 = +gq +L +0 = 10 +gq +L +FIG. 4. Efficiency as a function of the adimensional tempera- +ture ¯T ≡ kBT +ℏωm for different values of spontaneous decay rates +γ0 and for ξ = 99, +∆ +2π = 1 MHz, gq = +∆ +600, ΩL = +∆ +20. The +solid curve is obtained from the unitary evolution with Heff. +The dotted curves are numerical solutions of the open system +dynamics (master equation) with full Hamiltonian Hq. +In Fig. +(4) we repeat the numerical calculations of +the open system dynamics (same parameters), this time +for the efficiency. Again, we see that very low γ0’s are +consistent with the isentropic hypothesis, whereas higher +values of the spontaneous decay rates severely affect the +efficiency, specially for higher values of ¯T. Note that for +some parameters, the plotted efficiency is corrected to +η = +Eq +WL+Qem to adjust for the fact that the |e⟩ → |m⟩ +reservoir may also inject energy in the system in the form +of heat Qem. +The correction takes place whenever we +obtain Qem > 0. +To conclude, we have shown that the quantized nature +of a component of a charging circuit can significantly +enhance the isentropic charging of a quantum battery +when benchmarked against its classical counterpart. This +is a purely quantum effect due to the vacuum state of +the quantized component and the ability to selectively +manipulate quantum states in the Hilbert space. +We +have also shown that our protocol can achieve the same +full charging capacity of open system entropy producing +equivalent schemes. We have demonstrated the effect in + +5 +a typical setup of off-resonant Raman population transfer +in three-level λ−configuration where the power supply is +an external laser field and the quantized component is a +harmonic oscillator. This example is particularly useful +due to its broad presence in a variety of quantum opti- +cal setups such as trapped ions and atoms, cavity QED, +superconducting qubits, quantum dots and many other +equivalent experiments. +This +work +was +supported +by +CNPq +Projects +302872/2019-1, INCT-IQ 465469/2014-0, and FAPERJ +project E-26/202.576/2019. +TFFS and YVA thank +Capes for financial support. +∗ Corresponding author: mfsantos@if.ufrj.br +[1] K. V. 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Petruccione, The theory of open +quantum systems (Oxford University Press, 2002). + diff --git a/DNFRT4oBgHgl3EQfxDge/content/tmp_files/load_file.txt b/DNFRT4oBgHgl3EQfxDge/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7d5da5c4ca40f91d92f4ab86ff25a3545a2e595a --- /dev/null +++ b/DNFRT4oBgHgl3EQfxDge/content/tmp_files/load_file.txt @@ -0,0 +1,671 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf,len=670 +page_content='Vacuum enhanced charging of a quantum battery Tiago F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' Santos,1 Yohan Vianna de Almeida,1 and Marcelo F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' Santos1, ∗ 1Instituto de F´ısica, Universidade Federal do Rio de Janeiro, CP68528, Rio de Janeiro, Rio de Janeiro 21941-972, Brazil (Dated: February 1, 2023) Quantum batteries are quantum systems that store energy which can then be used for quantum tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' One relevant question about such systems concerns the differences and eventual advantages over their classical counterparts, whether in the efficiency of the energy transference, input power, total stored energy or other relevant physical quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' Here, we show how a purely quantum effect related to the vacuum of the electromagnetic field can enhance the charging of a quantum battery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' In particular, we demonstrate how an anti-Jaynes Cummings interaction derived from an off-resonant Raman configuration can be used to increase the stored energy of an effective two-level atom when compared to its classically driven counterpart, eventually achieving full charging of the battery with zero entropic cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' The quest for advanced quantum technologies or the ir- reversible role of measurements in quantum dynamics are examples of subjects that have stimulated the study of thermodynamics in the microscopic world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' An important recent topic of investigation involves the role played by quantum resources in the storage and use of energy by quantized systems [1–19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' For example, coherence and entanglement have been proven useful to speed up or to super-extend the charging of quantum batteries [20– 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' Experimental results have also shown advances to- wards the production of microscopic quantum thermal machines and quantum batteries [28–33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' Most results re- garding quantum properties influencing the performance of quantum batteries, however, focus on increasing the power of the process rather than enhancing the charg- ing capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' That is because the latter usually requires entropy producing mechanisms [7–12, 18] that have dele- terious effects in properties such as coherence and entan- glement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' In this work we investigate how the quantized nature of part of an entropy preserving charging circuit can in- fluence the charging of a quantum battery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' The circuit comprises a classical power source (p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=') and an auxiliary frequency changer (f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' We compare the variation of the internal energy stored in the battery and the efficiency of the work extraction from the p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=', both for a classical and quantum version of the f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' In both cases, the overall dynamics is unitary and, therefore, comes at zero entropic cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' In the classical scenario, both p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' and f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' are connected to the battery for a fixed amount of time, τc (“c” for classical), unitarily charging its ini- tially thermal state: ρB(τc) = Uc(τc)ρT BU −1 c (τc), where Uc(τc) is derived from the coupling Hamiltonian Hc = HB0 + Vp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' (t) + Vf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' (t), Vj(t) is the potential created by the circuit component j and HB0 is the free Hamilto- nian of the battery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' Thermal states are free resources in thermodynamics [34–36] and, therefore, ideal to establish the classical benchmark to be challenged by the quantum version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' The charging is measured by the variation ∆U of internal energy of the battery, where U = Tr[ρBHB0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' In the quantized version, Vf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' (t) is replaced by the inter- action Hamiltonian HB−f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' (t) and the initial state must include the f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' system which is also in a thermal state: ρ(0) = ρT B ⊗ ρT f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='. The variation of energy of the bat- tery is now given by ∆U = Tr{[ρB(τq) − ρT B]HB0} ( “q” for quantum) where ρB(τq) = Trf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='Uq(τq)ρ(0)U −1(τq) and Uq is the time evolution operator obtained from Hq = HB0 + Hf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='0 + HB−f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='c(t) + Vp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' Note that, in both cases we assume isolation from the environment and the charging does not produce any entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' For com- pleteness, we later add dissipative non-unitary terms to the dynamics to verify how our results are affected by the heat exchanged with surrounding reservoirs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' We investigate the classical protocol in a particular setup where the battery is an oscillating two-level sys- tem of frequency ωeg, the p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' generates an oscillat- ing potential of frequency ωL > ωeg and the f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='. gen- erates another potential of frequency ωq = ωL − ωeg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' This situation is commonly found in many different quantum optical experiments [37–43], where the battery consists of two non-degenerate ground states {|g⟩, |e⟩} (ωeg ≡ ωe − ωg > 0) of a real or artificial atom and two modes of the electromagnetic field play the role of power supply and f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='. The couplings are intermedi- ated by a third atomic level |m⟩ working as an ancilla as depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' (1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' Level |m⟩ should only con- tribute virtually to the transference of energy and has to be adiabatically eliminated from the dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' This is achieved when each of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' and f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' couples off- resonantly one of the lower levels of the battery to |m⟩ in a Raman configuration, where HB0 = ℏ � j=g,e,m ωjσjj (σjk ≡ |j⟩⟨k|), Vf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' (t) = ℏΩq(σemeiωqt + σmee−iωqt) and Vp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' (t) = ℏΩL(σgmeiωLt + σmge−iωLt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' If ∆ = ωmg − ωL = ωme − ωq ≫ ΩL, Ωq, the corresponding time evolution Uc(t) induces Rabi oscillations between levels |g⟩ and |e⟩ that are equivalent to directly cou- pling them through one effective classical field of cou- pling strength ¯Ω = ΩLΩq ∆ [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' The optimal charg- ing of the battery is then obtained for a full Rabi flip that swaps the populations pT g,e in the original ther- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='13640v1 [quant-ph] 31 Jan 2023 2 mal state ρT B = � j=g,e pT j σjj, where pT j = e ℏωj KBT ZB and ZB = � j e ℏωj KBT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' In this case, ∆Uc = ℏωeg[pT g − pT e ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' Note that this is the most that a unitary transforma- tion can charge an initially thermalized two-level bat- tery and corresponds to the ergotropy Ec of the resulting state, ρB(τc) = pT e σgg + pT g σee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' Ergotropy is defined as Eρ(τ) = � k,j rkEj(|⟨rk|Ej⟩|2 − δkj), where Ej are the eigenenergies of H0 in increasing magnitude, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=', Ei ≥ Ej for i > j, and rk are the eigenvalues of ρ(τ) in decreasing order, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=', ri ≤ rj for i > j [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' |m〉 |e〉 |g〉 Δ 𝛀L gq, 𝛀q 𝚪mg 𝚪gm 𝚪em 𝚪me |e〉 |g〉 N = 1 |e, 0〉 |e, 1〉 |g, 0〉 N = 2 N = k |e, 2〉 |e, k〉 |g, 1〉 |g, k-1〉 N = 1 N = 2 N = k (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' (a) off-resonant Raman configuration: the battery is a two-level atom ({|g⟩, |e⟩});' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' the p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' is a laser of frequency ωL (coupling ΩL), the f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' is another harmonic oscillator of frequency ωq and couplings Ωq (classical) and gq (quantum).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' Level |m⟩ is an ancilla that intermediates both couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' Each channel can also exchange heat with the surrounding reservoirs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='the battery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' (b) Selective scheme to charge the bat- tery: in each step N, a selective Rabi flip transfers energy from |g, N − 1⟩ to |e, N⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' If, now, the classical f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' is replaced by a quantized field, we need to add its free energy Hf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='0 = ℏωqˆb†ˆb to the Hamiltonian, where ˆb† creates an excitation, and replace Vf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' (t) by the interaction term HB−f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' = ℏgq(σemˆb†+σmeˆb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' Once again, for ∆ ≫ ΩL, gq, we elim- inate level |m⟩ and, as shown in [46–48], the dynamics of the Battery-f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' system becomes approximately given by the effective Hamiltonian (ℏ = 1) Heff = −g2 qN ∆ σgg − g2 qˆb†ˆb ∆ σee + ΩLgq ∆ (σgeˆb + σegˆb†), (1) Note that Heff also includes a small correction to the energy difference between levels |g⟩ and |e⟩, given by ℏ∆N eg = ℏ Ω2 L−g2 qN ∆ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' This term, of the same order of Heff, does not affect the conditions for eliminating |m⟩ and can be physically implemented by applying a d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' Stark shift to the atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' There are a few aspects of Heff useful for us: first, the a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' Stark shift correction to level |e⟩ depends on the number of excitations of the f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' and |e, 0⟩ is an eigen- state of Heff with eigenvalue 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' second, the Rabi oscil- lations occur in the joint Hilbert space of atom and f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=', splitting it into doublets {|g, n⟩, |e, n + 1⟩}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' This corre- sponds to the anti-Jaynes-Cummings (anti-JC) configu- ration where the p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' excites both the battery and the f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' Third, each doublet oscillates at its own Rabi frequency given by Ωn = � ∆2n/4 + G2n, where ∆n = r2Ω2 L(n+1−N) ∆ , Gn = rΩ2 L √n+1 ∆ and r ≡ gq ΩL , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' each doublet is detuned from resonance by an amount ∆n proportional to the number of excitations of the f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='. Such Hamiltonians were predicted and implemented in trapped ions, cavity QED and superconducting cir- cuits, and for r ≫ 1, they operate in a selective regime where ∆n ≫ Gn and the Rabi oscillation in all the dou- blets is highly detuned except if n = N − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' In this case, {|g, N −1⟩, |e, N⟩} oscillates resonantly (∆N−1 = 0, ΩN−1 = rΩ2 L √ N ∆ ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' Therefore, by properly choosing ∆N eg the battery population exchange is conditioned on the number of excitations of the f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' field as shown in [46, 47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' For example, for N = 1, after an interaction time τq = π∆ 2rΩ2 L , the population in the {|g, 0⟩, |e, 1⟩} subspace swaps while all other states only gain number dependent phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' That takes the initial state ρ(0) = ρT B ⊗ ρT f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' to ρ(τq) = pT e pT 0 |e, 0⟩⟨e, 0| + pT g pT 0 |e, 1⟩⟨e, 1| + pT e pT 1 |g, 0⟩⟨g, 0| + pT g pT 1 |g, 1⟩⟨g, 1| + ( � n>1 pT n|n⟩⟨n|) ⊗ ρT B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' (2) Here, ρT f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' = � n pT nσnn, pT n = e− nℏωq KBT (1 − e− ℏωq KBT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' A simple algebraic manipulation shows that this swap in- creases the charge of the battery by ∆Uq = (pT 0 pT g − pT e pT 1 )ℏωeg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' In this case, there is an advantage over ∆Uc if pT e pT g > 1−pT 0 1−pT 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' We can better understand this condition at low temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' When KBT ≪ ℏωq, ℏωm, the prob- abilities pT n are negligible for n > 1 and so is pT m and we can approximate 1 − pT 1 ≈ pT 0 and pT e ≈ 1 − pT g , mean- ing that ∆Uq > ∆Uc if pT e pT 0 pT g pT 1 ≈ e ℏωeg(ξ−1) KBT > 1, where ξ = ωq ωeg .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' This happens whenever ξ > 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' whenever the battery’s gap is smaller than one excitation of field ˆb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' In principle, the larger the value of ξ, the more accentu- ated the enhancement due to the vacuum of field ˆb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' This is a purely quantum effect due solely to the vacuum of the f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' Note, however, that the quantum protocol allows for the relaxation of the ξ > 1 condition and an even more enhanced charging, which is a much more powerful result, due to the selectivity of Heff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' In fact, similar Rabi flips can be sequentially applied, each one tuned to resonance by adjusting ∆N eg in consecutive subspaces (N = 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=') as pictorially shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' (1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' In principle, this se- quence must be infinite to maximize the charging of the battery but, in practice, pT n tends rapidly to zero unless T is very high, and only a few cycles are required to ap- proach maximum charging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' After the sequence, the final 3 state reads ρ(� j τqj) ≈ [pT e (1 − pT 0 )σgg + (pT g + pT e pT 0 )σee and the variation of internal energy is ∆Uq = ∆Uc + pT e pT 0 ℏωeg ≥ ∆Uc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' This shows an advantage for any pos- itive temperature and independent of ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' More than that, in the limit of ℏωq ≫ KBT, pT 0 → 1 and the quantized protocol fully charges the battery, independent of its ini- tial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' This is a purely quantum effect due to the vacuum of the f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' and consists in the main result of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' Not that similar charging can be obtained with open system entropy producing dynamics, such as opti- cal pumping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' Here, we match it in an entropy preserving protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' This sequence of cycles, however, can be cumbersome and, in practice, escape from the isentropic condition of no heat exchanged with external reservoirs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' Furthermore, the classical protocol is much faster, only requiring one Rabi flip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' One may wonder, then, if the quantized ad- vantage still holds under equivalent restrictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' To an- alyze this, we compute, from now on, single shot scenar- ios designed with a sole detuning adjustment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' The en- ergy variation is obtained by solving the Von-Neumann equation with Heff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' The separation of Heff in doublets makes it easy to derive the time evolution of the eigen- states of HB0 + Hf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' The anti-JC dynamics is similar to the JC and it is simple to show that an initial state |Ψ(0)⟩ = |g, n⟩ evolves to |Ψ(t)⟩ = e−i∆nt/2[(cos Ωnt + i∆n 2Ωn sin Ωnt)|g, n⟩ − iGn Ωn sin Ωnt|e, n + 1⟩].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' A similar ex- pression can be found for the initial state |e, n+1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' There- fore, after evolving for τq, the state of the battery changes to ρB(τq) = Trf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' [e−iHeff τq/ℏ(ρT B ⊗ ρT f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' )eiHeff τq/ℏ] = � pjσjj where pg = pT g − S(τq), pe = pT e + S(τq) and pm = pT m (due to the elimination of level |m⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' Here, S(τq) = �∞ n=0 An[pT g pT n(0) − pT e pT n+1] sin2 � Ωnτq 2 � , An = 1 1+ r2(n+1−N0)2 4(n+1) and r = gq ΩL (see Sup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' for full deriva- tion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' In this case, ∆Uq = ℏωegS(τq) and the battery’s ergotropy reads Eq = ℏωeg[pT e −pT g +2S(τq)] = 2∆Uq−Ec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' The quantized version will be advantageous whenever ∆Uq > Ec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' A quick inspection of S(τq) shows that, for single shots (ss), it is the non-selective regime of r ≪ 1 that optimizes the charging of the atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' In this case, all the doublets evolve almost resonantly, each of them contributing to enhance the charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' Because they oscillate at different Rabi frequencies, it is impossible to choose a τq,ss that simultaneously maximizes the energy transfer in all of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' The optimal interaction time, which depends on T, has to be numerically extracted by maximizing S(t) and, because higher excited states oscillate faster, it gets shorter for higher temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' (2) we plot the relative gain Kq ≡ ∆Uq ss−∆Uc ∆Uc = ∆Uq ss ∆Uc − 1 induced by the single shot quantized protocol as a function of ξ and for two temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' Note that, similar to the single shot selective case, Kq increases with T and requires ξ > 1 to represent positive gain over the classical counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' We also plot in the same figure the efficiency of the work extraction, defined as η ≡ Eq WL , where WL is the work injected by the power supply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' The first law of thermodynamics says that WL = ∆Uq + ∆Ufc where ∆Ufc = ℏωqS(τq) is the energy variation of the f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='. Therefore, the efficiency assumes the very simple formula η = 1 1+ξ 1+2Kq 1+Kq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' For a fixed value of ξ, the best efficiency, η = 2 1+ξ, is achieved when Kq ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' On the other hand, because ξ > 1 is a necessary condition for the advan- tage of the single shot quantum protocol and because Kq increases for larger values of ξ, it is clear that the best gains are achieved at lower efficiencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' This should be expected since ξ ≫ 1 means that most of the energy in- jected by the power supply is actually going to the f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='. Note that for each temperature, there is an ideal value of ξ if one wishes for the best gain at a given efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' Relative gain Kq = ∆Uq−∆Uc ∆Uc (blue, straight) and efficiency η = Eq WL (red, curved) as a function of parameter ξ = ωq ωeg for different values of the adimensional temperature ¯T = KBT ℏωm (≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='1 for solid and ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='4 for dashed lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' ∆ 2π = 1 MHz, gq = ∆ 600, ΩL = ∆ 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' So far, we have considered the isentropic injection of energy by the external source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' However, neither the battery nor the f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' are ever fully isolated from their environment and there will always be heat exchanged with the external reservoir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' From the battery’s perspec- tive, if both |g⟩ → |m⟩ and |e⟩ → |m⟩ transitions are dipole coupled, levels |g⟩ and |e⟩ must be of the same parity and, therefore, cannot be dipole coupled them- selves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' That means that the time scale for direct energy exchange between them is usually much slower than any other time scale of the problem and, in general, the cor- responding heat channel can be ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' Considering the standard weak coupling to thermal reservoirs, the overall dynamics of the system is, then, governed by a master equation of the form ˙ρ = − i ℏ[Hq, ρ] + L(ρ) [49], where L(ρ) = � s Γs[2LsρL† s − {L† sLs, ρ}], with s = gm, mg, em, me, +, −.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' The rates of the non-unitary parts are given by Γjm = γ0j(¯nj + 1), Γmj = γ0j¯nj, Γ− = 0 20 40 60 80 100 35 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='6 Kq n 30 E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='5 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='4 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='3 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='2 10 E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='0 0 20 40 60 80 100 54 γ0q(¯nq + 1), and Γ+ = γ0q¯nq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' Here, the γ0’s indicate the spontaneous decay rates and ¯n’s the average number of photons of the thermal reservoir at frequencies ωmj and ωq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' The respective jump operators are Ljk = σjk, L− = ˆb and L+ = ˆb†.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='0 T 0 10 20 30 40 50 60 Kq 0 = 0 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='01 gq L 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='1 gq L 0 = gq L 0 = 10 gq L FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' Relative gain as a function of the adimensional tem- perature ¯T ≡ kBT ℏωm for different values of spontaneous decay rates γ0 and for ξ = 99, ∆ 2π = 1 MHz, gq = ∆ 600, ΩL = ∆ 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' The solid curve is obtained from the unitary evolution with Heff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' The dotted curves are numerical solutions of the open system dynamics (master equation) with full Hamiltonian Hq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' The couplings to the thermal reservoirs establish at least four typical regimes to the problem, depending on their strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' The first one, already addressed, corre- sponds to γ0’s much smaller than the effective coupling gqΩL ∆ and kBT ≪ ℏωeg, ℏωq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' This is well approximated by the isentropic dynamics considered so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' However, we saw that the higher the temperature, the more advan- tageous the quantum protocol is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' This may not hold true when we take into consideration the heat exchanges with the reservoir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' As the spontaneous decay rates increase, a combination of effects begin to affect the charging of the battery and may even create optimal temperatures for better quantum gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' (3) we present Kq as a function of the adimen- sional temperature ¯T ≡ kBT ℏωm for different values of γ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' ¯T is relevant to the problem because it regulates the pop- ulation of level |m⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' Although each reservoir has its own spontaneous decay rate, they all produce similar effects on both Kq and η, therefore we have considered a single γ0 for all of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' The result was obtained by solving the full dynamics of the open quantum system and choos- ing the best τq,ss for each temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' In these plots, ωm 2π = 1012Hz, ∆ = 2πMHz = 600g = 20ΩL, ξ = 99, r = 1/30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' As previously discussed, for γ0 ≪ gqΩL ∆ we reach the unitary regime calculated with Hamiltonian (1) (solid curve), except for very high temperatures ( ¯T ∼ 2) when the population of level |m⟩ becomes too significant and start to affect the protocol as a whole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' As we in- crease γ0, effects such as decoherence of the f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' field and the augmented relaxation rates Γj begin to limit the quantum advantage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' These effects become particularly relevant when Γ’s rates approach the effective battery- f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' coupling gqΩL/∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' Note, however, that even for such values of dissipation, the quantum protocol can still pro- duce gains 30 times larger than its classical counterparts for ξ = 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' Finally, a fourth effect takes place for higher values of γ0 and at much higher temperatures: when Γ’s become of the order of ∆ the heat exchange eventually brings the transitions back into resonance in which case level |m⟩ cannot be adiabatically eliminated anymore and the charging scheme breaks down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='00 T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='0000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='0025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='0050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='0075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='0100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='0125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='0150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='0175 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='0200 0 = 0 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='01 gq L 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='1 gq L 0 = gq L 0 = 10 gq L FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' Efficiency as a function of the adimensional tempera- ture ¯T ≡ kBT ℏωm for different values of spontaneous decay rates γ0 and for ξ = 99, ∆ 2π = 1 MHz, gq = ∆ 600, ΩL = ∆ 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' The solid curve is obtained from the unitary evolution with Heff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' The dotted curves are numerical solutions of the open system dynamics (master equation) with full Hamiltonian Hq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' (4) we repeat the numerical calculations of the open system dynamics (same parameters), this time for the efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' Again, we see that very low γ0’s are consistent with the isentropic hypothesis, whereas higher values of the spontaneous decay rates severely affect the efficiency, specially for higher values of ¯T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' Note that for some parameters, the plotted efficiency is corrected to η = Eq WL+Qem to adjust for the fact that the |e⟩ → |m⟩ reservoir may also inject energy in the system in the form of heat Qem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' The correction takes place whenever we obtain Qem > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' To conclude, we have shown that the quantized nature of a component of a charging circuit can significantly enhance the isentropic charging of a quantum battery when benchmarked against its classical counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' This is a purely quantum effect due to the vacuum state of the quantized component and the ability to selectively manipulate quantum states in the Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' We have also shown that our protocol can achieve the same full charging capacity of open system entropy producing equivalent schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' We have demonstrated the effect in 5 a typical setup of off-resonant Raman population transfer in three-level λ−configuration where the power supply is an external laser field and the quantized component is a harmonic oscillator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' This example is particularly useful due to its broad presence in a variety of quantum opti- cal setups such as trapped ions and atoms, cavity QED, superconducting qubits, quantum dots and many other equivalent experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' This work was supported by CNPq Projects 302872/2019-1, INCT-IQ 465469/2014-0, and FAPERJ project E-26/202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content='576/2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' TFFS and YVA thank Capes for financial support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} +page_content=' ∗ Corresponding author: mfsantos@if.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFRT4oBgHgl3EQfxDge/content/2301.13640v1.pdf'} 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Vermeulen† +Gravity Exploration Institute, +School of Physics and Astronomy, Cardiff University, +Cardiff, CF24 3AA, United Kingdom +(Dated: January 10, 2023) +Gravitational waves modulate the apparent frequencies of other periodic signals. We propose to +use this effect to detect low-frequency gravitational waves by searching for correlated frequency +modulations in a large set of well-resolved gravitational wave signals. +We apply our proposed +method to the large number of gravitational wave signals from Galactic binary white dwarfs +that are expected to be detected with the planned space-based gravitational wave detector LISA. +We show that, given current projections for the number and properties of these sources and the +sensitivity of the instrument, this method would enable the detection of background gravitational +wave strain amplitudes of, e.g., A ≃ 10−10 at a frequency F ≃ 10−8 Hz. When using signals from +binary neutron stars such as those expected to be observed with proposed detectors like DECIGO, +we expect a sensitivity to gravitational waves competitive with that of current Pulsar Timing +Arrays. +This would allow the detection of gravitational waves from, e.g., super-massive black +hole binaries with chirp masses Mc ≳ 109 M⊙ at a distance D ≃ 10 Mpc. Our results show that +gravitational-wave detectors could be sensitive at frequencies outside of their designed bandwidth +using the same infrastructure. +This has the potential to open up unexplored and otherwise +inaccessible parts of the gravitational wave spectrum. +I. +INTRODUCTION +The field of gravitational-wave astronomy, as estab- +lished with the first direct detection of gravitational +waves (GWs) [1], is still in its infancy. So far, only GWs +with frequencies between ∼ 10 − 500 Hz produced by +the coalesecence of black holes and neutron stars with +masses ∼ 1 − 100 times the mass of our Sun have been +detected [2]. New detectors and techniques are being de- +veloped to probe different regions of the GW frequency +spectrum and to investigate numerous other potential +GW sources; e.g., rotating neutron stars [3], binary white +dwarfs (BWDs) [4], intermediate-mass and super-massive +binary black holes (SMBBHs) [5], a background of pri- +mordial GWs [6], and dark matter [7, 8]. +The sensitive bandwidth of laser interferometers (the +only proven type of GW detector), is typically limited +at low frequencies by spurious accelerations of the test +masses, and at high frequencies by quantum uncertainty +in the optical state and an intrinsically decreased re- +sponse to GWs with wavelengths shorter than the in- +terferometer’s arms. Laser interferometers can be very +sensitive at higher frequencies (∼ 1 − 100 MHz), using +cross-correlation and shorter arms [9, 10]. Increasing the +sensitivity at lower frequencies is not straightforward, +and even a space-based instrument such as LISA [11], +subject to greatly reduced environmental noise, will not +be sensitive to GWs below ∼ 10−5 Hz. While marginal +gains have been made in understanding and addressing +∗ StegmannJ@cardiff.ac.uk +† VermeulenSM@cardiff.ac.uk +the complex amalgam of low-frequency noise contribu- +tions encountered in laser interferometers (which include +fundamental quantum limits) [12], it seems unlikely that +their bandwidth will expand into lower frequencies by +more than an order of magnitude in the coming decades. +Other detection techniques to probe new areas of the +GW spectrum have been proposed and some have been +tried; none have proven successful in detecting GWs so +far. +At high frequencies (kHz – GHz) these include +techniques that exploit graviton-to-photon conversion +(known as the inverse Gertsenshtein effect) [13, 14], opti- +cally levitated sensors, resonant mass detectors [15], and +more [16]. At low frequencies, currently the only com- +petitive method to search for GWs is using sets of time- +resolved observations of pulsars, known as Pulsar Tim- +ing Arrays (PTAs), which are sensitive in the nHz – µHz +range [17–26]. GWs incident on the pulsar and/or the +detector produce deviations of the apparent frequency +or equivalently the arrival time of the radio pulses that +are correlated between different pulsars. This detection +technique thus exploits the interplay of electromagnetic +pulses with GWs which results in a modulation of the +pulse frequency. So far, after observing for ∼ 10 yr, PTAs +have not detected GWs [27–30]. +In this work we propose a new method for detecting +(low-frequency) GWs using interactions between GWs of +different frequencies. The basis of the method is the grav- +itational red- and blueshift induced by one GW onto the +other. This mechanism can also be viewed as one GW +perturbing the space-time along the direction of travel of +the other GW, and thus modulating the arrival times of +peaks and troughs of the other GW. Mathematically, the +effect can be described as a multiplication or mixing of +two GWs. From this description, it can be shown that the +arXiv:2301.02672v1 [gr-qc] 6 Jan 2023 + +2 +resulting GW signal contains Fourier components at the +sum and difference of the frequencies of the two waves, +with an amplitude proportional to the product of the am- +plitudes of the individual GWs. This elementary result +of the mixing of two waves, also known as heterodyn- +ing, has been used in the processing of electromagnetic +signals for over a century. Heterodyning effectively pro- +duces a frequency-shifted copy of one signal (known as a +sideband) in the frequency range of a readily detectable +second signal. As we show in this paper, this mechanism +can be used in GW astronomy, where GW signals de- +tectable with, e.g., laser interferometers can be used to +detect low-frequency background GWs. This method of +searching for low-frequency GWs is conceptually similar +to the technique used by PTAs, with the crucial difference +that instead of looking for disturbances in the periodic +signal of pulsars, we look for disturbances in a periodic +GW signal. The idea of looking for GW sidebands was +recently independently proposed by Bustamante-Rosell +et al. [31], when our paper was in preparation, but their +analysis and projections differ significantly from ours. +Our proposed method allows one to expand the sen- +sitive bandwidth of GW detectors into low-frequency +regimes using the detectors’ existing infrastructures. +Moreover, this method could enable a sensitivity to GWs +in a bandwidth where no other detection methods exist, +e.g., in the µHz regime where the sensitivity of space- +based laser interferometers and PTAs leaves a gap. +Although our method is applicable to general periodic +GW signals, we focus here on the example of future space- +based laser-interferometric GW detectors, i.e., LISA [11] +and DECIGO [32], which are expected to be able to ob- +serve large numbers of GW signals from BWDs and bi- +nary neutron stars (BNSs). Using projected parameters +of the detector and signals for these instruments, we show +that cross-correlation of many well-resolved GW signals +can provide sensitivity to secondary low-frequency GWs. +II. +THEORY +We consider a set of N ≫ 1 periodic GW sources +which could be simultaneously observed for a long time +(e.g., BWDs in our Galaxy that could be individually +resolved by LISA [11]). We further assume that these +sources emit quasi-monochromatic GWs, i.e., that their +frequency does not significantly change within the obser- +vation time T (see Sec. V for discussion of the implica- +tions of relaxing this assumption). In that case we can +write the GW signal (in units of strain) from the α-th +periodic source at distance dα as +hα(t) = aα cos[2πfαt + ϕα], +(α = 1, 2, . . . , N), +(1) +with constant frequency fα, amplitude aα, and initial +phase ϕα. We refer to these GWs as carrier signals and +to their sources as carrier sources. +If there is an incident GW from a secondary, more +distant source, this GW will perturb the spacetime at +the location of the carrier sources and at the location of +the observer. As a consequence, the frequency of the GW +carrier signals are no longer constant but are modulated +in time. For a background GW emitted by a distant point +source in the direction ˆ +N this frequency modulation of +the carrier signal is given by [33], +fα − fα(t) +fα += +ni +αnj +α +2(1 + ˆ +N · ˆnα) +� +hTT +ij (t) − hTT +ij (tα) +� +, +(2) +where ˆnα and ni +α is the unit vector from the observer +to the α-th carrier source and its components, respec- +tively, and tα = t−dα(1+ ˆ +N · ˆnα)/c is the retarded time +coordinate that accounts for the propagation of the car- +rier wave. Additionally, hTT +ij (t) and hTT +ij (tα) correspond +to the metric perturbation due to the incident GW at +the spacetime locations of the carrier source and the ob- +server, respectively (in the terminology of Pulsar Timing +Arrays (PTAs) [17, 18], the former is usually referred to +as the ‘Earth term’ and the latter as the ‘pulsar term’). +It can be shown that the single-sided frequency spec- +trum of the modulated signal can then be written as [31] +˜hα(f) ≃ aαeiϕαδ(fα − f) ++ 1 +2aαAIα,Lei(ϕα+ΦL)δ(f − fα + FL) ++ 1 +2aαAIα,Le−i(ϕα+ΦL)δ(f − fα − FL) ++ 1 +2aαAIα,Dei(ϕα+Φα,D)δ(f − fα + FD,α) ++ 1 +2aαAIα,De−i(ϕα+Φα,D)δ(f − fα − FD,α), +(3) +where Iα,L,D = (FL,D/fα) K( ˆ +N, ˆnα, hTT +ij , dα), and K is +a purely geometrical factor of order unity that accounts +for the polarisation, propagation direction, and propaga- +tion distance of the background and carrier GWs. The +first term in the spectrum given by Eq. (3) is the Fourier +component corresponding to the carrier signal at the fre- +quency f = fα. The modulation due to the background +GW at the location of the observer manifests as two +Fourier components with frequencies f = fα±FL (second +and third term in Eq. 3), which we will refer to as the +’local’ sideband terms. Similarly, the modulation of the +carrier signal due to the background GW at the location +of the carrier source produces sidebands with frequencies +f = fα ± FD,α (fourth and fifth term), which we will +refer to as the ‘distant’ sideband terms. Note that the +frequency and phase offsets, FL, ΦL, of the ‘local’ terms +are independent of the carrier (they are equal to the fre- +quency and phase of the modulating GW at the location +of the observer), whereas the ’distant’ terms have fre- +quency and phase offsets FD,α, Φα,D, which depend on +the location of the carrier source. +This mechanism, a sort of ‘GW heterodyning’ could +allow the indirect detection of low-frequency GWs that +may otherwise be undetectable when a GW detector is +not sensitive to signals down to a frequency F, but is + +3 +sensitive at much higher frequencies fα + F. Using this +method, the upconverted background signal amplitude +is Asideband = AaαKF/fα. +For example, if we take +the carrier signal to be the GWs emitted by a typical +BWD (such as the BWDs that LISA aims to detect), +with frequency fα ∼ 10−2 Hz, and we take the back- +ground signal to be GWs emitted by a SMBBH with +amplitude A ∼ 10−12 and frequency FL ∼ 10−8 Hz, +the background sideband signal appears at an amplitude +aαIα,L ∼ aα10−6. +This suppression relative to the carrier would mean the +background signal amplitude is below the typical noise +level of the detector. In the following section, we propose +a method to amplify the signal which utilises the coher- +ence of the modulation of multiple carrier signals. To this +end, we construct and add Np = N(N − 1)/2 ≫ 1 dif- +ferent cross-spectra (one for each pair of carrier sources) +such that the sideband terms sum up coherently to ex- +ceed the incoherent random noise. +III. +METHODS +We propose a cross-correlation method for detecting +a background gravitational wave signal that produces +phase modulation of carrier GW signals. We will later +use this method to make quantitative estimates of the +expected signal-to-noise ratio that can be obtained for +potential astrophysical GW sources using planned GW +detectors. +We consider the time-domain output signal of the GW +detector s(t) to be given by the sum of N carriers, all +modulated by a single background GW signal with fre- +quency F corresponding to either the ‘local’ (F = FL) or +the ‘distant’ (F = FD) term, and noise n(t) characteristic +of the detector +s(t) = +N +� +α=1 +hα(t) + n(t). +(4) +For any carrier, we can apply a demodulation and phase- +shift to the time-domain detector output and normalise +it by the modulation index and the carrier amplitude +sα(t) = +√ +2 +aαIα +e−i(2πfαt+ϕα) s(t). +(5) +This demodulation shifts the frequency of all Fourier +components in the output by an amount fα, such that +all sideband (heterodyne) signals are frequency shifted +to the frequency ±F of the modulating background GW +that produces them. Moreover, any heterodyne signals +from background GWs will now appear with a Fourier +amplitude equal to the background GW strain ampli- +tude that produces them. In general, the demodulation +frequency need not be constant in time, but could be ad- +justed over time to account for time-dependent changes +in the carrier frequency. Specifically, the demodulation +frequency and phase could be varied according to a pre- +determined carrier signal model, or they could be ad- +justed using feedback control (e.g., through maximising +the demodulated carrier amplitude) when the frequency +evolution is unknown a priori. After this frequency and +phase shift, we can apply an appropriate low-pass filter +to the data such that other terms, as long as they are +well-separated from the carrier and modulation sideband, +need not be considered [34]. +We consider the case where the time-domain detec- +tor output is discretised with a constant sampling fre- +quency fs for a total observation time T. Next, we take +the single-sided discrete Fourier transform of the detec- +tor output, which yields a discrete complex amplitude +spectrum Sj +α for each carrier signal, which will have the +form +Sj +α = AeiΦαδjl(F ) + +√ +2 +aαIα +� +ρj +α +T eiηj +α, +(6) +where the index j = 1, 2, . . . , Tfs/2 runs over the fre- +quency bins, l(F) [35] is the index of the bin that con- +tains the background signal (δjl is the Kronecker delta), +ρj +α is the noise power spectral density of the detector, +and ηj +α are the random noise phases (where both noise +parameters have undergone the frequency and phase shift +described by Eq. 5). +The spectrum Sj +α is unique for +each carrier signal. As background GWs would modu- +late all carrier signals coherently (i.e., the sideband phase +is deterministic), whereas the noise has a random phase, +cross-correlating different carrier signals is advantageous. +For each pair of carrier signals (αβ), a cross-spectrum +Sj +αβ = Sj +αSj∗ +β , can be constructed which has the form +Sj +αβ = A2ei(Φα−Φβ)δjl(F ) + +2 +aαaβIαIβ +� +ρj +αρj +β +T +ei(ηj +α−ηj +β), +(7) +where Φα − Φβ = Φαβ is the phase difference of the +modulating signal between the two carrier signals. From +this expression it can be seen that Φab is deterministic, +and ηj +α − ηj +β = ηj +αβ is random. Therefore, we can add +up signal terms from different cross-spectra coherently, +and the noise will average out. If we have N individ- +ually resolved carriers at our disposal we can construct +Np = N(N − 1)/2 different cross spectra and take a co- +herent weighted average of them +Sj = +�Np +(αβ) λj +αβSj +αβ e−iΦαβ +�Np +(αβ) λj +αβ +, +(8) +where λj +αβ are the weights of each cross-spectrum. Per- +forming this coherent summation is possible as long as +the relative modulation sideband phase Φαβ can be de- +termined for each carrier pair (αβ). For the modulation +produced by the background GW at the detector (‘local’ +term), Φαβ = 0 ∀ αβ. For the sideband due to the mod- +ulation produced at the source of the carrier GW signal + +4 +(‘distant’ term), Φαβ is a function of the relative posi- +tions of the background GW source and the carrier signal +sources. In this case, Φab can be taken as free parame- +ters that are fit to the data by maximising the total SNR +for a particular sideband frequency, which would yield an +upper estimate of the maximum background GW signal +power at a certain frequency. Alternatively, a hypothet- +ical background source position and frequency could be +assumed, which prescribes a certain set of Φαβ given the +geometry of the source positions, which would then yield +an upper limit of the estimated background GW strain +at that frequency and sky position. +Note that the coherent average is constructed such +that +the +expected +real +part +of +the +signal +bin +is +E +� +Re[Sl(F )] +� += A2. +The squared signal-to-noise ratio +can thus be defined for each bin +(SNRj)2 = +� +Re[Sj] +�2 +Var (Re[Sj]). +(9) +It can be shown that an optimal signal-to-noise ratio is +found by taking the weights [36] +λj +αβ = +Np +� +(γδ) +([Cj]−1)αβ,γδ ≃ +� +1 +σj +ασj +β +�2 += (aαaβIαIβ)2T 2 +4ρj +αρj +β +, +(10) +where Cj +αβ,δγ is the pair-wise cross-covariance matrix of +the cross-spectra Sj +αβ, Sj +δγ, and σj +α,β are the variances of +frequency bin j in each carrier spectrum (Eq. 6); the +approximation holds in the weak-signal limit [36]. The +SNR of a modulating background GW with frequency F +and amplitude A can now be evaluated +(SNRl(F ))2 ≃ A4 +2 +�Np +(αβ) +� +1 +σl(F ) +α +σl(F ) +β +�2 +. +(11) +The GW detector LISA is expected to observe a large +number of continuous, periodic GW signals from BWDs +in our Galaxy [4, 11, 37–42]. These BWDs could poten- +tially serve as carrier sources that allow for the detection +of low-frequency background GWs as described above. +The total number and properties of Galactic BWDs +is subject to large uncertainty. To obtain a quantitative +projection for the number, frequency, and amplitude of +BWD GW signals that may be detected with LISA, we +use an observationally driven parametric model of the +Galactic white dwarf population, constructed by Korol +et al. [42][43]. This model builds upon the spectroscopic +samples of single white dwarfs and BWDs from the Sloan +Digital Sky Survey (SDSS) and the Supernova Ia Progen- +itor surveY (SPY) to produce a synthetic population of +Galactic BWDs which are specified by their component +masses, orbital frequencies, sky positions, and orienta- +tions. These source parameters are then used to calculate +the GW signals of each BWD in the population. Part of +the BWDs would emit GWs at low frequencies f ≲ 3 mHz +and are predicted to be so numerous that they are not +TABLE I. Input parameters used for generating synthetic +populations of Galactic binary white dwarfs. +The parame- +ters ρKorol +WD,⊙, f Korol +BWD,4 AU, f Korol +BWD,amax, and αKorol are used as in- +put for the algorithm described by Korol et al. [42] to model +the sets of BWD carrier signals. +These parameters repre- +sent the local WD density, the fraction of binaries with semi- +major axes < 4 AU, the fraction of binaries with semi-major +axes less than the maximum separation detectable with LISA +(amax), and a power-law index specifying the BWD semi- +major axis distribution, respectively (see Korol et al. [42] +for details). The values of these parameters were chosen to +correspond to upper (Optimistic), median (Moderate), and +lower (Pessimistic) observational limits. We chose observa- +tion times T between 1.0 and 10.0 yr. N indicates the result- +ing number of BWDs which are individually resolvable with +LISA. +Model +Pessimistic Moderate Optimistic +ρKorol +WD,⊙ +[10−3 pc−3] +4.11 +4.49 +4.87 +f Korol +BWD,4 AU +0.112 +0.095 +0.078 +f Korol +BWD,amax +0.008 +0.009 +0.010 +αKorol +−1.18 +−1.30 +−1.45 +T +[yr] +1.0 +4.0 +10.0 +N +7.0 × 104 +1.1 × 105 +1.9 × 105 +individually resolvable but constitute a confusion-limited +foreground noise [39]. The rest, an estimated number of +∼ O(103 – 105) BWDs emit GWs at higher frequencies +and are expected to be sufficiently loud that they are in- +dividually resolvable; these are the BWDs which can be +used as carrier sources in our method. +We consider three models with different carrier source +and observation parameters, Pessimistic, Moderate, +and Optimistic. For these models, we synthesized three +BWD populations using different input parameters for +the model of Korol et al. [42]; specifically we vary the +local WD density ρKorol +WD,⊙, the WD binary fraction f Korol +BWD, +and the power-law index αKorol, which describes the +BWD semi-major axis distribution (see Korol et al. [42]). +On the observation side we use three different values for +the LISA mission lifetime T = 1.0, 4.0, and 10.0 yr, which +sets the length of observation. To get an upper and lower +limit for the resulting sensitivity to background GWs, we +choose the model parameters such that Pessimistic and +Optimistic models yield the lowest and highest number +of BWDs within the current observational uncertainty +while Moderate model corresponds to median values. +The parameter values of the three different models are +summarised in Table I. +In Figure 1, we show the amplitude spectral density +(ASD) of the BWD carriers for each model together with +LISA’s projected detector noise amplitude spectral den- +sity, as in [44], modified to account for the confusion +noise due to unresolved BWDs derived by Korol et al. +[42]. +Throughout this work we assume a BWD to be +individually resolvable if aα +� +T/ρα > 7, although the +precise threshold does not affect the resulting sensitiv- + +5 +FIG. 1. +Amplitude spectral densities aα +√ +T of gravita- +tional wave signals from individually resolvable binary white +dwarfs (BWDs) in three different models [42] as a function of +their frequency f = fα. +The solid line indicates the root +of the projected noise power spectral density √ρ of LISA +[42, 44]. +BWDs are assumed to be individually resolvable +if aα +� +T/ρα > 7. +ity due to the dominant contribution of loud sources (see +Section V). +IV. +RESULTS +We estimate the sensitivity to background gravita- +tional waves for the three models using our method, as +in Eq. (11). Figure 2 shows the amplitude A versus fre- +quency F of a background GW that could be detected +with SNR = 2, corresponding to a ≃ 95 % detection prob- +ability. +The differences between the Pessimistic and +Optimistic models are less than one order of magnitude +in A. Our method is sensitive to GWs with frequencies +as low as F ∼ 10−8 Hz. GWs of these frequencies could +be present in our Universe, e.g., as part of a (stochas- +tic) background of GWs emitted by numerous individual +sources [46]. At a frequency of F ≃ 10−8 Hz our method +would be sensitive to amplitudes A ≳ 10−10; GWs of +that amplitude at that frequency could, e.g., be emitted +by a very massive SMBBH with a chirp mass of several +∼ 1010 M⊙ at a distance D = 10 Mpc, which is the scale +of the Virgo cluster. No other method for detecting GWs +with frequencies between 10−6 and 10−5 Hz exists. +We also consider the more general case of a number +of carrier GW signals observed with any GW detector. +For this case we assume that all N carrier signals have a +similar frequency and are detected with the same SNR ∼ +aα +� +T/ρα = const. In Figure 3, we show the correlated +background GW amplitude that can be detected at an +SNR of one, as a function of the number and individual +SNR of the carrier signals. +We can apply this result to a proposed next-generation +GW detector such as DECIGO [47, 49, 50], which op- +erates in the dHz regime and is expected to observe +GWs from a large number of compact binary stars. As- +suming DECIGO observes GW signals from a popula- +tion of N = 105 binary neutron stars (BNSs) each ob- +served with an SNR of ∼ 104 [47] at a typical frequency +of fα = 0.1 Hz, it would be possible to detect back- +ground GWs from SMBBHs with chirp masses of about +∼ 109 M⊙ (at a fiducial distance D = 10 Mpc and fre- +quency F = 10−8 Hz). This would make the sensitivity +of DECIGO to low-frequency GWs competitive with that +of current PTAs (cf. Figure 2). +For reference, we also indicate in Figure 3 the sensi- +tivity that could be obtained using ∼ 105 carrier signals +with an SNR ∼ 102 from compact binary coalescences, +as expected to be detected using both Einstein Telescope +(ET) and Cosmic Explorer (CE) [48]. These carrier sig- +nals would have frequencies between 10 and 103 Hz and +could be observed for a duration T ≲ 103 s, which means +the minimum detectable background GW frequency us- +ing our method is F ∼ 10−3 Hz. Coherent background +GW signals may be searched for using non-coincident +carrier signals with a slight modification of the method +described in Sec. III; a frequency-dependent phase correc- +tion (φcorr = 2πTdiffF) must be applied to each carrier’s +demodulated spectrum (Eq. 6), for a time difference be- +tween the signals Tdiff. In case the background GW signal +has a coherence time much shorter than the total obser- +vation time for all signals (i.e., the detector’s lifetime), +only coincident carrier signals can be cross-correlated to +gain sensitvity. +The sensitivity of our method is fundamentally lim- +ited to frequencies F ≳ 1/T, as for lower frequencies +the background signal cannot be distinguished from the +carrier [31]. The same low-frequency limit due to obser- +vation time exists for PTAs. The high-frequency limit of +our method is set by the Nyquist frequency of the detec- +tor output sampling, fs/2, where for LISA fs ∼ 1 Hz +[31]. PTAs have a much smaller sensitive bandwidth due +to the low observation cadence of radio telescopes (once +every several days or less). +V. +DISCUSSION +There are several effects that could in practice degrade +the sensitivity that would be obtained using our method. +Of particular concern is phase noise imparted by the +data acquisition system of the gravitational-wave detec- +tor. As this noise would appear as modulations of the car- +rier signal, it would obfuscate any background GWs that +produce the same effect. Phase noise in the data acqui- +sition system, due to, e.g., timing jitter of the sampling + +Noise ASD +Moderate BWD ASD +Optimistic BWDASD +Pessimistic BWD ASD +10-16 +-18 +10 +10 +20 +10 +10 +10-3 +10-2 +10-1 +Frequency f [Hz]6 +FIG. 2. Sensitivity to low-frequency gravitational waves (GWs) that can be obtained by searching for correlated modulations +in a set of well-resolved GW signals from binary white dwarfs (BWDs), as expected to be detected with LISA. For reference, +we show the expected GW amplitudes of super-massive binary black holes with chirp masses ranging from 108 to 1011 M⊙ at a +fiducial distance D = 10 Mpc. We also show sensitivity curves from Pulsar Timing Arrays (PPTA [27]; EPTA [29]; NANOGrav +[45]). The detection threshold (SNR = 2) is chosen to allow a consistent comparison to reported PTA sensitivities. In practice, +we expect our method to show a reduction in sensitivity around F ≃ 1/yr ≃ 32 nHz as seen for PTAs, where it would be difficult +to distinguish a background GW from the Doppler modulation due the annual motion of LISA around the sun. The sensitivity +of our method is limited to frequencies F ≳ 1/T (e.g., 32 nHz in the Pessimistic model), below which the sensitivity is limited +by the finite width of the frequency bins. +clocks, would produce irreducible correlated noise in the +demodulated cross-spectra of different carriers. This ef- +fect might only be reduced by cross-correlating data ob- +tained with different uncorrelated oscillators. Similarly, +stochastic phase noise intrinsic to the carrier GW sig- +nal would reduce sensitivity to background GWs. In this +case the effect on the sensitivity is limited as this noise +will be uncorrelated between carriers and will be reduced +in the average cross-spectrum (Eq. 8). +In addition to these effective stochastic fluctuations +of the carrier signal, there could be deterministic fre- +quency changes of the carrier and background GWs. If +the frequency of the background GWs changes signifi- +cantly over the measurement time, i.e., if the GW back- +ground power spectral density is non-stationary, the co- +herent signal power would be spread over multiple fre- +quency bins, leading to a lower SNR in each bin. +An +SMBBH background source might undergo a significant +frequency evolution as its orbital period decays due to +energy loss by GW emission. Figure 4 shows that this +frequency change ˙F (‘chirp’) would not not be significant +for SMBBHs (Mc ≳ 109) over the duration of observa- +tion T ≃ 1 – 10 yr. Figure 4 also shows the expected fre- +quency changes of the LISA and DECIGO carrier signals. +In particular, it shows that most DECIGO BNSs undergo +significant frequency evolution over the duration of the +detected signal. As discussed in Sec. III, these frequency +changes could be compensated for at the demodulation +stage. +Non-stationarity of the background GW PSD has an- +other effect; the frequency change over a time equal +to the typical light travel time between the carrier +source and observer determines the frequency-space sep- +aration of the ‘local’ and ‘distant’ sideband terms, i.e., +|FL − FD| ∝ dα ˙F/c. If these terms are not separated in +the spectrum, i.e., when |FL −FD| ≲ 1/T, coherent sum- +mation of the ‘local’ terms of different cross-spectra is +still possible but the ‘distant’ terms would add a small in- +coherent noise-like contribution to any signal bin. The in- +set of Figure 4 shows that given typical light travel times +between BWDs and the LISA detector of dα/c ≃ 10−1 – +101 kpc/c [41], both separated and non-separated side- +bands could be observed for background SMBBH GW +sources. On the other hand, DECIGO will observe car- +rier signals from BNSs at much larger distances, e.g., +dα ≃ 104 kpc for a GW170817-like event [52], and there- +fore ‘local’ and ’distant’ sidebands produced by a back- +ground SMBBH source (Mc ≳ 109 M⊙) would be well- +separated in DECIGO data. +We note that for the sensitivity projections for LISA, + +10-8 +10l1 Mo + Strain amplitude A +1010Mo +Pessimistic +12 +10 +109 Mo +Moderate +M +Optimistic. +NANOGrav +08 M +10-14 +EPTA +PPTA +10-8 +10-7 +10-6 +10-5 +Frequency F [Hz]7 +FIG. 3. +Order-of-magnitude estimate for the sensitivity to +background gravitational waves (GWs) by cross-correlating a +generic set of a number of GW signals N that are each de- +tected with a certain SNR (‘Carrier SNR’). The sensitivity +(given by the colour scale) is expressed as the product of the +background amplitude A times the typical frequency ratio of +the background and carrier signals fα/F, where the detection +threshold corresponds to an SNR equal to one. Furthermore, +we indicate the sensitivity that could be obtained using a set +of GW signals in the dHz regime from binary neutron stars as +carriers, which could be done using data from DECIGO [47], +and similarly the sensitivity using carrier signals detected us- +ing ET and CE [48]. We also show the sensitivity that could +be obtained using the average SNR of binary white dwarf +signals detected by LISA (in the Moderate model), as expli- +cated in Fig. 2. For these detectors we assume typical carrier +frequencies of fα ≃ 0.1 Hz (DECIGO), 10 Hz (ET/CE), and +10−3 Hz (LISA). For reference, we show contour lines that cor- +respond to GW amplitudes from super-massive binary black +holes with chirp masses ranging from 109 to 1011 M⊙ at a +fiducial distance D = 10 Mpc, with a background frequency +F = 10−8 Hz, and a carrier frequency fα = 0.1 Hz. +the number N of individually resolvable BWDs in our +models (see Table I) is larger by a factor up to ∼ 10 com- +pared to previous estimates from Galaxy models com- +bined with a binary population model [4, 39, 41, 53, 54] +which reflects the large uncertainty of current predictions +about the detectable BWD population. However, the ex- +act total number of BWDs does not significantly affect +the estimated sensitivity because the ∼ O(103) loudest +BWDs signals provide the dominant contribution to the +sensitivity. This is shown in Figure 5; where we plot the +normalised cumulative contribution of BWDs to the total +SNR. It can be seen that several 102 to 103 BWDs are +enough to achieve similar sensitivities to the total BWD +population. +FIG. 4. +Timescale f/ ˙f = (5/96)(c3/GMc)5/3(πf)−8/3 at +which the frequency f of a compact binary with chirp +mass Mc significantly increases due to energy loss through +gravitational-wave emission. Coloured boxes indicate the pa- +rameter regions of background super-massive binary black +holes (SMBBHs), LISA binary white dwarfs (BWDs), and +DECIGO binary neutron stars (BNSs). This shows that LISA +BWDs and most of the SMBBHs would not undergo signifi- +cant frequency changes within the observation time T ≃ 1 – +10 yr, whereas most DECIGO BNSs would. The inset shows +whether the SMBBHs would exhibit significant frequency +changes within typical light travel times between a carrier +source and the observer, i.e., whether ‘local’ and ‘distant’ +sidebands overlap or not. For this figure we take the max- +imum GW frequency emitted by SMBBHs to correspond to +the Innermost Stable Circular Orbit f ≲ 1 kHz (M⊙/Mc) eval- +uated for equal-mass binaries [51], which causes the diagonal +cut-off. +VI. +CONCLUSION +In this work, we have outlined a method to use a set +of carrier gravitational wave sources to search for cor- +related frequency modulations caused by low-frequency +background GWs. +In this method demodulated cross- +spectra of carrier sources are added coherently and with +optimal weights such that any modulation common to +the carrier sources is amplified with respect to random +detector noise. +We considered the case of using our method to search +for low-frequency GWs in data from LISA, which is ex- +pected to detect GWs from a large number of Galac- +tic binary white dwarfs. The projected sensitivity that +could thus be obtained (Figure 2) ranges from strain am- +plitudes of A ∼ 10−10 at F ∼ 10−8 Hz to ∼ 10−7 at +∼ 10−5 Hz, and would cover a part of the GW spectrum +where no other detection methods are currently available. + +105 +100 +10-1 +109 Mo× +104 +10-2 +DECIGO +10- +3 +Carrier SNR +103 +10-4 +ET&CE +10-5 +102 +X +1011 Ma +10-6 +LISA +10-7 +101 +10-8 +100 +10-9 +101 +103 +105 +107 +Number of carriers N1011 +109 +Mc [Mo] +SMBBHS +107 +100 +106 +kpc +SMBBHS +kpc +kpc +kpc +c +c +105 +10-9 +10-7 +10-5 +f [Hz] +12 +Insignificant chirp +Significant chirp +10yr +f/f>T +f/f +Key +Feature +Consistency +Encoder +Predictor +Loss +Data +Augmentation +Query +Encoder +Decoder +Reconstruction Loss +S +a +Stack of input +Actor +frames +SAC +Forward Pass +Training +Critic +Q(s,a) +Weight sharing/ 00 (2023) 1–13 +4 +progress in last few years. Auto-encoders [19] [48] learn the state representation by compressing the observation +into low-dimensional state that is sufficient to reconstruct the observation. These have been used to improve the +performance of RL algorithms as demonstrated in [17][49] [50] [24]. On the other hand, contrastive learning [22] +[51] learns the class-relevant feature representations by maximizing the agreement between the augmented versions +of the same observation. It has been shown to greatly improve the sample efficiency of RL algorithms as in [10]. +Similarly, recent studies have shown that the right kind of data augmentation techniques can improve the sample +efficiency and generalization capabilities of RL algorithms learning task-relevant features which remain unaffected by +distractions introduced by the augmentation [9] [52]. This can be further enforced by making the encoder minimize +the consistency loss as suggested in [23]. In short, learning suitable feature representation plays a significant role in +improving the performance of RL algorithms by increasing sample efficiency, improving generalization and stability. +The work presented in this paper contributes to this field by proposing a novel loss function that leads to superior +learning performance for continuous control tasks as will be demonstrated later in this paper. +3. Method +This section provides details of the proposed CRC-RL model that uses a novel heterogeneous loss function to ex- +tract useful information from visual images to be used for learning optimal policy using an end-to-end RL framework. +The discussion is organized in the following subsections. The architecture of the proposed model is described next. +3.1. The Model Architecture +The overall architecture of the proposed model is shown in Figure 1. The observation is available in the form +of images which are stacked together to act as input to the model. Stacking of frames is a heuristic approach to +incorporate temporal information in the learning process [53]. The observations obtained from the environment is +stored in a replay buffer D and a batch is sampled from this replay buffer during the training process. A Siamese Twin +encoder model is employed for extracting features from the input images. These two encoders, termed as query and +key encoders, are used for computing contrastive and consistency losses. The query encoder with a decoder is used for +computing the reconstruction loss. A combination of these three losses, known as the CRC loss, is used for updating +the parameters of the query encoder and decoder network. The input images are augmented before applying to the key +encoder. The features obtained from the query encoder is used for policy estimation using soft-actor-critic algorithm +[45]. The parameters of the query encoder and decoder networks are updated using error signals obtained from their +own outputs as well as from the RL algorithm. Since the encoder networks are getting influenced by the RL policy +algorithm, the features learnt in the process are action-dependent. This aspect will be analyzed in more detail in the +experiment section presented later in this paper. The weights of the key encoder network is the exponential moving +average of the query encoder weights. The proposed CRC loss function used for learning the feature embeddings is +discussed in the next subsection. +3.2. The loss function for feature extraction +The query encoder is trained using the proposed CRC loss function which is a combination of the following three +loss components as described below. +3.2.1. Contrastive loss +In contrastive learning, we have a query q observation and a set of key observation samples K = {k0, k1, ...} +consisting of both positive samples (k+) and negative samples (K \ k+). The positive samples are those that belong to +the same class as that of the query observation and the rest are considered to be the negative samples. The goal is to +learn embeddings such that q is relatively more similar to the positive keys k+ than the negative keys in the latent space. +The query and key observations, generated by applying data augmentation on sampled observations, are encoded using +the query and key encoder respectively. The contrastive loss depends on the output of both the encoders (Siamese +Twin) represented by the symbols s and s′ respectively. The idea behind using the contrastive loss is that the different +augmentations of the same image will have the same underlying information and hence their high-level representations +will be mapped together in the latent space. The similarity between the query and the key embeddings is computed +using the bilinear inner-product qTWk > 0 where W is a symmetric matrix of parameters to be estimated [54] along +4 + +/ 00 (2023) 1–13 +5 +with other parameters during the training process. The objective of training is to reduce this similarity measure so that +the query embeddings become more distinct from the key embeddings over time (qTWk ≈ 0 ⇒ q ⊥ k, W > 0). This +is achieved by minimizing the InfoNCE loss [55] given by: +Lq = log +exp(qTWk+) +� +ki∈K exp(qTWki) +(1) +3.2.2. Reconstruction loss +A well-trained encoder-decoder network is expected to reconstruct the input image at the output of the decoder +network. The reconstruction loss is computed based on the inaccuracy in the reconstructed image. A convolutional +encoder fθ maps an input observation x ∈ Rm×n×3 to a lower-dimensional latent vector s ∈ Rl, and a deconvolutional +decoder gφ then reconstructs s back to ˆx ∈ Rm×n×3 such that +fθ : x → s +(2) +gφ : s → ˆx +(3) +Both the encoders and decoder are trained simultaneously by maximizing the expected log-likelihood. The recon- +struction loss checks how well the image has been reconstructed from the input. The reconstruction loss forces the +update such that the latent representation preserves the core attributes of the input data. An L2 penalty is imposed on +the learned representation s and a weight-decay is imposed on the decoder parameters to incorporate the regularization +affects as proposed in [56]. +Lr = Ex∼D[log pθ(x|s) + λs∥s∥2 + λθ∥θ∥2] +(4) +where λs, and λθ are hyper-parameters. +3.2.3. Consistency loss +The consistency loss depends on the output of both the query and key encoder fθ and f ′ +θ. Here, the query encoder +takes the original non-augmented observation x and the key encoder uses the augmented observation ˜x as input. The +output of the Key encoder s′ is then used as an input to a feature predictor module, which is nothing but an MLP, +to estimate the non-augmented embedding ˆs. The consistency loss is designed to minimize the error between the +non-augmented embedding s and the augmented embedding s′, thereby enabling the encoder to learn essential task- +relevant features while ignoring irrelevant distractions (such as background clutter or texture). This eliminates the +need of using negative samples for the computation of consistency loss. The consistency loss function can, therefore, +be mathematically written as: +Lc(ˆs, s, θ) = Ex∼D[∥ˆs − s∥2] +(5) +3.3. The CRC loss function +It is our conjecture that each of the above three loss functions enables the encoder to extract non-redundant and +complementary information from the higher dimensional input. Thus, a combination of these three should improve the +overall RL performance in learning optimum policy. The resulting loss function, called CRC loss, has the following +mathematical form: +LCRC = c1Lq + c2Lr + c3Lc +(6) +where ci > 0, � +i ci = 1, i = 1, 2, 3 are hyper-parameters that control the relative importance of individual components. +The RL model for policy learning takes the query encoder output as its input. The SAC algorithm used for learning +policy is also allowed to affect the query encoder weights fθ during the backward gradient update step. At regular +intervals, the key encoder f ′ +θ weights are updated using the exponential moving average (EMA) of the weights of the +query encoder fθ. The feature learning and policy learning takes place in jointly in parallel. The latent representations +learned by the query encoder fθ receives gradients from both the CRC loss and the SAC algorithm losses. This makes +the feature representations action-dependent, an aspect which will be analyzed in some more detail in the next section. +SAC algorithm is given in Algorithm 1 for the reference of the readers. +5 + +/ 00 (2023) 1–13 +6 +Algorithm 1 SOFT Actor-Critic Algorithm +1: Input: +Initial policy parameters θ, Q-function parameters φ1, φ2, empty replay buffer +D +2: Set target parameters equal to main parameters φtarg,1 ← φ1, φtarg,2 ← φ2 +3: repeat +▷ Observe state s and select action +4: +5: +a ∼ πθ(.|s) +6: +Execute a in the environment +7: +Observe next state s′ reward r, and done signal d to indicate whether s′ is +terminal. +8: +Store (s, a, r, s′, d) in replay buffer D +9: +10: +if s′ is terminal, reset environment state then +11: +it’s time to update then +12: +for j in range (however many updates) do +13: +Randomly sample a batch of transitions, B = (s, a, r, s′, d) from D +14: +Compute targets for the Q function +y(r, s′, d) = r + γ(1 − d)(min +i=1,2 Qφtarg,i(s′ − ˜a′) − α log πθ( ˜a′|s′)), ˜a′ ∼ πθ(.|s′) +15: +Update Q-function by one step of gradient descent using +▽φi +1 +| B | +� +(s,a,r,s′,d)∈B +(Qφi(s, a) − γ(r, s′, d))2 +for i= 1, 2 +16: +Update policy by one step of gradient ascent using +▽φ +1 +| B | +� +s∈B +(min +i=1,2 Qφi(s, ˜aθ(s)) − α log πθ( ˜aθ(s)|s)) +where, ˜aθ(s) is a sample from πθ(.|s) which is differentiable w.r.t θ via the +re-parameterization trick. +17: +Update target networks with +φtarg,i ← ρφtarg,i + (1 − ρ)φi +for i = 1, 2 +18: +end for +19: +end if +20: until Convergence +6 + +/ 00 (2023) 1–13 +7 +Table 1. Hyper-parameters used for DMControl experiments. Most hyper-parameters values are unchanged across environments with the exception +for action repeat, learning rate, and batch size. +Hyper-parameter +Value +Pre transform image size +(100, 100) +Image size +(84, 84) +Action repeat +8 +Frame stack +3 +Transform +Random crop +Replay buffer capacity +100000 +Initial steps +1000 +Batch size +512 +Hidden layers +1024 +Evaluation episodes +10 +Optimizer +Adam +Learning rate ( fθ, πψ, Qφ) +1e-3 +Learning rate (α) +1e-4 +Critic target update frequency +2 +Convolution layers +4 +Number of filters +32 +Latent dimension +50 +Discount (γ) +0.99 +Initial temperature +0.1 +Table 2. Mean episodic reward (with standard deviation) over 10 evaluation runs on DMControl environments after training for 100k environment +steps. The best scores are shown in bold letters. +100K Step +Our Method +CURL [10] +SODA [23] +PLANET [59] +DREAMER [60] +SAC+AE [17] +PIXEL +STATE +% Increase +Scores +SAC [45] +SAC [45] +over CURL +FINGER, SPIN +793±36 +767±56 +363±185 +136±216 +341±70 +740±64 +179±66 +811±46 +3.38 +CARTPOLE, SWINGUP +813±45 +582±146 +474±143 +297±39 +326±27 +311±11 +419±40 +835±22 +39.6 +REACHER, EASY +636±301 +538±233 +- +20±50 +314±155 +274±14 +145±30 +746±25 +18.2 +CHEETAH, RUN +355±31 +299±48 +- +138±88 +235±137 +267±24 +197±15 +616±18 +18.7 +WALKER, WALK +490±52 +403±24 +635±48 +224±48 +277±12 +394±22 +42±12 +891±82 +21.5 +BALL IN CUP, CATCH +832±81 +769±43 +539±111 +0±0 +246±174 +391±82 +312±63 +746±91 +8.19 +4. Experimental Results and Discussions +The proposed CRC-RL model architecture takes its inspiration from the original CURL implementation by Laskin +et al. [10]. The original model is extended by incorporating additional decoder and feature predictor to facilitate +computing the CRC loss function as described in the previous section. The model is implemented using PyTorch [57] +deep learning framework and the source code is made available on GitHub https://github.com/darshitajain/CRC-RL- +for the convenience of readers and facilitate reproduction of results furnished in this paper. The reinforcement learning +framework for policy estimation makes use of the publicly released implementation of the SAC algorithm by Yarats +et al. [58]. The query encoder and decoder architecture is similar to the ones used in the above work. The query +encoder weights are tied between the actor and critic so that they both use the same encoder to embed input image +observations. The feature predictor module is a MLP network which consists of cascaded linear layers and ReLU +activation function. The complete list of hyper-parameters is shown in Table 4. A number of experiments are carried +out to establish the efficacy of the proposed model. The design choices are justified through several ablation studies +as discussed below. +4.1. Performance Comparison +The performance of the proposed CRC-RL model is compared with the current state-of-the-art methods on the +challenging Deep mind control suite (DMControl) environments [29]. The outcome is shown in Table 4 and 3 after +7 + +/ 00 (2023) 1–13 +8 +CURL +CRC-RL +(a) Cartpole-Swingup +(b) Cheetah-Run +(c) Walker-walk +Figure 2. t-SNE visualization of latent feature embeddings obtained from query encoder at 49K training steps. Colors correspond to cluster labels +in the action space. One can observe that CRC-RL leads to more pristine clusters with less outliers compared to CURL. +training for 100K and 500K environment steps respectively. It can be observed that the proposed CRC-RL model out- +performs the current state-of-the-art methods, such as CURL [10], SODA [23], PlaNet [59], Dreamer [60], SAC+AE +[17], pixel-based SAC [45] on most of the DMControl environments, thereby establishing the superiority of our ap- +proach. The environments shown in Table 3 are difficult compared to those shown in Table 4 and hence require longer +training time. In this case, our proposed model outperforms the baseline CURL model in only 300K training steps. +Table 3. Mean episodic score (with standard deviation) for 10 evaluation runs on DMControl environments obtained after training for 500k +environment steps. The best scores are shown in bold letters. +Environment +Our Method* +CURL +% Increase over CURL +QUADRUPED, WALK +88 ± 51 +39±22 +125.6 +HOPPER, HOP +61 ± 33 +10±17 +510 +WALKER, RUN +306 ± 5 +245±32 +24.8 +FINGER TURN, HARD +423 ± 78 +207±32 +104.3 +* shows values for 300K training steps +4.2. t-SNE Visualizations +To better understand the relationship between the learned latent representations and the action generated by the +RL policy, we generate the two-dimensional t-SNE plots of feature embeddings obtained from the query encoder for 3 +different environments as shown in Figure 2. These features are assigned with the corresponding action labels gener- +ated by partitioning the action space into five clusters by using the k-mean clustering algorithm. As one can observe, +the proposed CRC-RL model leads to more pristine clusters with lesser outliers compared to the baseline CURL [10] +algorithm for some amount of training. Compared to the Cartpole environment, other two are comparatively more +complex and require larger amount of time for training. This shows that the proposed model leads to better correlation +between the feature embeddings and agent actions. This aspect has not been empirically investigated extensively in +the existing literature and thus, the current work makes a novel contribution by filling this void. +4.3. Feature Correlation Heat Maps +Another study is performed to validate our hypothesis that the proposed CRC loss contribute new information +resulting in learning new feature representations which are distinct from those obtained using individual losses. In +this study, the correlation matrices between the latent features obtained with the baseline CURL algorithm (that uses +8 + +15 +0 +1 +2 +10 +3 +4 +5 +0 +-5 +-10 +-10 +-5 +0 +5 +1020 +0 +1 +2 +15 +3 +4 +10 +5 +0 +-5 +10 +-10 +0 +5 +10 +1510 +0 +1 +2 +. +. +3 +5 +4 +. +0 +-5 +-10 +-15 +-10 +-5 +0 +5 +10 +1515 +0 +2 +10 +3 +4 +5 +00 +0 - +-5 +心 +-10 +-10 +-5 +0 +5 +107.5 +5.0 +2.5 +0 +0.0 +2 +m +-2.5 +4 +5.0 +7.5 +10.0 +-10 +-5 +0 +5 +1015 +0 +1 +2 +3 +10 +4 +5 +0 +-5 +. +-10 +-15 +-10 +-5 +0 +5 +10/ 00 (2023) 1–13 +9 +(a) +(b) +(c) +Figure 3. Effect of incorporating various loss components. CRC loss function performs better than other combinations for more +difficult environments such as ’Cheetah-Run’ and ’Walker-walk’. +(a) Cartpole-Swingup +(b) Cheetah-Run +(c) Walker-walk +Figure 4. Feature correlation heat-maps for three environments showing the correlation between the latent features obtained with CURL and +CRC-RL models. The models are trained for 49K environment steps and 200 latent features are used to generate this plot. +contrastive loss) and that with the proposed CRC-RL model (that uses CRC loss) is plotted as heat-maps as shown in +Figure 4.3. These matrices are generated by collecting 200 sample embeddings from both models trained with 49000 +environment steps. Since, each image sample is encoded into a 50 × 1 feature vector, the features generated by the +above two methods are grouped into two feature matrices (say, F1 and F2) of size 200 × 50. The correlation between +these two feature matrices results in a 400x400 matrix is visualized as a heat map in the above figure where the darker +regions shows higher correlation and lighter regions show lower correlation. It is observed that off-diagonal regions +have lower correlation (lighter regions) indicating that the two feature embeddings (F1 and F2 ) are very distinct from +each other. The diagonal regions are highly correlated (darker regions) as they correspond to the features from the +same method. Another interesting finding of this study is that these heat maps show increasingly complex patterns for +difficult environments such as ’Walker-walk’ or ’Cheetah-Run’ compared to simpler environments such as ’Cartpole- +Swingup’. These patterns evolve over time and stabilize as the training performance saturates. This is an interesting +insight that may provide clue to better understand the relationship between features and action policies learned in an +end-to-end RL framework. +4.4. Ablation Study +Three separate ablation studies are carried out to justify the design choices made in this paper as described below. +9 + +Cartpole, Swingup +Mean episodic reward +800 +600 +400 +200 +Environment Step +10k +20k +30k +40k +0Cheetah, Run +400 +Mean episodic reward +300 +200 +100 +Environment Step +20k +40k +60k +80k +0Walker, Walk +Mean episodic reward +500 +400 +300 +200 +100 +Environment Step +10k +20k +30k +40k +0CURL+Consistency +CURL +CRC-RL +CURL+Reconstruction1.0 +0.8 +112 +128 +144 +0.6 +160 +176 +192 +208 +0.4 +224 +240 +256 +272 +0.2 +288 +304 +320 +352 +0.0 +368 +3841.0 +0.8 +0.6 +128 +144 +160 +0.4 +176 +192 +208 +0.2 +224 +240 +256 +272 +0.0 +288 +304 +320 +0.2 +336 +352 +368 +0.4 +384 +05 +m1.0 +0.8 +0.6 +88 +96 +0.4 +104 +112 +120 +128 +0.2 +136 +144 +152 +160 +- +0.0 +168 +176 +184 +192 +0.2 +4 +6 +4 +00 +80/ 00 (2023) 1–13 +10 +4.4.1. Usefulness of CRC loss function +First study validates the usefulness of the proposed CRC loss function comprising of contrastive, reconstruction +and consistency losses. The outcome is shown in Figure 4.2. We start with the baseline CURL model [10] that +uses contrastive loss to learn the feature presentations. Then this model is trained with a combined loss function of +contrastive and reconstruction loss and finally, with the CRC loss function comprising of contrastive, reconstruction +and consistency losses. The inclusion of these losses require modifying the existing CURL model leading to the +formation of the CRC-RL model proposed in this paper. The figure shows that CRC-RL performs better than the +other two for the benchmark problems from the Deepmind control suite (DMC), namely, ‘Cheetah-Run’ or ‘Walker- +walk’. These two are comparatively difficult problems to solve compared to simpler problems such as the ‘Cartpole- +Swingup’ problem which does not benefit from the proposed CRC-RL model. This observation clearly establishes the +usefulness of the proposed approach. This becomes more evident in the second ablation study discussed in the next +section below. +4.4.2. Relative weights of loss components in the CRC loss function +In this study, the relative weights of three loss components, namely, contrastive, reconstruction and consistency +loss, are varied and its effect on the validation performance is compared as shown in Figure 5. The weights are varied +with the constraint of forming a convex sum. In other words, ci > 0, i ∈ 1, 2, 3 and � +i ci = 1. The figure shows +that the individual loss functions do not always provide the best performance. The best performance is obtained by +a combination of all the three losses. Having equal weights (c1 = c2 = c3 = 0.33) for all the three losses have a +regularizing effect on model performance in the sense that the performance is traded for better generalizability. This +is evident from the fact that the validation performance curve with this combination of weights lie somewhere in the +middle of the all the curves. The generalization capability of the proposed model is demonstrated in the third ablation +study discussed next. +4.4.3. Generalization Capability of the CRC-RL model +In order to test the generalization capabilities of the proposed CRC-RL model, another experiment is performed +where the RL models are trained on images augmented with random crop effect and then validated on images aug- +mented with Video-Easy and Color-Hard artifacts [23]. The outcome is shown in Figure 6. Comparing with the +validation plots in Figure 5, it is observed that the overall performance has come down significantly due to these com- +plex augmentations which makes it difficult for the model to generalize when trained only with images augmented +with only random crop artefact. However, even in this case, the RL model trained with CRC loss function provides +the best or closer-to-the-best evaluation performance compared to the models that use individual loss components for +training. This demonstrates the superior generalization capability of the proposed CRC-RL model over the existing +models that use one of the above loss functions for training. It also corroborates the earlier finding mentioned in the +previous subsection that the equal weights for individual loss components in the CRC loss function have a regularizing +effect on the model performance. +5. Conclusions +The paper addresses the problem of feature representation learning in end-to-end reinforcement learning models +with visual observations. Specifically, a new loss function, called CRC loss, is proposed to learn action-dependent +features leading to superior performance in learning optimal action policies. This loss function is a combination of +three different loss functions, namely, the image reconstruction loss, the contrastive loss and the consistency loss. +Through empirical analysis including latent feature visualization, an attempt is made to generate new insights that can +better explain the relationship between the features being learnt and the actions being taken by the RL agent. The +resulting architecture is shown to outperform the existing state-of-the-art methods in solving the challenging DMC +problems by a significant margin thereby forming a new benchmark in this field. The future work will involve carrying +out more in-depth analysis and evaluation of the individual loss components on the overall performance as well as on +the quality of features being learned. +10 + +/ 00 (2023) 1–13 +11 +(a) +(b) +(c) +Figure 5. Effect of varying weighting parameters for different loss functions on the Evaluation performance. c1, c2 and c3 are +the weights to the contrastive loss, the reconstruction loss and the consistency loss respectively in the CRC loss function. The +environments used are: (a) Cartpole-Swingup, (b) Cheetah-Run and (c) Walker-walk. Varying these parameters have a regularizing +effect on the training performance. A smoothing factor of 0.5 is applied to the plot. +(a) +(b) +(c) +(d) +Figure 6. Generalization capabilities of CRC-RL algorithm. c1, c2 and c3 are the weights to the contrastive loss, the reconstruction +loss and the consistency loss respectively in the CRC loss function. The environments used are: (a) Cartpole-Swingup, (b) Cheetah- +Run and (c) Ball-in-Cup-Catch and (d) Walker-walk. The RL models are trained on images with random-crop augmentation and +evaluated on images with Video-Easy and Color-Hard augmentations. Compared to the individual losses, CRC loss provide best +or second-best evaluation performance for these new augmentations thereby establishing the superior generalization capabilities of +the CRC-RL model. +11 + +Cartpole-Swingup +Showing first 1o runs +c1: 0.33， c2: 0.33， c3: 0.33 + c1: 0.01， c2: 0.89, c3: 0.1 + c1: 0， c2: 0， c3: 1 c1: 0， c2: 1， c3: 0 + c1: 1， c2: 0， c3: 0 +Mean Episodic Reward +800 +600 +400 +200 +Training Steps +0 +0 +20k +40k +60k +80kCheetah-Run +Showing first 10 runs + c1: 0， c2: 0， c3: 1 + c1: 0. c2: 1. c3: 0 +c1: 1， c2: 0， c3: 0 +c1: 0.01, c2: 0.89, c3: 0.1 + c1: 0.33, c2: 0.33,c3: 0.33 +Mean Episodic Reward +400 +300 +200 +100 +Training Steps +0 +0 +20k +40k +60k +80kWalker-walk +Showing first 10 runs + c1: 0， c2: 0， c3: 1 + — c1: 0， c2: 1， c3: 0 + c1: 1， c2: 0， c3: 0 +c1: 0.01, c2: 0.1, c3: 0.89 + c1: 0.2, c2: 0.6, c3: 0.2 + c1: 0.33, c2: 0.33, c3: 0.33 +700 +Mean Episodic Reward +600 +500 +400 +300 +200 +100 +Training Step +0 +0 +20k +40k +60k +80kCartpole-Swingup:Video-Easy +Mean Episodic Reward +500 +400 +300 +200 +100 +Training Steps +0 +50k +100k +150k +200kCheetah-Run: Video-Easy +Mean Episodic Reward +250 +200 +150 +100 +50 +Training Steps +0 +50k +100k +150k +200kBall-in-Cup-Catch: Video-Easy +Mean Episodic Reward +500 +400 +300 +200 +100 +Training step +50k +100k +150k +200kWalker-walk: Video-Easy +Mean Episodic Reward +300 +200 +100 +Training Step +50k +100k +150k +200kCartpole-Swingup: Color-Hard +Mean Episodic Reward +600 +500 +400 +300 +200 +100 +Training Steps +0 +50k +100k +150k +200kCheetah-Run: Color-Hard +Mean Episodic Reward +250 +200 +150 +100 +W +50 +Training Steps +50k +100k +150k +200kBall-in-Cup-Catch: Color-Hard +600 +Mean Episodic Reward +500 +400 +300 +200 +100 +Training step +0 +50k +100k +150k +200kWalker-walk: Color-Hard +Showing first 1o runs + c1: 0， c2: 0， c3: 1 + c1: 0， c2: 1， c3: 0 +c1: 1，c2: 0， c3: 0 + c1: 0.33， c2: 0.33, c3: 0.33 +Mean Episodic Reward +500 +400 +300 +200 +100 +Training Step +50k +100k +150k +200k/ 00 (2023) 1–13 +12 +References +[1] V. 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Antiga, et al., Pytorch: An imperative +style, high-performance deep learning library, Advances in neural information processing systems 32. +[58] D. Yarats, I. Kostrikov, R. Fergus, Image augmentation is all you need: Regularizing deep reinforcement learning from pixels, in: International +Conference on Learning Representations, 2020. +[59] D. Hafner, T. Lillicrap, I. Fischer, R. Villegas, D. Ha, H. Lee, J. Davidson, Learning latent dynamics for planning from pixels, in: International +conference on machine learning, PMLR, 2019, pp. 2555–2565. +[60] D. Hafner, T. Lillicrap, J. Ba, M. Norouzi, Dream to control: Learning behaviors by latent imagination, arXiv preprint arXiv:1912.01603. +13 + diff --git a/KdFRT4oBgHgl3EQfDjfR/content/tmp_files/load_file.txt b/KdFRT4oBgHgl3EQfDjfR/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..583961d282031858850e34b5d72d199d2837e1dc --- /dev/null +++ b/KdFRT4oBgHgl3EQfDjfR/content/tmp_files/load_file.txt @@ -0,0 +1,889 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf,len=888 +page_content='00 (2023) 1–13 CRC-RL: A Novel Visual Feature Representation Architecture for Unsupervised Reinforcement Learning Darshita Jain, Anima Majumder, Samrat Dutta and Swagat Kumar Darshita Jain, Anima Majumder, Samrat Dutta and Swagat Kumar are with Tata Consultancy Services, Bangalore, India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' E-mail: darshita.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content='jain@tcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content='com, anima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content='majumder@tcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content='com, samrat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content='dutta@tcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content='com, swagat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content='kumar@tcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content='com* Abstract This paper addresses the problem of visual feature representation learning with an aim to improve the performance of end-to-end reinforcement learning (RL) models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' Specifically, a novel architecture is proposed that uses a heterogeneous loss function, called CRC loss, to learn improved visual features which can then be used for policy learning in RL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' The CRC-loss function is a com- bination of three individual loss functions, namely, contrastive, reconstruction and consistency loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' The feature representation is learned in parallel to the policy learning while sharing the weight updates through a Siamese Twin encoder model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' This encoder model is augmented with a decoder network and a feature projection network to facilitate computation of the above loss com- ponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' Through empirical analysis involving latent feature visualization, an attempt is made to provide an insight into the role played by this loss function in learning new action-dependent features and how they are linked to the complexity of the problems being solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' The proposed architecture, called CRC-RL, is shown to outperform the existing state-of-the-art methods on the challenging Deep mind control suite environments by a significant margin thereby creating a new benchmark in this field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' Keywords: CRC-RL, Contrastive learning, Image Reconstruction, Image Consistency, Reinforcement Learning, CRC loss, Feature representation learning 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' Introduction In the recent past, deep reinforcement learning (DRL) algorithms have been successfully used to learn action policies directly from visual observations, thereby finding application in several interesting areas such as gaming [1, 2], robotics [3, 4, 5, 6] and autonomous vehicles [7, 8] etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' This success is mostly driven by the agent’s ability to jointly learn feature representation and policy decisions by using long-term credit-assignment capabilities of reinforcement learning algorithms in an end-to-end fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' In spite of this success, RL algorithms are known to be sample-inefficient and suffer from poor generalization for high-dimensional observations such as images [9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' There are several approaches to address these concerns, including methods such as, transfer learning [11, 12], meta-learning [13] [14] and active learning [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' Feature representation learning [16] is an alternative, and sometimes complementary, to these approaches which aims at learning useful features that can simplify the intended task, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=', classification or prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' This paper primarily focuses on this later approach as it is now widely accepted that the problem of sample-inefficiency in RL can be solved to a great extent by learning suitable feature representation which is shown to expedite the policy learning process [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' The feature representations are learned using self-supervised methods which are increasingly becoming popular as they obviate the need for manually generated labeled datasets thereby simplifying the practical deployment of deep learning models [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' Some of these approaches include auto-encoders [19], GANs [20] [21], contrastive learning [22] and data augmentation techniques [9] [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' The features thus obtained 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content='13473v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content='CV] 31 Jan 2023 NON SOLUS ELSEVIERbleonlineatwww.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content='sciencedirect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content='con ScienceDirect/ 00 (2023) 1–13 2 have been shown to greatly improve the sample efficiency and generalizability of RL methods as demonstrated in [24] [10] [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' Rather than decoupling the representation learning from policy learning as done in [25] [23] [17], we continue working with end-to-end models because of their simplicity and aim at improving their performance by performing auxiliary tasks as demonstrated in [10] [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' Since the feature representations are learned along side the policy de- cisions in an end-to-end fashion, the features learned are actually action-dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' This is because, the backward gradient flow from the policy-learning algorithm is allowed to update the encoder weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' This makes the learned feature vectors strongly correlated to the actions being taken by the agent [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' Given this hindsight, we are motivated by two factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' First, it is our belief that the quality of the features learnt could be improved by using a better loss function leading to improved RL performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' Secondly, we are keen to develop a better understanding of the rela- tionship between the feature and action spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' Towards fulfilling these objectives, we propose a new heterogeneous loss function called CRC loss for feature representation learning by combining three different loss functions, namely, contrastive loss [10], reconstruction loss and consistency loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' The reconstruction loss obtained with an auto-encoder model helps in learning compact features that is sufficient to reconstruct the original observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' On the other hand, the consistency loss [23] helps in learning features that are invariant to image augmentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' In other words, by mini- mizing the consistency loss, the encoder is encouraged to learn task-relevant features while ignoring irrelevant aspects (such as, background color) thereby avoiding observational over-fitting [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' Similarly, the contrastive loss helps in learning class-invariant features from augmented images by contrasting them against a batch of negative samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' In that sense, these three loss functions contribute complementary information and hence, should improve the feature representation learning when combined together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' In order to implement feature training with this loss function, a new architecture inspired by CURL [10], is proposed that uses a Siamese Twin encoder model, a decoder network and a feature predictor to generate these losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' Through empirical analysis including feature visualizations, it is shown that the feature representations learnt by the CRC loss function is different from those learnt with the baseline CURL model, indicating the role played by the CRC-loss in learning new action-dependent features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' In addition, visual- ization of correlation matrices between latent features generated by this model show increasingly complex patterns for complex environments with higher-dimensional action spaces, thereby providing a clue about how features are inherently linked with action in an end-to-end RL model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' Through rigorous experiments on the challenging Deep Mind Control suite environments [29], it is shown that the proposed CRC-RL model outperforms the existing state- of-the-art methods by a significant margin, thereby establishing the efficacy of the approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' The design choices for the proposed model are justified through extensive ablation studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' In short, the major contributions made in this paper are as follows: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' A new self-supervised feature representation architecture along with a novel loss function is proposed to im- prove the performance of RL models in learning policies directly from image observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' Through empirical analysis involving latent feature visualization, an attempt has been made to provide insights into the relationship between the action and feature space thereby providing better standing of the role played of the new loss function in learning action-dependent features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' The resulting architecture is shown to outperform existing state-of-the-art methods on the challenging DMC environments by a significant margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' The rest of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' Related works are reviewed in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' The proposed architec- ture is described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' The experimental results are discussed and analyzed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' The paper ends with the concluding section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' Related Works This section provides an overview of related literature in the following subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' Deep RL architectures for policy learning Reinforcement learning algorithms learn the optimal policy for a given task by maximizing a measure of cumula- tive discounted future reward for the task while balancing between exploration (of new possibilities) and exploitation (of post experiences) [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' This cumulative discounted reward function, represented as Q or value function, is not 2 / 00 (2023) 1–13 3 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' CRC-RL Architecture: It consists of a Siamese Twin encoder along with a decoder and a feature predictor network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' The query encoder together with the decoder forms an auto-encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' The query encoder is used for learning policy using SAC algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' Observations are data- augmented to form query and key observations, which are then encoded into latent features by the respective encoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' Only the query encoder weights are updated during the training step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' The weights of key encoder are exponential moving average of query encoder weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' known a priori and, is used to evaluate a given action taken by the agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' Depending on how this function is estimated and desirable actions are derived from it, the RL-based methods can be broadly classified into two categories: value- based methods and policy-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' The value-based methods aim at estimating the Q-function and then derive action from this by using a greedy policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' On the other hand, policy- based methods directly estimate the policy function by maximizing a given objective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' The traditional Q-learning algorithm estimates the Q function iteratively by using an approximate dynamic programming formulation based on Bellman’s equation starting from an initial estimate [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' The original Q-learning algorithm could be applied to problems with discrete state (observation) and action spaces and hence, suffered from the curse-of-dimensionality problem with higher dimensions and range of values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' This limitation is overcome by using a deep network to estimate Q function that can take arbitrary observation inputs, thereby, greatly enhancing the capabilities of RL algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' The resulting approach is known as Deep Q Networks (DQN) [32] [33] which has been applied successfully to a wide range of problems while achieving super- human level performances in a few cases, such as ATARI video games [34], Go [35] etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' The success of DQN has spawned a new research field known as deep reinforcement learning (DRL) attracting a large following of researchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' Readers are referred to [36] for a survey of this field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' The DQN models were subsequently extended to continuous action spaces by using policy gradient methods that used a parameterized policy function to maximize DQN output using gradient ascent methods [37] [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' This has opened the doors for solving various robotics problems that use continuous values such as joint angles, joint velocities or motor torques as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' Since then, a number of methods have been proposed to improve the performance of RL algorithms and have been applied successfully to different robotic problems - manipulation [39] [40], grasping [41] [42], navigation [43] etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' Some of the notable methods include actor-critic models - A2C and A3C [44], soft actor-critic (SAC) [45] and proximal policy optimization (PPO) [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' In spite of the success of these methods, (deep) reinforcement learning algorithms, in general, suffer from limi- tations such as poor sampling efficiency leading to longer training time, poor generalization and instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' The work presented in this paper aims to address some of these concerns by focusing on learning better task-relevant features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' Feature Representation Learning in RL It is now widely accepted that learning policies from low-dimensional feature vectors based on physical states is much more sample efficient compared to learning directly from image pixels [10] [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' Hence, it is imperative to learn suitable state representations from image observations that will reduce the search space thereby improving the sample efficiency and stability of RL algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' The field of self-supervised representation learning has seen great 3 Siamese Twins Contrastive Loss S > Key Feature Consistency Encoder Predictor Loss Data Augmentation Query Encoder Decoder Reconstruction Loss S a Stack of input Actor frames SAC Forward Pass Training Critic Q(s,a) Weight sharing/ 00 (2023) 1–13 4 progress in last few years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' Auto-encoders [19] [48] learn the state representation by compressing the observation into low-dimensional state that is sufficient to reconstruct the observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' These have been used to improve the performance of RL algorithms as demonstrated in [17][49] [50] [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' On the other hand, contrastive learning [22] [51] learns the class-relevant feature representations by maximizing the agreement between the augmented versions of the same observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' It has been shown to greatly improve the sample efficiency of RL algorithms as in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' Similarly, recent studies have shown that the right kind of data augmentation techniques can improve the sample efficiency and generalization capabilities of RL algorithms learning task-relevant features which remain unaffected by distractions introduced by the augmentation [9] [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' This can be further enforced by making the encoder minimize the consistency loss as suggested in [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' In short, learning suitable feature representation plays a significant role in improving the performance of RL algorithms by increasing sample efficiency, improving generalization and stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' The work presented in this paper contributes to this field by proposing a novel loss function that leads to superior learning performance for continuous control tasks as will be demonstrated later in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' Method This section provides details of the proposed CRC-RL model that uses a novel heterogeneous loss function to ex- tract useful information from visual images to be used for learning optimal policy using an end-to-end RL framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' The discussion is organized in the following subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' The architecture of the proposed model is described next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' The Model Architecture The overall architecture of the proposed model is shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' The observation is available in the form of images which are stacked together to act as input to the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' Stacking of frames is a heuristic approach to incorporate temporal information in the learning process [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' The observations obtained from the environment is stored in a replay buffer D and a batch is sampled from this replay buffer during the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' A Siamese Twin encoder model is employed for extracting features from the input images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' These two encoders, termed as query and key encoders, are used for computing contrastive and consistency losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' The query encoder with a decoder is used for computing the reconstruction loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' A combination of these three losses, known as the CRC loss, is used for updating the parameters of the query encoder and decoder network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' The input images are augmented before applying to the key encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' The features obtained from the query encoder is used for policy estimation using soft-actor-critic algorithm [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' The parameters of the query encoder and decoder networks are updated using error signals obtained from their own outputs as well as from the RL algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' Since the encoder networks are getting influenced by the RL policy algorithm, the features learnt in the process are action-dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' This aspect will be analyzed in more detail in the experiment section presented later in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' The weights of the key encoder network is the exponential moving average of the query encoder weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' The proposed CRC loss function used for learning the feature embeddings is discussed in the next subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' The loss function for feature extraction The query encoder is trained using the proposed CRC loss function which is a combination of the following three loss components as described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' Contrastive loss In contrastive learning, we have a query q observation and a set of key observation samples K = {k0, k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content='} consisting of both positive samples (k+) and negative samples (K \\ k+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' The positive samples are those that belong to the same class as that of the query observation and the rest are considered to be the negative samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' The goal is to learn embeddings such that q is relatively more similar to the positive keys k+ than the negative keys in the latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' The query and key observations, generated by applying data augmentation on sampled observations, are encoded using the query and key encoder respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' The contrastive loss depends on the output of both the encoders (Siamese Twin) represented by the symbols s and s′ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' The idea behind using the contrastive loss is that the different augmentations of the same image will have the same underlying information and hence their high-level representations will be mapped together in the latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' The similarity between the query and the key embeddings is computed using the bilinear inner-product qTWk > 0 where W is a symmetric matrix of parameters to be estimated [54] along 4 / 00 (2023) 1–13 5 with other parameters during the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' The objective of training is to reduce this similarity measure so that the query embeddings become more distinct from the key embeddings over time (qTWk ≈ 0 ⇒ q ⊥ k, W > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' This is achieved by minimizing the InfoNCE loss [55] given by: Lq = log exp(qTWk+) � ki∈K exp(qTWki) (1) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' Reconstruction loss A well-trained encoder-decoder network is expected to reconstruct the input image at the output of the decoder network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' The reconstruction loss is computed based on the inaccuracy in the reconstructed image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' A convolutional encoder fθ maps an input observation x ∈ Rm×n×3 to a lower-dimensional latent vector s ∈ Rl, and a deconvolutional decoder gφ then reconstructs s back to ˆx ∈ Rm×n×3 such that fθ : x → s (2) gφ : s → ˆx (3) Both the encoders and decoder are trained simultaneously by maximizing the expected log-likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' The recon- struction loss checks how well the image has been reconstructed from the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' The reconstruction loss forces the update such that the latent representation preserves the core attributes of the input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' An L2 penalty is imposed on the learned representation s and a weight-decay is imposed on the decoder parameters to incorporate the regularization affects as proposed in [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' Lr = Ex∼D[log pθ(x|s) + λs∥s∥2 + λθ∥θ∥2] (4) where λs, and λθ are hyper-parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' Consistency loss The consistency loss depends on the output of both the query and key encoder fθ and f ′ θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' Here, the query encoder takes the original non-augmented observation x and the key encoder uses the augmented observation ˜x as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' The output of the Key encoder s′ is then used as an input to a feature predictor module, which is nothing but an MLP, to estimate the non-augmented embedding ˆs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' The consistency loss is designed to minimize the error between the non-augmented embedding s and the augmented embedding s′, thereby enabling the encoder to learn essential task- relevant features while ignoring irrelevant distractions (such as background clutter or texture).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' This eliminates the need of using negative samples for the computation of consistency loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' The consistency loss function can, therefore, be mathematically written as: Lc(ˆs, s, θ) = Ex∼D[∥ˆs − s∥2] (5) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' The CRC loss function It is our conjecture that each of the above three loss functions enables the encoder to extract non-redundant and complementary information from the higher dimensional input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' Thus, a combination of these three should improve the overall RL performance in learning optimum policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' The resulting loss function, called CRC loss, has the following mathematical form: LCRC = c1Lq + c2Lr + c3Lc (6) where ci > 0, � i ci = 1, i = 1, 2, 3 are hyper-parameters that control the relative importance of individual components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' The RL model for policy learning takes the query encoder output as its input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' The SAC algorithm used for learning policy is also allowed to affect the query encoder weights fθ during the backward gradient update step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' At regular intervals, the key encoder f ′ θ weights are updated using the exponential moving average (EMA) of the weights of the query encoder fθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' The feature learning and policy learning takes place in jointly in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' The latent representations learned by the query encoder fθ receives gradients from both the CRC loss and the SAC algorithm losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' This makes the feature representations action-dependent, an aspect which will be analyzed in some more detail in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' SAC algorithm is given in Algorithm 1 for the reference of the readers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' 5 / 00 (2023) 1–13 6 Algorithm 1 SOFT Actor-Critic Algorithm 1: Input: Initial policy parameters θ, Q-function parameters φ1, φ2, empty replay buffer D 2: Set target parameters equal to main parameters φtarg,1 ← φ1, φtarg,2 ← φ2 3: repeat ▷ Observe state s and select action 4: 5: a ∼ πθ(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content='|s) 6: Execute a in the environment 7: Observe next state s′ reward r, and done signal d to indicate whether s′ is terminal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' 8: Store (s, a, r, s′, d) in replay buffer D 9: 10: if s′ is terminal, reset environment state then 11: it’s time to update then 12: for j in range (however many updates) do 13: Randomly sample a batch of transitions, B = (s, a, r, s′, d) from D 14: Compute targets for the Q function y(r, s′, d) = r + γ(1 − d)(min i=1,2 Qφtarg,i(s′ − ˜a′) − α log πθ( ˜a′|s′)), ˜a′ ∼ πθ(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content='|s′) 15: Update Q-function by one step of gradient descent using ▽φi 1 | B | � (s,a,r,s′,d)∈B (Qφi(s, a) − γ(r, s′, d))2 for i= 1, 2 16: Update policy by one step of gradient ascent using ▽φ 1 | B | � s∈B (min i=1,2 Qφi(s, ˜aθ(s)) − α log πθ( ˜aθ(s)|s)) where, ˜aθ(s) is a sample from πθ(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content='|s) which is differentiable w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content='t θ via the re-parameterization trick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' 17: Update target networks with φtarg,i ← ρφtarg,i + (1 − ρ)φi for i = 1, 2 18: end for 19: end if 20: until Convergence 6 / 00 (2023) 1–13 7 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' Hyper-parameters used for DMControl experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' Most hyper-parameters values are unchanged across environments with the exception for action repeat, learning rate, and batch size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' Hyper-parameter Value Pre transform image size (100, 100) Image size (84, 84) Action repeat 8 Frame stack 3 Transform Random crop Replay buffer capacity 100000 Initial steps 1000 Batch size 512 Hidden layers 1024 Evaluation episodes 10 Optimizer Adam Learning rate ( fθ, πψ, Qφ) 1e-3 Learning rate (α) 1e-4 Critic target update frequency 2 Convolution layers 4 Number of filters 32 Latent dimension 50 Discount (γ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content='99 Initial temperature 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content='1 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' Mean episodic reward (with standard deviation) over 10 evaluation runs on DMControl environments after training for 100k environment steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' The best scores are shown in bold letters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' 100K Step Our Method CURL [10] SODA [23] PLANET [59] DREAMER [60] SAC+AE [17] PIXEL STATE % Increase Scores SAC [45] SAC [45] over CURL FINGER, SPIN 793±36 767±56 363±185 136±216 341±70 740±64 179±66 811±46 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content='38 CARTPOLE, SWINGUP 813±45 582±146 474±143 297±39 326±27 311±11 419±40 835±22 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content='6 REACHER, EASY 636±301 538±233 20±50 314±155 274±14 145±30 746±25 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content='2 CHEETAH, RUN 355±31 299±48 138±88 235±137 267±24 197±15 616±18 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content='7 WALKER, WALK 490±52 403±24 635±48 224±48 277±12 394±22 42±12 891±82 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content='5 BALL IN CUP, CATCH 832±81 769±43 539±111 0±0 246±174 391±82 312±63 746±91 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content='19 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' Experimental Results and Discussions The proposed CRC-RL model architecture takes its inspiration from the original CURL implementation by Laskin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' The original model is extended by incorporating additional decoder and feature predictor to facilitate computing the CRC loss function as described in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' The model is implemented using PyTorch [57] deep learning framework and the source code is made available on GitHub https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content='com/darshitajain/CRC-RL- for the convenience of readers and facilitate reproduction of results furnished in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' The reinforcement learning framework for policy estimation makes use of the publicly released implementation of the SAC algorithm by Yarats et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' The query encoder and decoder architecture is similar to the ones used in the above work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' The query encoder weights are tied between the actor and critic so that they both use the same encoder to embed input image observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' The feature predictor module is a MLP network which consists of cascaded linear layers and ReLU activation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' The complete list of hyper-parameters is shown in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' A number of experiments are carried out to establish the efficacy of the proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' The design choices are justified through several ablation studies as discussed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' Performance Comparison The performance of the proposed CRC-RL model is compared with the current state-of-the-art methods on the challenging Deep mind control suite (DMControl) environments [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' The outcome is shown in Table 4 and 3 after 7 / 00 (2023) 1–13 8 CURL CRC-RL (a) Cartpole-Swingup (b) Cheetah-Run (c) Walker-walk Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' t-SNE visualization of latent feature embeddings obtained from query encoder at 49K training steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' Colors correspond to cluster labels in the action space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' One can observe that CRC-RL leads to more pristine clusters with less outliers compared to CURL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' training for 100K and 500K environment steps respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' It can be observed that the proposed CRC-RL model out- performs the current state-of-the-art methods, such as CURL [10], SODA [23], PlaNet [59], Dreamer [60], SAC+AE [17], pixel-based SAC [45] on most of the DMControl environments, thereby establishing the superiority of our ap- proach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' The environments shown in Table 3 are difficult compared to those shown in Table 4 and hence require longer training time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' In this case, our proposed model outperforms the baseline CURL model in only 300K training steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' Mean episodic score (with standard deviation) for 10 evaluation runs on DMControl environments obtained after training for 500k environment steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' The best scores are shown in bold letters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' Environment Our Method* CURL % Increase over CURL QUADRUPED, WALK 88 ± 51 39±22 125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content='6 HOPPER, HOP 61 ± 33 10±17 510 WALKER, RUN 306 ± 5 245±32 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content='8 FINGER TURN, HARD 423 ± 78 207±32 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content='3 shows values for 300K training steps 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' t-SNE Visualizations To better understand the relationship between the learned latent representations and the action generated by the RL policy, we generate the two-dimensional t-SNE plots of feature embeddings obtained from the query encoder for 3 different environments as shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' These features are assigned with the corresponding action labels gener- ated by partitioning the action space into five clusters by using the k-mean clustering algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' As one can observe, the proposed CRC-RL model leads to more pristine clusters with lesser outliers compared to the baseline CURL [10] algorithm for some amount of training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' Compared to the Cartpole environment, other two are comparatively more complex and require larger amount of time for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' This shows that the proposed model leads to better correlation between the feature embeddings and agent actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' This aspect has not been empirically investigated extensively in the existing literature and thus, the current work makes a novel contribution by filling this void.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' Feature Correlation Heat Maps Another study is performed to validate our hypothesis that the proposed CRC loss contribute new information resulting in learning new feature representations which are distinct from those obtained using individual losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' In this study, the correlation matrices between the latent features obtained with the baseline CURL algorithm (that uses 8 15 0 1 2 10 3 4 5 0 5 10 10 5 0 5 1020 0 1 2 15 3 4 10 5 0 5 10 10 0 5 10 1510 0 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' 3 5 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' 0 5 10 15 10 5 0 5 10 1515 0 2 10 3 4 5 00 0 - 5 心 10 10 5 0 5 107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content='0 2 m 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content='5 4 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content='0 10 5 0 5 1015 0 1 2 3 10 4 5 0 5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' 10 15 10 5 0 5 10/ 00 (2023) 1–13 9 (a) (b) (c) Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' Effect of incorporating various loss components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' CRC loss function performs better than other combinations for more difficult environments such as ’Cheetah-Run’ and ’Walker-walk’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' (a) Cartpole-Swingup (b) Cheetah-Run (c) Walker-walk Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' Feature correlation heat-maps for three environments showing the correlation between the latent features obtained with CURL and CRC-RL models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' The models are trained for 49K environment steps and 200 latent features are used to generate this plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' contrastive loss) and that with the proposed CRC-RL model (that uses CRC loss) is plotted as heat-maps as shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' These matrices are generated by collecting 200 sample embeddings from both models trained with 49000 environment steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' Since, each image sample is encoded into a 50 × 1 feature vector, the features generated by the above two methods are grouped into two feature matrices (say, F1 and F2) of size 200 × 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' The correlation between these two feature matrices results in a 400x400 matrix is visualized as a heat map in the above figure where the darker regions shows higher correlation and lighter regions show lower correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' It is observed that off-diagonal regions have lower correlation (lighter regions) indicating that the two feature embeddings (F1 and F2 ) are very distinct from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' The diagonal regions are highly correlated (darker regions) as they correspond to the features from the same method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' Another interesting finding of this study is that these heat maps show increasingly complex patterns for difficult environments such as ’Walker-walk’ or ’Cheetah-Run’ compared to simpler environments such as ’Cartpole- Swingup’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' These patterns evolve over time and stabilize as the training performance saturates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' This is an interesting insight that may provide clue to better understand the relationship between features and action policies learned in an end-to-end RL framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' Ablation Study Three separate ablation studies are carried out to justify the design choices made in this paper as described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' 9 Cartpole, Swingup Mean episodic reward 800 600 400 200 Environment Step 10k 20k 30k 40k 0Cheetah, Run 400 Mean episodic reward 300 200 100 Environment Step 20k 40k 60k 80k 0Walker, Walk Mean episodic reward 500 400 300 200 100 Environment Step 10k 20k 30k 40k 0CURL+Consistency CURL CRC-RL CURL+Reconstruction1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content='8 112 128 144 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content='6 160 176 192 208 0.' metadata={'source': 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208 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content='2 224 240 256 272 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content='0 288 304 320 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content='2 336 352 368 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content='4 384 05 m1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content='6 88 96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content='4 104 112 120 128 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content='2 136 144 152 160 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content='0 168 176 184 192 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content='2 4 6 4 00 80/ 00 (2023) 1–13 10 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' Usefulness of CRC loss function First study validates the usefulness of the proposed CRC loss function comprising of contrastive, reconstruction and consistency losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' The outcome is shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' We start with the baseline CURL model [10] that uses contrastive loss to learn the feature presentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' Then this model is trained with a combined loss function of contrastive and reconstruction loss and finally, with the CRC loss function comprising of contrastive, reconstruction and consistency losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' The inclusion of these losses require modifying the existing CURL model leading to the formation of the CRC-RL model proposed in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' The figure shows that CRC-RL performs better than the other two for the benchmark problems from the Deepmind control suite (DMC), namely, ‘Cheetah-Run’ or ‘Walker- walk’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' These two are comparatively difficult problems to solve compared to simpler problems such as the ‘Cartpole- Swingup’ problem which does not benefit from the proposed CRC-RL model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' This observation clearly establishes the usefulness of the proposed approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' This becomes more evident in the second ablation study discussed in the next section below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' Relative weights of loss components in the CRC loss function In this study, the relative weights of three loss components, namely, contrastive, reconstruction and consistency loss, are varied and its effect on the validation performance is compared as shown in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' The weights are varied with the constraint of forming a convex sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' In other words, ci > 0, i ∈ 1, 2, 3 and � i ci = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' The figure shows that the individual loss functions do not always provide the best performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' The best performance is obtained by a combination of all the three losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' Having equal weights (c1 = c2 = c3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content='33) for all the three losses have a regularizing effect on model performance in the sense that the performance is traded for better generalizability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' This is evident from the fact that the validation performance curve with this combination of weights lie somewhere in the middle of the all the curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' The generalization capability of the proposed model is demonstrated in the third ablation study discussed next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' Generalization Capability of the CRC-RL model In order to test the generalization capabilities of the proposed CRC-RL model, another experiment is performed where the RL models are trained on images augmented with random crop effect and then validated on images aug- mented with Video-Easy and Color-Hard artifacts [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' The outcome is shown in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' Comparing with the validation plots in Figure 5, it is observed that the overall performance has come down significantly due to these com- plex augmentations which makes it difficult for the model to generalize when trained only with images augmented with only random crop artefact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' However, even in this case, the RL model trained with CRC loss function provides the best or closer-to-the-best evaluation performance compared to the models that use individual loss components for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' This demonstrates the superior generalization capability of the proposed CRC-RL model over the existing models that use one of the above loss functions for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' It also corroborates the earlier finding mentioned in the previous subsection that the equal weights for individual loss components in the CRC loss function have a regularizing effect on the model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' Conclusions The paper addresses the problem of feature representation learning in end-to-end reinforcement learning models with visual observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' Specifically, a new loss function, called CRC loss, is proposed to learn action-dependent features leading to superior performance in learning optimal action policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' This loss function is a combination of three different loss functions, namely, the image reconstruction loss, the contrastive loss and the consistency loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' Through empirical analysis including latent feature visualization, an attempt is made to generate new insights that can better explain the relationship between the features being learnt and the actions being taken by the RL agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' The resulting architecture is shown to outperform the existing state-of-the-art methods in solving the challenging DMC problems by a significant margin thereby forming a new benchmark in this field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' The future work will involve carrying out more in-depth analysis and evaluation of the individual loss components on the overall performance as well as on the quality of features being learned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' 10 / 00 (2023) 1–13 11 (a) (b) (c) Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' Effect of varying weighting parameters for different loss functions on the Evaluation performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' c1, c2 and c3 are the weights to the contrastive loss, the reconstruction loss and the consistency loss respectively in the CRC loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' The environments used are: (a) Cartpole-Swingup, (b) Cheetah-Run and (c) Walker-walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' Varying these parameters have a regularizing effect on the training performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' A smoothing factor of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content='5 is applied to the plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' (a) (b) (c) (d) Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' Generalization capabilities of CRC-RL algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' c1, c2 and c3 are the weights to the contrastive loss, the reconstruction loss and the consistency loss respectively in the CRC loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' The environments used are: (a) Cartpole-Swingup, (b) Cheetah- Run and (c) Ball-in-Cup-Catch and (d) Walker-walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' The RL models are trained on images with random-crop augmentation and evaluated on images with Video-Easy and Color-Hard augmentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' Compared to the individual losses, CRC loss provide best or second-best evaluation performance for these new augmentations thereby establishing the superior generalization capabilities of the CRC-RL model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content=' 11 Cartpole-Swingup Showing first 1o runs c1: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQfDjfR/content/2301.13473v1.pdf'} +page_content='33， c2: 0.' metadata={'source': 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a/PtFQT4oBgHgl3EQfYzaZ/content/tmp_files/2301.13313v1.pdf.txt b/PtFQT4oBgHgl3EQfYzaZ/content/tmp_files/2301.13313v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..0d90244c23140ff4902b215b3a450be1edf606ef --- /dev/null +++ b/PtFQT4oBgHgl3EQfYzaZ/content/tmp_files/2301.13313v1.pdf.txt @@ -0,0 +1,862 @@ +Proceedings of Machine Learning Research vol XX:1–12, 2023 +Incorporating Recurrent Reinforcement Learning into Model +Predictive Control for Adaptive Control in Autonomous Driving +Yuan Zhang +YZHANG@CS.UNI-FREIBURG.DE +Joschka Boedecker +JBOEDECK@CS.UNI-FREIBURG.DE +Neurorobotics Lab, University of Freiburg, Germany +Chuxuan Li +LICHUXUAN@AIR.TSINGHUA.EDU.CN +Guyue Zhou +ZHOUGUYUE@AIR.TSINGHUA.EDU.CN +Institute for AI Industry Research (AIR), Tsinghua University, China +Abstract +Model Predictive Control (MPC) is attracting tremendous attention in the autonomous driving task +as a powerful control technique. The success of an MPC controller strongly depends on an accurate +internal dynamics model. However, the static parameters, usually learned by system identification, +often fail to adapt to both internal and external perturbations in real-world scenarios. In this paper, +we firstly (1) reformulate the problem as a Partially Observed Markov Decision Process (POMDP) +that absorbs the uncertainties into observations and maintains Markov property into hidden states; +and (2) learn a recurrent policy continually adapting the parameters of the dynamics model via +Recurrent Reinforcement Learning (RRL) for optimal and adaptive control; and (3) finally eval- +uate the proposed algorithm (referred as MPC-RRL) in CARLA simulator and leading to robust +behaviours under a wide range of perturbations. +Keywords: Autonomous Driving, Model Predictive Control, Recurrent Neural Network, Rein- +forcement Learning +1. INTRODUCTION +Model Predictive Control (MPC) has become the primary control method for enormous fields, e.g. +autonomous driving (Reiter et al.) and robotics (Song and Scaramuzza, 2020). As a model-based +method, MPC largely depends on an accurate dynamics model of the system, xk+1 = f(xk, uk; θ), +where x, u represent state and control respectively and θ is the parameter of the model. The parame- +ter θ is assumed to be determined by prior knowledge or system identification method that learns the +parameter from a collection of experience. However, the awareness of parameter θ consistently fails +due to the perturbations emerging from all sources in the autonomous driving task. In detail, both +internal (e.g. car mass, drag coefficient) and external (e.g. road friction, planning route) parameters +may vary in the driving process. Also, the accurate values of the parameters are difficult to collect. +Therefore, an MPC controller with a fixed parameter θ may degenerate the control performance in +the autonomous driving task. +Learning-based MPC is receiving increasing attention as it focuses on automatic adaption to +varying environmental parameters (Hewing et al., 2020; Spielberg et al., 2021). The standard ap- +proach is to incorporate an additional module which aims to tune the parameters of the MPC for +optimal control. Bayesian Optimization (BO) and Reinforcement Learning (RL) are two significant +mythologies to learn this module. BO aims at optimizing the closed-loop cost J(θ) and generates +© 2023 Y. Zhang, J. Boedecker, C. Li & G. Zhou. +arXiv:2301.13313v1 [cs.LG] 30 Jan 2023 + +MODEL PREDICTIVE CONTROL - RECURRENT REINFORCEMENT LEARNING +the most suitable parameters of an MPC. However, the update of parameters is considerably slow +to react in a dynamically changing environment. RL is capable of modifying the parameter at each +time step. However, the perturbation in the dynamics is non-stationary, thus violating the Markov +property required in RL theory. +This paper follows the RL method to boost the MPC controller’s adaptability. We firstly re- +formulate the problem as a Partially Observed Markov Decision Process (POMDP) to ease the +non-stationary property under environmental perturbations. The original state in MPC is viewed as +an observation in the POMDP formulation. Meanwhile, a hidden state represents the system’s ac- +tual state, including the perturbation information. We add a recurrent policy on top of the MPC and +facilitate its learning with 2 objectives: cumulative reward maximization and system identification +loss. The whole system achieves optimal and adaptive control in autonomous driving simulations +compared with pure MPC and RL methods. +(a) The kinematic bicycle model (Kong et al., +2015). +(b) CARLA simulator. +Figure 1: Model and simulation in the autonomous driving task. +2. RELATED WORK +As mentioned in Section 1, Bayesian Optimization (BO) is a parallel approach to improve the pa- +rameters of an MPC. Marco et al. (2016) adopts an LQR formulation and learns the parameters of +Q,R matrix with entropy search method. Bansal et al. (2017) also utilizes an LQR model but learns +the parameters of the transition function to control a quadcopter. Both methods update the parame- +ters θ at the end of an episode as it requires evaluating the cost J(θ), which leads to slow responses +in case of environmental perturbations. +Comparatively, Reinforcement Learning (RL) can modify MPC parameters at each time step to +quickly adapt to dynamic environments. Zarrouki et al. (2021) and Song and Scaramuzza (2020) +both learn a policy that can improve parameters of MPC’s cost function, while Gros and Zanon +(2020); Amos et al. (2018) aim to modify both transition and cost function’s parameters. Neverthe- +less, none of these methods consider the non-stationary scenario in an autonomous driving system, +which violates the essential Markov property in RL. +There are other research directions to combine RL with MPC. Brito et al. (2021) utilizes RL +to learn a sub-goal so as to reduce the optimization horizon of the MPC. Farshidian et al. (2019); +2 + +Y.O +y +r=0 +C +XMODEL PREDICTIVE CONTROL - RECURRENT REINFORCEMENT LEARNING +Zhong et al. (2013) replace the terminal cost in MPC by the value function in RL to avoid inaccurate +human-designed objectives. While previous works focus on improving the efficiency of MPC with +the aid of RL, in our work we are more interested in ensuring adaption to perturbed environments. +3. PRELIMINARIES +3.1. Vehicle Dynamics Modelling +The kinematic bicycle model (Kong et al., 2015) is a simplified vehicle model targeted for au- +tonomous vehicles, whose continuous time equation is +� +����� +˙p +˙q +˙ψ +˙v +β +� +����� += +� +������ +v cos(ψ + β) +v sin(ψ + β) +v +lr sin β +a +tan−1 � +lr +lf+lr tan δf +� +� +������ +(1) +where state x = [a, b, ψ, v, β] includes a, b: the coordinates of the mass center, ψ: the head- +ing angle of the vehicle, v: the speed at the mass center and β: the angle of the current velocity +w.r.t. the longitudinal axis of the vehicle; control u = [a, δf] consists of a: the acceleration at +the mass center and δf: the steering angle of the front wheel. Other than that, lr and lf are the +vehicle’s inertial parameters. An intuitive illustration of the kinematic bicycle model can be seen in +Figure 1(a)subfigure. +However, such a model cannot be directly adopted in existing autonomous driving platform +(e.g. CARLA (Dosovitskiy) and APOLLO (Gao et al., 2022)) where the output control u normally +consists of steering st, throttle th, brake br instead of acceleration a and angle δf. Xia (2019) +suggests utilizing a neural network to represent the dynamics, which has the following form +˙x = +� +����� +˙p +˙q +˙ψ +˙v +˙β +� +����� += +� +����� +v cos(ψ + β) +v sin(ψ + β) +f0(v, β; θ) +f1(v, β, st, th, br; θ) +f2(v, β, st, th, br; θ) +� +����� += f(x, u; θ), +(2) +where f0, f1, f2 are neural networks parameterized with θ, which can be approximated by sys- +tem identification methods. Furthermore, the discrete dynamic function becomes xt+1 = xt + +f(xt, ut; θ)∆t, with the state xt = [pt, qt, ψt, vt, βt]T and the control ut = [stt, tht, brt]T . The +control interval ∆t equals 0.1 seconds in the experiments. +3.2. Model Predictive Control Formulation +Model Predictive Control (MPC) is an advanced optimization method for non-linear optimal control +problems with constraints. To control an autonomous vehicle moving along a reference trajectory +G = (g1, g2, ..., g|G|) (gi are waypoints on the trajectory) with a target speed V, an MPC problem +can be formulated as follows: +3 + +MODEL PREDICTIVE CONTROL - RECURRENT REINFORCEMENT LEARNING +min +x,u +lH(xH, G) + +H−1 +� +t=0 +l(xt, ut, G, V) +s.t. +∀t, +xt+1 = xt + f(xt, ut; θ)∆t +− 1 ≤ stt = ut[0] ≤ 1 +0 ≤ tht = ut[1] ≤ 1 +0 ≤ brt = ut[2] ≤ 1 +x0 = xinit +(3) +where x = (x0, ..., xH) and u = (u0, ..., uH−1) represent the state and control sequences to +be optimized respectively, and xinit is the initial state of the autonomous vehicle. The stage cost +function l(xt, ut, G, V) = cposition×D(xt, G)+cspeed×(xt[3]−V)2+ccontrol×(ut[0]2+ut[1]2+ +ut[2]2 + ut[1] × ut[2]), where D(xt, G) = mink∈{1,2,...,|G|} ∥xt[: 2] − gk)∥2 is the distance function +to the nearest waypoint. The terminal cost function lH(xH, G) = ∥xH[: 2] − g|G|)∥2. Among +them, cposition, cspeed, ccontrol are coefficients to balance different parts in the cost function and set +to 0.04, 0.002, 0.0005 in the experiments. To solve this non-linear MPC problem efficiently, the +iLQR method (Li, 2004) can be applied accordingly. +3.3. Partially Observed Markov Decision Process (POMDP) +A Partially Observable Markov Decision Process (POMDP) is a generalized mathematical frame- +work of an MDP to deal with the unobserved state issue. +It is formulated as a 7-tuple +⟨S, A, P, r, Ω, O, γ⟩, where S, A and Ω stand for the state, action and observation space respec- +tively, and r(st, at) : S × A → R is the reward function at time step t. Define ∆|S|, ∆|A|, ∆|Ω| +be the probability measure on S, A and Ω respectively, then P(st+1|st, at) : S × A → ∆|S| is the +transition function, and the future rewards are discounted by the discount factor γ ∈ [0, 1]. The +most crucial concept in POMDP is that agents can only obtain the observation ot with probability +O(ot|st, at−1) : S × A → ∆|Ω|, instead of receiving the entire state st. +The received partial observation is not sufficient for agents to make decisions. Instead, the +agent needs to maintain a belief state bt(st) : ∆|S| (bt for short) to estimate a complete knowledge +of the system. There exists an update equation for the belief state given previous belief state bt−1, +action at−1 and new observation ot: bt = ηO(ot|st, at−1) � +st−1 P(st|st−1, at−1)bt−1, where η is +a normalization factor to ensure probability measure. The policy function in a POMDP is usually +defined as Π = +� +π(at|bt) : ∆|S| → ∆|A| +� +and the objective of this agent can be formulated as an +optimization problem, +J∗ = max +π∈Π Eπ,P,O +�+∞ +� +t=0 +γtr(st, at)|b0 +� +. +(4) +where b0 is an initial guess on the belief state. The expectation is on policy π, transition function P +and observation function O. The belief state bt can be iteratively calculated. +4 + +MODEL PREDICTIVE CONTROL - RECURRENT REINFORCEMENT LEARNING +!" +#" +$" +%" +#&'( = * #&, %&; $" +#-.-/ = #" +MPC Controller +t = 0 +!( +#( +$( +%( +#&'( = * #&, %&; $( +#-.-/ = #( +MPC Controller +t = 1 +!3 +#3 +$3 +%3 +#&'( = * #&, %&; $3 +#-.-/ = #3 +MPC Controller +t = 2 +5" +5( +53 +6($|9(!); ;) +Figure 2: A POMDP Formulation of autonomous driving with an MPC controller +4. MPC-RRL Framework +In this section, we will firstly reformulate the autonomous driving task with an MPC controller as +a POMDP problem. Furthermore, within this formulation, we learn a recurrent policy with RL to +pursue optimal and adaptive control, thus called MPC-RRL for short. +4.1. POMDP Formulation +As mentioned in Section 1, it is challenging for the original dynamic of the MPC (Equation 2) to +react to both internal and external variations of the system, thus decreasing the performance in real- +life scenarios. Comparatively, a POMDP Formulation can ease this problem. Viewing the state xt +in MPC’s dynamic function as an observation ot in a POMDP, one can maintain a hidden state st +in a POMDP that can reflect the perturbations in the current system. The observation distribution +O(xt|st) represents the probability of the MPC controller receiving state xt given hidden state st. +Furthermore, we model MPC’s system parameters θt as the output action in this POMDP. The +MPC controller calculates the control action ut given the state xt and the system parameters θt, +referred as ut = MPC(xt, θt). After executing ut on the environment, the hidden state transfers +to st+1 with probability P(st+1|st, ut) = P(st+1|st, MPC(xt, θt)) and reward rt = r(st, ut) is +returned. +The optimal action aims to modify the parameters θt at each time step so that MPC can achieve +an optimal control ut even under perturbed environments. As mentioned in Section 3.3, an agent +needs to maintain a belief state bt on the current hidden state st so that it can generate the action +θt to influence the MPC results by following the policy π(θt|bt). This whole POMDP formulation +allows the MPC controller to dynamically change the dynamics’ parameters, leading to an optimal +and adaptive control if the belief state bt and the action θt is appropriately generated. The overall +framework is illustrated in Figure 2. +5 + +MODEL PREDICTIVE CONTROL - RECURRENT REINFORCEMENT LEARNING +4.2. Recurrent Policy +In this section, we will focus on the most essential sub-modules in the framework, belief state bt +and neural policy π(θt|bt), on how they are represented, learned and deployed. In general, these +two modules are combined in one single recurrent neural network and learned with 2 objectives +for optimality and adaptability respectively. The combination is referred as recurrent policy for +simplicity. +4.2.1. POLICY REPRESENTATION +We leverage neural networks to represent both belief state and neural policy, with parameters τ and +ω respectively. A belief tracker bt = F(bt−1, xt; τ) is designed to update the belief state given new +observation xt at each time step, while the neural policy is represented by π(θt|bt; ω). Following the +general practice in POMDP problem (Wang et al., 2017; Hausknecht and Stone, 2017), these two +networks can be combined into one recurrent neural network. Recurrent neural network (Hochreiter +and Schmidhuber, 1997) possess a general representation of bt, θt = R(xt, bt−1; λ), which perfectly +absorbs F and π in one network R with parameter λ. The whole network can be optimized and +utilized altogether (Young et al., 2012). +4.2.2. TRAINING DETAILS +As introduced in last section, the recurrent policy R with parameters λ is the only module to be +learned. We design two learning objectives focusing on (i) encouraging MPC to generate optimal +control sequences; (ii) adapting the transition model to realities separately. +The first objective is to maximize the discounted cumulative reward, as introduced in Equation 4, +J1 = max +λ +Eπ,P,O +�+∞ +� +t=0 +γtr(st, θt)|b0 +� +. +(5) +This objective encourages the neural policy to output the parameters beneficial to the control be- +haviours with respect to a higher return. +The second objective is inspired by the system identification loss in control theory, +J2 = min +λ E +�+∞ +� +t=0 +[xt + f(xt, ut; θt)∆t − xt+1]2 +� +. +(6) +This objective pushes the policy to imitate the realistic dynamics’ parameters when it deviates from +the previous approximations. +Combining these two objectives by J = J1 + αJ2 (α is a coefficient to balance these 2 objec- +tives), we can successfully learn a policy aiming optimal and robust MPC control. In Section 5 we +will further illustrate the effectiveness of this method and the different roles of these 2 objectives. +Since the expectation in Equation 5 and Equation 6 is expensive to calculate directly, the RL area +usually interacts with the environments and learns from the sampling experience. In this paper, we +adopt PPO (Schulman et al., 2017) as the basic RL algorithm as it is one of the best performing +on-policy RL algorithms suitable to the training of dynamics belief states. +6 + +MODEL PREDICTIVE CONTROL - RECURRENT REINFORCEMENT LEARNING +4.2.3. INFERENCE DETAILS +The parameters λ are fixed during inference. Given an observation of the vehicle xt and the belief +state of last step bt−1, the agent directly generates the dynamics parameter θt and updates the belief +state bt by executing bt, θt = R(xt, bt−1; λ). The MPC controller utilizes the dynamics parameter θ +and initial state xt to generate the control value ut, which is further sent to the autonomous driving +environment. +5. Experiments +In this section, we evaluate our proposed MPC-RRL framework on the CARLA simulator. We first +show the superior performance with respect to goal error and route error in the main results and +further analyze the controller’s behaviours in the ablation study and policy study. +5.1. CARLA Simulator +We utilize CARLA simulator (Dosovitskiy) (See Figure 1(b)subfigure for a rendering example) to +evaluate autonomous driving performances of all baselines. CARLA simulator is a popular simu- +lation platform to train and evaluate different components of an autonomous driving system (per- +ception, planning, control, etc.). CARLA is grounded on Unreal Engine to run the simulation with +changeable configurations. Thus it is convenient to modify both internal (car mass, tire friction) and +external (planning route, road friction) factors of the system, which facilitates our evaluation. We +run CARLA in a synchronous mode, which ensures the reproducibility of the experiments but may +fail to simulate some real-world situations, e.g. missing observations and delayed control, which +will be studied in further research. +5.1.1. TASK SETUP +For each episode of the autonomous driving task, a starting and goal point is generated. The aim +of a control method is to reach the goal point following a reference trajectory. The episode ends if +the vehicle reaches the goal or experiences a collision. The performance is evaluated by the goal +error and the route error together. The goal error is defined by the distance between the vehicle’s +final position and the goal point, while the route error is calculated by the cumulative displacement +between the vehicle’s actual trajectory and the reference trajectory during the driving process. +We divide the self-driving task into training and testing phase. We first train the RL policy +described in Section 4.2.2 and freeze the policy’s parameters in the testing phase. To evaluate the +adaptability of each controller, we modify a system parameter (e.g. car’s final ratio, tire’s friction) +to a different value. A decent controller should be able to adapt to such model mismatches between +training and testing environments. The specific parameters for evaluation and their value during the +training and testing phase are in Table 1. Notably, for each parameter, we testify on 3 perturbed +values, with one value slightly different from the training value and the other two deviating largely. +5.1.2. BASELINES +MPC Controller +The MPC controller strictly follows the MPC formulation as described in Sec- +tion 3.2. Some practical implementation details should be considered to apply MPC on the CARLA +simulator, which will be further clarified in Section 5.1.3. +7 + +MODEL PREDICTIVE CONTROL - RECURRENT REINFORCEMENT LEARNING +Table 1: Verified parameters and their values during training and testing phase. +Parameter +Training Value +Testing Value +Explanation +final ratio +4.0 +2.0, 5.0, 10.0 +Transmission ratio from engine to wheels +moi +1.0 +0.4, 1.3, 1.9 +Moment of inertia of the vehicle +tire friction +3.5 +0.5, 2.25, 4.0 +Friction factor of all wheels +damping rate +0.25 +5e-3, 5e-1, 5e1 +Damping rate of all wheels +drag coefficient +0.15 +1e-4, 2e-1, 100 +Drag coefficient of the vehicle’s car body +town +Town01 +Town04, Town02, Town06 +Carla’s default town maps +MPC-RRL Controller +The MPC-RRL controller is precisely the framework introduced in Sec- +tion 4. For a fair comparison, we adopt the same MPC controller as Paragraph 5.1.2 as the based +controller in the framework. +RRL Controller +To fairly illustrate the power of combing RRL and MPC, we also add a baseline +with the recurrent policy only and trained with reinforcement learning methods. In this baseline, +the input of the policy is observation x, and the output is direct control action u, including steering, +throttle, and brake instead of MPC’s parameters in comparison with MPC-RRL. +5.1.3. PRACTICAL IMPLEMENTATION DETAILS +To apply the proposed MPC-RRL framework on the CARLA simulator, we explain some practical +implementations on both MPC and RL parts in this section. +Vehicle Dynamics Model +(i) Regarding the dynamics model in MPC, we utilize 3 neural +networks f0, f1, f2 parameterized with a 65-dim variable θ to represent it. +f0(v, β; θ) += +vsinβ +θ[0] +following the original format of the vehicle model (Equation 1) can predict the head- +ing angle ψ. +For f1, f2, we require them to have such properties: f1(v, β, st, th, br; θ) = +f1(v, −β, −st, th, br; θ), f2(v, β, st, th, br; θ) += +−f2(v, −β, −st, th, br; θ) so that the sym- +metry of velocity angle β and steering control st maintain. +To achieve that, we rewrite +f1 as [f +′ +1(v, β, st, th, br; θ) + f +′ +1(v, −β, −st, th, br; θ)]/2 and f2 as [f +′ +2(v, β, st, th, br; θ) − +f +′ +2(v, −β, −st, th, br; θ)]/2. Currently, only f +′ +1 and f +′ +2 are represented with parameters θ, which +is a 2-layer feed-forward neural network in our experiment; (ii) Notably, the prediction of the veloc- +ity v should always be positive, for which we further rewrite f1 as [f +′ +1(v, β, st, th, br; θ) × (2√v + +f +′ +1(v, β, st, th, br; θ))+f +′ +1(v, −β, −st, th, br; θ)×(2√v+f +′ +1(v, −β, −st, th, br; θ))]/(2∆t) so that +v + ∆t × f1(v, β, st, th, br; θ) ≥ 0 is always true; (iii) For system identification, we manually con- +trol the vehicles running in the CARLA simulator and collect 19000 transitions to train the f0, f1, f2 +models. The mean-squared-error loss and Adam optimizer (Kingma and Ba, 2017) are used to learn +the parameters θ. +Model Predictive Control +(i) As mentioned in Section 3.2, the stage cost function l(xt, ut, G, V) +is written as cposition × mink∈{1,2,...,|G|} ∥xt[: 2] − gk)∥2 + cspeed × (xt[3] − V)2 + ccontrol × +(ut[0]2 + ut[1]2 + ut[2]2 + ut[1] × ut[2]). The minimization term is not a perfect choice for +gradient-based optimization methods. +Instead, we utilize a differential function D(xt, G) = +− log +� |G| +� +k=1 +exp −∥xt[: 2] − gk)∥2 +� +as the distance function, which can be viewed as an approxi- +8 + +MODEL PREDICTIVE CONTROL - RECURRENT REINFORCEMENT LEARNING +2.0 +5.0 +10 +final_ratio +0 +10 +20 +30 +40 +50 +60 +70 +80 +goal_error (m) +0.4 +1.3 +1.9 +moi +0 +10 +20 +30 +40 +50 +60 +70 +80 +goal_error (m) +0.5 +2.25 +4.0 +tire_friction +0 +10 +20 +30 +40 +50 +60 +70 +80 +goal_error (m) +0.005 +0.5 +5.0 +damping_rate +0 +10 +20 +30 +40 +50 +60 +70 +80 +goal_error (m) +0.0001 +0.2 +100 +drag_coefficient +0 +10 +20 +30 +40 +50 +60 +70 +80 +goal_error (m) +Town04 +Town02 +Town06 +town +0 +10 +20 +30 +40 +50 +60 +70 +80 +goal_error (m) +2.0 +5.0 +10 +final_ratio +0 +10 +20 +30 +40 +50 +60 +70 +80 +route_error (m) +MPC +MPC-RRL +0.4 +1.3 +1.9 +moi +0 +10 +20 +30 +40 +50 +60 +70 +80 +route_error (m) +0.5 +2.25 +4.0 +tire_friction +0 +10 +20 +30 +40 +50 +60 +70 +80 +route_error (m) +0.005 +0.5 +5.0 +damping_rate +0 +10 +20 +30 +40 +50 +60 +70 +80 +route_error (m) +0.0001 +0.2 +100 +drag_coefficient +0 +10 +20 +30 +40 +50 +60 +70 +80 +route_error (m) +Town04 +Town02 +Town06 +town +0 +10 +20 +30 +40 +50 +60 +70 +80 +route_error (m) +Figure 3: The average goal error and route error of the autonomous vehicle under perturbed testing +environments. The x-axis is the modified parameter value during testing. The y-axis shows the +corresponding goal error and route error respectively. +mation to the minimization calculation (Chen et al., 2017); (ii) In this experiment, we incorporate +the constraints on the controls as a sine activation function on the original controls. +POMDP Setup +(i) Reward function is a combination of goal error eg and route error er, informed +as rt = e−eg/steps ∗ e−er; (ii) State function can be acquired from the CARLA simulator. Among +them, position p, q, heading angle φ and v is returned directly, while velocity angle β is calculated +by arctan v y/v x and smoothed closed to 0; (iii) Action space of the recurrent policy is 65-dim, +including 1 parameter in f0, 32 parameters in f +′ +1 and f +′ +2 respectively. +Recurrent Policy +(i) Recurrent policy adopts a LSTM-based neural network (Hochreiter and +Schmidhuber, 1997) with one 256-dim recurrent layer; (ii) Hyperparameters are set equally among +all baselines for a fair comparison. The roll-out step of the PPO is a small value of 32 for the sta- +tionary hidden state of the recurrent policy, and the system identification coefficient equals 0.01. +The detailed hyperparameters are exhibited in the open-sourced code base 1 due to the page limit. +5.2. Main Results +We firstly train MPC-RRL and RRL controllers in training environments until convergence and then +include MPC controller into the testing phase. However, we find that the RRL controller fails to +learn proper driving behaviours even in the training scenario (i.e. The mean episode goal error +exceeds 100), which implies the difficulties and inefficiencies of pure RL algorithms in such a +complex control task. We therefore exclude RRL in the testing phase and only compare MPC and +MPC-RRL. From Figure 3, both MPC-based methods MPC and MPC-RRL present an acceptable +adaptive performance under minor perturbations of environments. However, when the perturbed +value significantly deviates from the training setup, MPC controller fails to control vehicles towards +the goal (a large goal error) or following the reference trajectory while MPC-RRL can still adapt to +the changes in most cases, except 2.0 in final ratio 4.0 in tire friction and 100 in drag coefficient. +1. Code base for this work: https://github.com/mikezhang95/mpc rl +9 + +MODEL PREDICTIVE CONTROL - RECURRENT REINFORCEMENT LEARNING +5.3. Ablation Study +In this section we execute an ablation study on the two most essential designs in the framework: +RNN structure (RNN for short) and system identification loss (SI Loss for short). ”- RNN” replaces +recurrent policy with a feed-forward policy, and ”- SI Loss” trains with cumulative reward maxi- +mization alone. As Table 2 shows, RNN plays a more fundamental role in terms of the goal error. +Incorporating both designs leads to the best performance. +Table 2: The median goal error and average rank of all ablation settings. Less goal error turns to +lower rank. +Ablation Settings +final +ratio +moi +tire +friction +damping +rate +drag +coefficient +town +AVG RANK +RNN +SI Loss ++ ++ +9.09 +8.44 +9.13 +8.33 +23.50 +8.03 +1.3 ++ +- +10.44 +9.43 +9.97 +9.19 +10.68 +8.83 +2.5 +- ++ +10.82 +9.40 +9.94 +9.28 +10.74 +8.94 +3.0 +- +- +10.65 +9.40 +9.98 +9.12 +23.89 +8.80 +2.8 +5.4. Study on Recurrent Policy +In this section, we further analyze the learned recurrent policy to find out why it successfully resists +perturbations in the environment. We plot a graph on the average absolute value of the vehicle’s +acceleration and how it varies by the recurrent neural policy. Figure 4 clearly explains that accel- +eration goes up with the rise of final ratio and goes down with the rise of drag coefficient and +tire friction. These trends all show that our MCP-RRL framework can adapt well to environmental +perturbations, thus leading to better control performance. +2.0 +5.0 +10.0 +final_ratio +2.48 +2.50 +2.52 +2.54 +2.56 +2.58 +2.60 +Acceleration (m/s^2) +0.0001 +0.2 +100 +drag_coefficient +2.25 +2.26 +2.27 +2.28 +2.29 +2.30 +2.31 +2.32 +Acceleration (m/s^2) +0.5 +2.25 +4.0 +tire_friction +1.9 +2.0 +2.1 +2.2 +2.3 +Acceleration (m/s^2) +Figure 4: The average absolute value of the autonomous vehicle’s acceleration under perturbed +testing environments. +6. CONCLUSION +In this paper, we propose an MPC-RRL algorithm to handle the problem of perturbed parameters in +the autonomous driving task, which is proven to be effective theoretically and empirically. This is +the first work to combine RRL and MPC under a POMDP Formulation, which could be potentially +beneficial to develop more robust control methods. +In future work, we will combine this work with domain randomization to better generalize the +algorithm on unseen environments. Furthermore, MPC-RRL is expected to reveal a more robust +performance than vanilla MPC on real-world cars, which will be evaluated in further experiments. +10 + +MODEL PREDICTIVE CONTROL - RECURRENT REINFORCEMENT LEARNING +Acknowledgments +This project is funded by the European Union’s Horizon 2020 research and innovation program +under the Marie Skłodowska-Curie grant agreement No. 953348. We would also like to thank Jasper +Hoffman regarding the usage of SLURM system and Jun Hou for his exploration on CARLA. +References +Brandon Amos, Ivan Dario Jimenez Rodriguez, Jacob Sacks, Byron Boots, and J Zico Kolter. Dif- +ferentiable mpc for end-to-end planning and control. page 12, 2018. +Somil Bansal, Roberto Calandra, Ted Xiao, Sergey Levine, and Claire J. Tomlin. +Goal-driven +dynamics learning via bayesian optimization, September 2017. +Bruno Brito, Michael Everett, Jonathan P. How, and Javier Alonso-Mora. Where to go next: Learn- +ing a subgoal recommendation policy for navigation among pedestrians, February 2021. +Jianyu Chen, Wei Zhan, and Masayoshi Tomizuka. Constrained iterative lqr for on-road autonomous +driving motion planning. In 2017 IEEE 20th International Conference on Intelligent Transporta- +tion Systems (ITSC), pages 1–7, 2017. doi: 10.1109/ITSC.2017.8317745. +Alexey Dosovitskiy. 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In 2013 IEEE Symposium on Adaptive Dynamic +Programming and Reinforcement Learning (ADPRL), pages 100–107, April 2013. doi: 10.1109/ +ADPRL.2013.6614995. +12 + diff --git a/PtFQT4oBgHgl3EQfYzaZ/content/tmp_files/load_file.txt b/PtFQT4oBgHgl3EQfYzaZ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8b268fdf5f74a98dce9da64848011a7c3a1efa3b --- /dev/null +++ b/PtFQT4oBgHgl3EQfYzaZ/content/tmp_files/load_file.txt @@ -0,0 +1,549 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf,len=548 +page_content='Proceedings of Machine Learning Research vol XX:1–12, 2023 Incorporating Recurrent Reinforcement Learning into Model Predictive Control for Adaptive Control in Autonomous Driving Yuan Zhang YZHANG@CS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='UNI-FREIBURG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='DE Joschka Boedecker JBOEDECK@CS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='UNI-FREIBURG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='DE Neurorobotics Lab, University of Freiburg, Germany Chuxuan Li LICHUXUAN@AIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='TSINGHUA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='EDU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='CN Guyue Zhou ZHOUGUYUE@AIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='TSINGHUA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='EDU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='CN Institute for AI Industry Research (AIR), Tsinghua University, China Abstract Model Predictive Control (MPC) is attracting tremendous attention in the autonomous driving task as a powerful control technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' The success of an MPC controller strongly depends on an accurate internal dynamics model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' However, the static parameters, usually learned by system identification, often fail to adapt to both internal and external perturbations in real-world scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' In this paper, we firstly (1) reformulate the problem as a Partially Observed Markov Decision Process (POMDP) that absorbs the uncertainties into observations and maintains Markov property into hidden states;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' and (2) learn a recurrent policy continually adapting the parameters of the dynamics model via Recurrent Reinforcement Learning (RRL) for optimal and adaptive control;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' and (3) finally eval- uate the proposed algorithm (referred as MPC-RRL) in CARLA simulator and leading to robust behaviours under a wide range of perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' Keywords: Autonomous Driving, Model Predictive Control, Recurrent Neural Network, Rein- forcement Learning 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' INTRODUCTION Model Predictive Control (MPC) has become the primary control method for enormous fields, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' autonomous driving (Reiter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=') and robotics (Song and Scaramuzza, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' As a model-based method, MPC largely depends on an accurate dynamics model of the system, xk+1 = f(xk, uk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' θ), where x, u represent state and control respectively and θ is the parameter of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' The parame- ter θ is assumed to be determined by prior knowledge or system identification method that learns the parameter from a collection of experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' However, the awareness of parameter θ consistently fails due to the perturbations emerging from all sources in the autonomous driving task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' In detail, both internal (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' car mass, drag coefficient) and external (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' road friction, planning route) parameters may vary in the driving process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' Also, the accurate values of the parameters are difficult to collect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' Therefore, an MPC controller with a fixed parameter θ may degenerate the control performance in the autonomous driving task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' Learning-based MPC is receiving increasing attention as it focuses on automatic adaption to varying environmental parameters (Hewing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' Spielberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' The standard ap- proach is to incorporate an additional module which aims to tune the parameters of the MPC for optimal control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' Bayesian Optimization (BO) and Reinforcement Learning (RL) are two significant mythologies to learn this module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' BO aims at optimizing the closed-loop cost J(θ) and generates © 2023 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' Zhang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' Boedecker, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' Li & G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' Zhou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='13313v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='LG] 30 Jan 2023 MODEL PREDICTIVE CONTROL - RECURRENT REINFORCEMENT LEARNING the most suitable parameters of an MPC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' However, the update of parameters is considerably slow to react in a dynamically changing environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' RL is capable of modifying the parameter at each time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' However, the perturbation in the dynamics is non-stationary, thus violating the Markov property required in RL theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' This paper follows the RL method to boost the MPC controller’s adaptability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' We firstly re- formulate the problem as a Partially Observed Markov Decision Process (POMDP) to ease the non-stationary property under environmental perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' The original state in MPC is viewed as an observation in the POMDP formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' Meanwhile, a hidden state represents the system’s ac- tual state, including the perturbation information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' We add a recurrent policy on top of the MPC and facilitate its learning with 2 objectives: cumulative reward maximization and system identification loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' The whole system achieves optimal and adaptive control in autonomous driving simulations compared with pure MPC and RL methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' (a) The kinematic bicycle model (Kong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' (b) CARLA simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' Figure 1: Model and simulation in the autonomous driving task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' RELATED WORK As mentioned in Section 1, Bayesian Optimization (BO) is a parallel approach to improve the pa- rameters of an MPC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' Marco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' (2016) adopts an LQR formulation and learns the parameters of Q,R matrix with entropy search method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' Bansal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' (2017) also utilizes an LQR model but learns the parameters of the transition function to control a quadcopter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' Both methods update the parame- ters θ at the end of an episode as it requires evaluating the cost J(θ), which leads to slow responses in case of environmental perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' Comparatively, Reinforcement Learning (RL) can modify MPC parameters at each time step to quickly adapt to dynamic environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' Zarrouki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' (2021) and Song and Scaramuzza (2020) both learn a policy that can improve parameters of MPC’s cost function, while Gros and Zanon (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' Amos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' (2018) aim to modify both transition and cost function’s parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' Neverthe- less, none of these methods consider the non-stationary scenario in an autonomous driving system, which violates the essential Markov property in RL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' There are other research directions to combine RL with MPC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' Brito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' (2021) utilizes RL to learn a sub-goal so as to reduce the optimization horizon of the MPC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' Farshidian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' 2 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='O y r=0 C XMODEL PREDICTIVE CONTROL - RECURRENT REINFORCEMENT LEARNING Zhong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' (2013) replace the terminal cost in MPC by the value function in RL to avoid inaccurate human-designed objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' While previous works focus on improving the efficiency of MPC with the aid of RL, in our work we are more interested in ensuring adaption to perturbed environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' PRELIMINARIES 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' Vehicle Dynamics Modelling The kinematic bicycle model (Kong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=', 2015) is a simplified vehicle model targeted for au- tonomous vehicles, whose continuous time equation is � ����� ˙p ˙q ˙ψ ˙v β � ����� = � ������ v cos(ψ + β) v sin(ψ + β) v lr sin β a tan−1 � lr lf+lr tan δf � � ������ (1) where state x = [a, b, ψ, v, β] includes a, b: the coordinates of the mass center, ψ: the head- ing angle of the vehicle, v: the speed at the mass center and β: the angle of the current velocity w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' the longitudinal axis of the vehicle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' control u = [a, δf] consists of a: the acceleration at the mass center and δf: the steering angle of the front wheel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' Other than that, lr and lf are the vehicle’s inertial parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' An intuitive illustration of the kinematic bicycle model can be seen in Figure 1(a)subfigure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' However, such a model cannot be directly adopted in existing autonomous driving platform (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' CARLA (Dosovitskiy) and APOLLO (Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=', 2022)) where the output control u normally consists of steering st, throttle th, brake br instead of acceleration a and angle δf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' Xia (2019) suggests utilizing a neural network to represent the dynamics, which has the following form ˙x = � ����� ˙p ˙q ˙ψ ˙v ˙β � ����� = � ����� v cos(ψ + β) v sin(ψ + β) f0(v, β;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' θ) f1(v, β, st, th, br;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' θ) f2(v, β, st, th, br;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' θ) � ����� = f(x, u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' θ), (2) where f0, f1, f2 are neural networks parameterized with θ, which can be approximated by sys- tem identification methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' Furthermore, the discrete dynamic function becomes xt+1 = xt + f(xt, ut;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' θ)∆t, with the state xt = [pt, qt, ψt, vt, βt]T and the control ut = [stt, tht, brt]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' The control interval ∆t equals 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='1 seconds in the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' Model Predictive Control Formulation Model Predictive Control (MPC) is an advanced optimization method for non-linear optimal control problems with constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' To control an autonomous vehicle moving along a reference trajectory G = (g1, g2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=', g|G|) (gi are waypoints on the trajectory) with a target speed V, an MPC problem can be formulated as follows: 3 MODEL PREDICTIVE CONTROL - RECURRENT REINFORCEMENT LEARNING min x,u lH(xH, G) + H−1 � t=0 l(xt, ut, G, V) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' ∀t, xt+1 = xt + f(xt, ut;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' θ)∆t − 1 ≤ stt = ut[0] ≤ 1 0 ≤ tht = ut[1] ≤ 1 0 ≤ brt = ut[2] ≤ 1 x0 = xinit (3) where x = (x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=', xH) and u = (u0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=', uH−1) represent the state and control sequences to be optimized respectively, and xinit is the initial state of the autonomous vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' The stage cost function l(xt, ut, G, V) = cposition×D(xt, G)+cspeed×(xt[3]−V)2+ccontrol×(ut[0]2+ut[1]2+ ut[2]2 + ut[1] × ut[2]), where D(xt, G) = mink∈{1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=',|G|} ∥xt[: 2] − gk)∥2 is the distance function to the nearest waypoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' The terminal cost function lH(xH, G) = ∥xH[: 2] − g|G|)∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' Among them, cposition, cspeed, ccontrol are coefficients to balance different parts in the cost function and set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='04, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='002, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='0005 in the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' To solve this non-linear MPC problem efficiently, the iLQR method (Li, 2004) can be applied accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' Partially Observed Markov Decision Process (POMDP) A Partially Observable Markov Decision Process (POMDP) is a generalized mathematical frame- work of an MDP to deal with the unobserved state issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' It is formulated as a 7-tuple ⟨S, A, P, r, Ω, O, γ⟩, where S, A and Ω stand for the state, action and observation space respec- tively, and r(st, at) : S × A → R is the reward function at time step t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' Define ∆|S|, ∆|A|, ∆|Ω| be the probability measure on S, A and Ω respectively, then P(st+1|st, at) : S × A → ∆|S| is the transition function, and the future rewards are discounted by the discount factor γ ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' The most crucial concept in POMDP is that agents can only obtain the observation ot with probability O(ot|st, at−1) : S × A → ∆|Ω|, instead of receiving the entire state st.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' The received partial observation is not sufficient for agents to make decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' Instead, the agent needs to maintain a belief state bt(st) : ∆|S| (bt for short) to estimate a complete knowledge of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' There exists an update equation for the belief state given previous belief state bt−1, action at−1 and new observation ot: bt = ηO(ot|st, at−1) � st−1 P(st|st−1, at−1)bt−1, where η is a normalization factor to ensure probability measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' The policy function in a POMDP is usually defined as Π = � π(at|bt) : ∆|S| → ∆|A| � and the objective of this agent can be formulated as an optimization problem, J∗ = max π∈Π Eπ,P,O �+∞ � t=0 γtr(st, at)|b0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' (4) where b0 is an initial guess on the belief state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' The expectation is on policy π, transition function P and observation function O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' The belief state bt can be iteratively calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' 4 MODEL PREDICTIVE CONTROL - RECURRENT REINFORCEMENT LEARNING !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='" #" $" %" #&\'( = * #&, %&;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' $" #-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='-/ = #" MPC Controller t = 0 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=" ( #( $( %( #&'( = * #&, %&;" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' $( #-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='-/ = #( MPC Controller t = 1 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content="3 #3 $3 %3 #&'( = * #&, %&;" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' $3 #-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='-/ = #3 MPC Controller t = 2 5" 5( 53 6($|9(!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=') Figure 2: A POMDP Formulation of autonomous driving with an MPC controller 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' MPC-RRL Framework In this section, we will firstly reformulate the autonomous driving task with an MPC controller as a POMDP problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' Furthermore, within this formulation, we learn a recurrent policy with RL to pursue optimal and adaptive control, thus called MPC-RRL for short.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' POMDP Formulation As mentioned in Section 1, it is challenging for the original dynamic of the MPC (Equation 2) to react to both internal and external variations of the system, thus decreasing the performance in real- life scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' Comparatively, a POMDP Formulation can ease this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' Viewing the state xt in MPC’s dynamic function as an observation ot in a POMDP, one can maintain a hidden state st in a POMDP that can reflect the perturbations in the current system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' The observation distribution O(xt|st) represents the probability of the MPC controller receiving state xt given hidden state st.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' Furthermore, we model MPC’s system parameters θt as the output action in this POMDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' The MPC controller calculates the control action ut given the state xt and the system parameters θt, referred as ut = MPC(xt, θt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' After executing ut on the environment, the hidden state transfers to st+1 with probability P(st+1|st, ut) = P(st+1|st, MPC(xt, θt)) and reward rt = r(st, ut) is returned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' The optimal action aims to modify the parameters θt at each time step so that MPC can achieve an optimal control ut even under perturbed environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' As mentioned in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='3, an agent needs to maintain a belief state bt on the current hidden state st so that it can generate the action θt to influence the MPC results by following the policy π(θt|bt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' This whole POMDP formulation allows the MPC controller to dynamically change the dynamics’ parameters, leading to an optimal and adaptive control if the belief state bt and the action θt is appropriately generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' The overall framework is illustrated in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' 5 MODEL PREDICTIVE CONTROL - RECURRENT REINFORCEMENT LEARNING 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' Recurrent Policy In this section, we will focus on the most essential sub-modules in the framework, belief state bt and neural policy π(θt|bt), on how they are represented, learned and deployed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' In general, these two modules are combined in one single recurrent neural network and learned with 2 objectives for optimality and adaptability respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' The combination is referred as recurrent policy for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' POLICY REPRESENTATION We leverage neural networks to represent both belief state and neural policy, with parameters τ and ω respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' A belief tracker bt = F(bt−1, xt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' τ) is designed to update the belief state given new observation xt at each time step, while the neural policy is represented by π(θt|bt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' Following the general practice in POMDP problem (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' Hausknecht and Stone, 2017), these two networks can be combined into one recurrent neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' Recurrent neural network (Hochreiter and Schmidhuber, 1997) possess a general representation of bt, θt = R(xt, bt−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' λ), which perfectly absorbs F and π in one network R with parameter λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' The whole network can be optimized and utilized altogether (Young et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=', 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' TRAINING DETAILS As introduced in last section, the recurrent policy R with parameters λ is the only module to be learned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' We design two learning objectives focusing on (i) encouraging MPC to generate optimal control sequences;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' (ii) adapting the transition model to realities separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' The first objective is to maximize the discounted cumulative reward, as introduced in Equation 4, J1 = max λ Eπ,P,O �+∞ � t=0 γtr(st, θt)|b0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' (5) This objective encourages the neural policy to output the parameters beneficial to the control be- haviours with respect to a higher return.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' The second objective is inspired by the system identification loss in control theory, J2 = min λ E �+∞ � t=0 [xt + f(xt, ut;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' θt)∆t − xt+1]2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' (6) This objective pushes the policy to imitate the realistic dynamics’ parameters when it deviates from the previous approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' Combining these two objectives by J = J1 + αJ2 (α is a coefficient to balance these 2 objec- tives), we can successfully learn a policy aiming optimal and robust MPC control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' In Section 5 we will further illustrate the effectiveness of this method and the different roles of these 2 objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' Since the expectation in Equation 5 and Equation 6 is expensive to calculate directly, the RL area usually interacts with the environments and learns from the sampling experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' In this paper, we adopt PPO (Schulman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=', 2017) as the basic RL algorithm as it is one of the best performing on-policy RL algorithms suitable to the training of dynamics belief states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' 6 MODEL PREDICTIVE CONTROL - RECURRENT REINFORCEMENT LEARNING 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' INFERENCE DETAILS The parameters λ are fixed during inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' Given an observation of the vehicle xt and the belief state of last step bt−1, the agent directly generates the dynamics parameter θt and updates the belief state bt by executing bt, θt = R(xt, bt−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' The MPC controller utilizes the dynamics parameter θ and initial state xt to generate the control value ut, which is further sent to the autonomous driving environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' Experiments In this section, we evaluate our proposed MPC-RRL framework on the CARLA simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' We first show the superior performance with respect to goal error and route error in the main results and further analyze the controller’s behaviours in the ablation study and policy study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' CARLA Simulator We utilize CARLA simulator (Dosovitskiy) (See Figure 1(b)subfigure for a rendering example) to evaluate autonomous driving performances of all baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' CARLA simulator is a popular simu- lation platform to train and evaluate different components of an autonomous driving system (per- ception, planning, control, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' CARLA is grounded on Unreal Engine to run the simulation with changeable configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' Thus it is convenient to modify both internal (car mass, tire friction) and external (planning route, road friction) factors of the system, which facilitates our evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' We run CARLA in a synchronous mode, which ensures the reproducibility of the experiments but may fail to simulate some real-world situations, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' missing observations and delayed control, which will be studied in further research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' TASK SETUP For each episode of the autonomous driving task, a starting and goal point is generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' The aim of a control method is to reach the goal point following a reference trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' The episode ends if the vehicle reaches the goal or experiences a collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' The performance is evaluated by the goal error and the route error together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' The goal error is defined by the distance between the vehicle’s final position and the goal point, while the route error is calculated by the cumulative displacement between the vehicle’s actual trajectory and the reference trajectory during the driving process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' We divide the self-driving task into training and testing phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' We first train the RL policy described in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='2 and freeze the policy’s parameters in the testing phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' To evaluate the adaptability of each controller, we modify a system parameter (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' car’s final ratio, tire’s friction) to a different value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' A decent controller should be able to adapt to such model mismatches between training and testing environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' The specific parameters for evaluation and their value during the training and testing phase are in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' Notably, for each parameter, we testify on 3 perturbed values, with one value slightly different from the training value and the other two deviating largely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' BASELINES MPC Controller The MPC controller strictly follows the MPC formulation as described in Sec- tion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' Some practical implementation details should be considered to apply MPC on the CARLA simulator, which will be further clarified in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' 7 MODEL PREDICTIVE CONTROL - RECURRENT REINFORCEMENT LEARNING Table 1: Verified parameters and their values during training and testing phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' Parameter Training Value Testing Value Explanation final ratio 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='0, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='0, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='0 Transmission ratio from engine to wheels moi 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='4, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='3, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='9 Moment of inertia of the vehicle tire friction 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='5, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='25, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='0 Friction factor of all wheels damping rate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='25 5e-3, 5e-1, 5e1 Damping rate of all wheels drag coefficient 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='15 1e-4, 2e-1, 100 Drag coefficient of the vehicle’s car body town Town01 Town04, Town02, Town06 Carla’s default town maps MPC-RRL Controller The MPC-RRL controller is precisely the framework introduced in Sec- tion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' For a fair comparison, we adopt the same MPC controller as Paragraph 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='2 as the based controller in the framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' RRL Controller To fairly illustrate the power of combing RRL and MPC, we also add a baseline with the recurrent policy only and trained with reinforcement learning methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' In this baseline, the input of the policy is observation x, and the output is direct control action u, including steering, throttle, and brake instead of MPC’s parameters in comparison with MPC-RRL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' PRACTICAL IMPLEMENTATION DETAILS To apply the proposed MPC-RRL framework on the CARLA simulator, we explain some practical implementations on both MPC and RL parts in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' Vehicle Dynamics Model (i) Regarding the dynamics model in MPC, we utilize 3 neural networks f0, f1, f2 parameterized with a 65-dim variable θ to represent it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' f0(v, β;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' θ) = vsinβ θ[0] following the original format of the vehicle model (Equation 1) can predict the head- ing angle ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' For f1, f2, we require them to have such properties: f1(v, β, st, th, br;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' θ) = f1(v, −β, −st, th, br;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' θ), f2(v, β, st, th, br;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' θ) = −f2(v, −β, −st, th, br;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' θ) so that the sym- metry of velocity angle β and steering control st maintain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' To achieve that, we rewrite f1 as [f ′ 1(v, β, st, th, br;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' θ) + f ′ 1(v, −β, −st, th, br;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' θ)]/2 and f2 as [f ′ 2(v, β, st, th, br;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' θ) − f ′ 2(v, −β, −st, th, br;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' θ)]/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' Currently, only f ′ 1 and f ′ 2 are represented with parameters θ, which is a 2-layer feed-forward neural network in our experiment;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' (ii) Notably, the prediction of the veloc- ity v should always be positive, for which we further rewrite f1 as [f ′ 1(v, β, st, th, br;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' θ) × (2√v + f ′ 1(v, β, st, th, br;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' θ))+f ′ 1(v, −β, −st, th, br;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' θ)×(2√v+f ′ 1(v, −β, −st, th, br;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' θ))]/(2∆t) so that v + ∆t × f1(v, β, st, th, br;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' θ) ≥ 0 is always true;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' (iii) For system identification, we manually con- trol the vehicles running in the CARLA simulator and collect 19000 transitions to train the f0, f1, f2 models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' The mean-squared-error loss and Adam optimizer (Kingma and Ba, 2017) are used to learn the parameters θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' Model Predictive Control (i) As mentioned in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='2, the stage cost function l(xt, ut, G, V) is written as cposition × mink∈{1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=',|G|} ∥xt[: 2] − gk)∥2 + cspeed × (xt[3] − V)2 + ccontrol × (ut[0]2 + ut[1]2 + ut[2]2 + ut[1] × ut[2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' The minimization term is not a perfect choice for gradient-based optimization methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' Instead, we utilize a differential function D(xt, G) = − log � |G| � k=1 exp −∥xt[: 2] − gk)∥2 � as the distance function, which can be viewed as an approxi- 8 MODEL PREDICTIVE CONTROL - RECURRENT REINFORCEMENT LEARNING 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='0 10 final_ratio 0 10 20 30 40 50 60 70 80 goal_error (m) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='9 moi 0 10 20 30 40 50 60 70 80 goal_error (m) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='25 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='0 tire_friction 0 10 20 30 40 50 60 70 80 goal_error (m) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='0 damping_rate 0 10 20 30 40 50 60 70 80 goal_error (m) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='0001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='2 100 drag_coefficient 0 10 20 30 40 50 60 70 80 goal_error (m) Town04 Town02 Town06 town 0 10 20 30 40 50 60 70 80 goal_error (m) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='0 10 final_ratio 0 10 20 30 40 50 60 70 80 route_error (m) MPC MPC-RRL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='9 moi 0 10 20 30 40 50 60 70 80 route_error (m) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='25 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='0 tire_friction 0 10 20 30 40 50 60 70 80 route_error (m) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='0 damping_rate 0 10 20 30 40 50 60 70 80 route_error (m) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='0001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='2 100 drag_coefficient 0 10 20 30 40 50 60 70 80 route_error (m) Town04 Town02 Town06 town 0 10 20 30 40 50 60 70 80 route_error (m) Figure 3: The average goal error and route error of the autonomous vehicle under perturbed testing environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' The x-axis is the modified parameter value during testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' The y-axis shows the corresponding goal error and route error respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' mation to the minimization calculation (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=', 2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' (ii) In this experiment, we incorporate the constraints on the controls as a sine activation function on the original controls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' POMDP Setup (i) Reward function is a combination of goal error eg and route error er, informed as rt = e−eg/steps ∗ e−er;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' (ii) State function can be acquired from the CARLA simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' Among them, position p, q, heading angle φ and v is returned directly, while velocity angle β is calculated by arctan v y/v x and smoothed closed to 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' (iii) Action space of the recurrent policy is 65-dim, including 1 parameter in f0, 32 parameters in f ′ 1 and f ′ 2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' Recurrent Policy (i) Recurrent policy adopts a LSTM-based neural network (Hochreiter and Schmidhuber, 1997) with one 256-dim recurrent layer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' (ii) Hyperparameters are set equally among all baselines for a fair comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' The roll-out step of the PPO is a small value of 32 for the sta- tionary hidden state of the recurrent policy, and the system identification coefficient equals 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' The detailed hyperparameters are exhibited in the open-sourced code base 1 due to the page limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' Main Results We firstly train MPC-RRL and RRL controllers in training environments until convergence and then include MPC controller into the testing phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' However, we find that the RRL controller fails to learn proper driving behaviours even in the training scenario (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' The mean episode goal error exceeds 100), which implies the difficulties and inefficiencies of pure RL algorithms in such a complex control task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' We therefore exclude RRL in the testing phase and only compare MPC and MPC-RRL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' From Figure 3, both MPC-based methods MPC and MPC-RRL present an acceptable adaptive performance under minor perturbations of environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' However, when the perturbed value significantly deviates from the training setup, MPC controller fails to control vehicles towards the goal (a large goal error) or following the reference trajectory while MPC-RRL can still adapt to the changes in most cases, except 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='0 in final ratio 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='0 in tire friction and 100 in drag coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' Code base for this work: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='com/mikezhang95/mpc rl 9 MODEL PREDICTIVE CONTROL - RECURRENT REINFORCEMENT LEARNING 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' Ablation Study In this section we execute an ablation study on the two most essential designs in the framework: RNN structure (RNN for short) and system identification loss (SI Loss for short).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' ”- RNN” replaces recurrent policy with a feed-forward policy, and ”- SI Loss” trains with cumulative reward maxi- mization alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' As Table 2 shows, RNN plays a more fundamental role in terms of the goal error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' Incorporating both designs leads to the best performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' Table 2: The median goal error and average rank of all ablation settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' Less goal error turns to lower rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' Ablation Settings final ratio moi tire friction damping rate drag coefficient town AVG RANK RNN SI Loss + + 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='09 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='44 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='13 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='33 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='50 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='03 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='3 + 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='44 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='43 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='97 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='19 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='68 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='83 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='5 + 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='82 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='40 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='94 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='28 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='74 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='94 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='65 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='40 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='98 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='12 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='89 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='80 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' Study on Recurrent Policy In this section, we further analyze the learned recurrent policy to find out why it successfully resists perturbations in the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' We plot a graph on the average absolute value of the vehicle’s acceleration and how it varies by the recurrent neural policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' Figure 4 clearly explains that accel- eration goes up with the rise of final ratio and goes down with the rise of drag coefficient and tire friction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' These trends all show that our MCP-RRL framework can adapt well to environmental perturbations, thus leading to better control performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='0 final_ratio 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='48 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='50 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='52 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='54 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='56 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='58 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='60 Acceleration (m/s^2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='0001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='2 100 drag_coefficient 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='26 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='27 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='28 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='29 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='30 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='31 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='32 Acceleration (m/s^2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='25 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='0 tire_friction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='3 Acceleration (m/s^2) Figure 4: The average absolute value of the autonomous vehicle’s acceleration under perturbed testing environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' CONCLUSION In this paper, we propose an MPC-RRL algorithm to handle the problem of perturbed parameters in the autonomous driving task, which is proven to be effective theoretically and empirically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' This is the first work to combine RRL and MPC under a POMDP Formulation, which could be potentially beneficial to develop more robust control methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' In future work, we will combine this work with domain randomization to better generalize the algorithm on unseen environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' Furthermore, MPC-RRL is expected to reveal a more robust performance than vanilla MPC on real-world cars, which will be evaluated in further experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' 10 MODEL PREDICTIVE CONTROL - RECURRENT REINFORCEMENT LEARNING Acknowledgments This project is funded by the European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska-Curie grant agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' 953348.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' We would also like to thank Jasper Hoffman regarding the usage of SLURM system and Jun Hou for his exploration on CARLA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' References Brandon Amos, Ivan Dario Jimenez Rodriguez, Jacob Sacks, Byron Boots, and J Zico Kolter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' Dif- ferentiable mpc for end-to-end planning and control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' page 12, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' Somil Bansal, Roberto Calandra, Ted Xiao, Sergey Levine, 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' In 2013 IEEE Symposium on Adaptive Dynamic Programming and Reinforcement Learning (ADPRL), pages 100–107, April 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='1109/ ADPRL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content='6614995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} +page_content=' 12' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFQT4oBgHgl3EQfYzaZ/content/2301.13313v1.pdf'} diff --git a/QdE2T4oBgHgl3EQfsAgY/content/tmp_files/2301.04054v1.pdf.txt b/QdE2T4oBgHgl3EQfsAgY/content/tmp_files/2301.04054v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..033c3ac53b09bf2d6c231fe15f826cb420710eb8 --- /dev/null +++ b/QdE2T4oBgHgl3EQfsAgY/content/tmp_files/2301.04054v1.pdf.txt @@ -0,0 +1,924 @@ +Surface-Sensitive Raman Scattering by Transferable Nanoporous +Plasmonic Membranes +Roman M. Wyss,1, 2 Günter Kewes,1 Martin Frimmer,3 Karl-Philipp Schlichting,4 Markus +Parzefall,3 Eric Bonvin,3 Martin F. Sarott,5 Morgan Trassin,5 Lala Habibova,1 Giorgia +Marcelli,1 Jan Vermant,2 Lukas Novotny,3 Mads C. Weber,6 and Sebastian Heeg1, ∗ +1Institut für Physik und IRIS Adlershof, +Humboldt-Universität zu Berlin, 12489 Berlin, Germany +2Soft Materials, Department of Materials, +ETH Zürich, 8093 Zürich, Switzerland +3Photonics Lab, ETH Zürich, 8093 Zürich, Switzerland +4Laboratory of Thermodynamics in Emerging Technologies +Department of Mechanical and Process Engineering, +ETH Zürich, 8092 Zürich, Switzerland +5Laboratory for Multifunctional Ferroic Materials, +Department of Materials, ETH Zürich, 8093 Zürich, Switzerland +6Institut des Molécules et Matériaux du Mans, +UMR 6283 CNRS, Le Mans Université, 72085 Le Mans, France +(Dated: +January 11, 2023) +1 +arXiv:2301.04054v1 [cond-mat.mtrl-sci] 10 Jan 2023 + +Abstract +Raman spectroscopy is a powerful technique to characterize materials. It probes non-destructively +chemical composition, crystallinity, defects, strain and coupling phenomena. However, the Raman +response of surfaces or thin films is often weak and obscured by dominant bulk signals. +Here +we overcome this limitation by placing a transferable porous gold membrane (PAuM) on top of +the surface of interest. Slot-like nanopores in the membrane act as plasmonic slot antennas and +enhance the Raman response of the surface or thin film underneath. Simultaneously, the PAuM +suppresses the penetration of the excitation laser into the bulk, efficiently blocking the bulk Raman +signal. Using graphene as a model surface, we show that these two simultaneous effects lead to +an increase in the surface-to-bulk Raman signal ratio by three orders of magnitude. We find that +90 % of the Raman enhancement occurs within the top 2.5 nm of the material, demonstrating truly +surface-sensitive Raman scattering. To validate our approach, we analyze the surface of a LaNiO3 +thin film. We observe a Raman mode splitting for the LaNiO3 surface-layer, which is spectroscopic +evidence that the surface structure differs from the bulk. This result underpins that PAuM give +direct access to Raman signatures of surfaces and their structural properties. +KEYWORDS +Raman spectroscopy; surface; surface-sensitive Raman scattering; plasmonic nanopore; com- +plex oxide thin film. +INTRODUCTION +Raman spectroscopy, the inelastic scattering of light by vibrations or phonons, is a +widespread analytical tool to study and characterize materials [1]. Owing to its versatil- +ity, simplicity and specificity, Raman spectroscopy is used in material science [2], i.e. to +study phase transitions [3, 4], catalysis [5] or novel 2D materials [6], or even in pharmaceu- +tics and biomedical diagnostics [7]. In principle, Raman spectroscopy is ideal to study the +structure of surfaces, since their atomic registry differs from the bulk of the material and +may additionally be modified by terminations. This leads to changes in the frequency of +the Raman active vibrations or to peak-splitting as a result of a change insymmetry [8–10]. +However, the study of surfaces and thin films by Raman spectroscopy is notoriously diffi- +2 + +cult as light typically penetrates several micrometers into the material. The overall Raman +response is, therefore, dominated by the bulk, while the Raman signals of the surface are +orders of magnitude weaker and mostly go undetected. Hence, obtaining Raman signals of +thin films requires a minimal thickness of several tens to hundreds of nanometers. Raman +signatures of surfaces are often not observed at all [11, 12]. +One way to address the issue of vanishing surface- or thin film Raman signals is ultravio- +let (UV) Raman spectroscopy. Here, a UV laser excites the sample instead of a laser in the +visible or near infrared range. UV-Raman takes advantage of the shallow penetration-depth +of the UV-light into many materials [13, 14] and is not impeded by autofluorescence effects. +However, the penetration of the laser in the material is still in the order of hundreds of +nanometers, and can only be reduced further for materials with suitable band gaps [14]. +Moreover, the low damage threshold of many materials to UV light limits the application of +UV Raman [15]. Probing surface and thin-film Raman signals has also been addressed by +mathematical decomposition of a large stack of spectra. To do so, multiple spectra of a sam- +ple are measured under varying conditions. An example can be altering the laser focal point +with respect the the sample surface. Subsequently, a statistical analysis allows to decom- +pose the spectra into substrate/bulk and surface/thin film contributions [16]. However, this +method requires an already detectable signal of the surface or the thin film. Furthermore, +the measurement times are often beyond practical use. +A general strategy to enhance Raman signals is plasmon-enhanced Raman scattering +(PERS) [17, 18]. Here, the enhancement in PERS occurs in the vicinity of metallic nanos- +tructures and arises from the near-fields of localized surface plasmon resonances in the metal. +Particularly striking enhancement occurs in a nanoscale gap between two metal nanostruc- +tures, also referred to as plasmonic hotspot. This configuration enables the detection of +molecules adsorbed at the hotspot down to the single molecule level, embodying surface- +enhanced Raman scattering (SERS) [19–22]. Even though SERS can be realized with a +large number of different nanoparticle geometries, its use to enhance the Raman signals of +a surface or thin film is limited: there is no geometry that efficiently interfaces a plasmonic +hotspot between two metallic structures with a flat and extended surface or film. +Tip- +enhanced Raman spectroscopy (TERS), where a plasmonic hotspot at the apex of a metal +tip scans over a surface, partially solves this problem [23, 24]. The enhancement, however, +occurs only at one spot and is weaker than for gap type geometries. The largest drawback +3 + +of TERS is that bulk Raman signals are recorded together with the TERS signal. On top, +TERS remains a complex and challenging technique, such that its use to probe surfaces or +thin films is rarely reported. +Overall, the ideal plasmonic structure to study surfaces and thin films with Raman spec- +troscopy consists of a flat gap type plasmonic hotspot with simultaneous bulk Raman signal +suppression. Plasmonic nanoslots, rectangular nanoscale voids in a thin metallic film (i.e. +nanoporous membrane), fulfill these conditions. Upon resonant excitation, these slots act +as plasmonic slot antennas that harbour localized and enhanced near-fields, which rapidly +decay outside the pore within few nanometers. Placed on a material, the slot’s near-fields in- +teract primarily with the material surface. Away from the nanopore the metallic membrane +reflects incident fields and bulk Raman signals, which effectively suppresses the bulk Raman +signal. +Recently, transferable and easy-to-manufacture porous gold membranes (PAuM) +with nanoscale pores acting as plasmonic slot antennas were introduced [25]. It was shown +that individual pores feature local Raman enhancement factors up to 104 to 105 and sustain +high excitation powers (106 W cm−2), which makes them the ideal plasmonic structure to +study surfaces and thin films with Raman spectroscopy. +Here, we use porous gold membranes to enhance the surface Raman signal and to simul- +taneously suppress the bulk signal of the sample under investigation. Using wavelength- +dependent Raman spectroscopy, we show that PAuM enhance the surface-to-bulk Raman +signal ratio by up to three orders of magnitude. Combining experiment and simulation, +we reveal that the enhancement decays exponentially in the material such that 90 % of the +enhanced signal occurs within the top 2.5 nm. Hence, our approach enables highly surface- +sensitive Raman spectroscopy for weak or bulk-obscured Raman signals. We directly apply +this technique to study the surface of a 20 nm LaNiO3 thin film. We find Raman signatures +of the surface that differ from the bulk of the 20 nm film, in line with theoretical predictions +and experimental observations using scanning tunneling microscope (STM) [26]. Our work, +therefore, underscores the power of PAuM-supported Raman spectroscopy of surfaces and +thin films. +4 + +RESULTS +Our paper is structured as follows: First, we introduce the PAuM manufacturing and its +working principle. Second, we demonstrate the surface enhancement and bulk suppression of +Raman signals by a wavelength-dependent study with graphene as a model surface. Third, +we unravel the depth dependence of the Raman response in experiment and simulation. To +do so, we probe graphene sheets buried at various depths from the surface. Finally, we +showcase the use of PAuM to enhance the Raman response of a 20 nm LaNiO3 thin film. +Manufacturing and Working Principle of PAuM +Figure 1 (a) illustrates the key idea of this work: Without PAuM, a laser penetrates +several multiples of its wavelength into the material, limited by absorption and focal depth. +The Raman scattered signal, therefore, originates primarily from within the bulk. +The +surface Raman signal remains weak or non-detectable due to the vanishing small scattering +volume of the surface. In contrast, using PAuM, the surface Raman signal is drastically +enhanced compared the bulk Raman signal. This results from two simultaneous effects: 1) +local plasmonic enhancement within the metallic nanopores and 2) the suppression of the +laser penetration into the bulk and of residual bulk Raman signal reaching the detection +pathway. Our non-continuous membranes are formed by evaporation of a 20 nm gold film +on a SiO2/Si wafer. The membrane is subsequently transferred onto the sample [25] (see +Methods and Supporting Information S1). +To characterize PAuM, we transfer a 1 cm2-sized membrane on a Si/Si3N4 chip bearing +arrays of 4 µm holes (Methods) as shown in the optical microscope image in Fig. 1(b), top. +The scanning electron microscope (SEM) images Fig. 1(b), middle and lower panel, reveal +that the PAuM spans over the circular hole as a freestanding, mechanically stable membrane +[25]. The pores in the PAuM are visible in the SEM images as dark spots and come in various +shapes and sizes. The majority of pores is round or slot-like and below 100 nm in size (See +Supporting Information S1). A recent study demonstrated that the nanopores in the PAuM +act as plasmonic nanoslot antennas [25]. The nanopores harbour intense light fields upon +excitation at their plasmonic resonance. The energy of the plasmonic resonance depends one +the shape and aspect ratio of the individual pores. The highest field enhancement occurs +5 + +FIG1 +a +Surface +Bulk +Raman Shift +Raman Intensity +Surface +Bulk +Raman Shift +Raman Intensity +Bulk +Surface +PAuM +500 +600 +700 +800 +15 +30 +45 +pores in +PAuM +Transmission (%) +wavelength +sim. no pores +b +c +1 cm +1 µm +100 nm +FIG. 1. Principle of surface-sensitive Raman scattering enabled by PAuM. a) Nanopores +in the gold membrane enhance the Raman signal of the surface while suppressing the bulk Raman +signals. b) Photographic image (top) of a 20 nm PAuM transferred on a Si/Si3N4 partially sus- +pended over 4 µm holes (See Methods), forming freestanding membrane-like structures, as shown +in the Scanning electron microscope (SEM) image (middle). The nanopores are visible as dark +irregular, slot-like and circular features (bottom). c) Optical transmission of a freestanding PAuM +measured as function of wavelength. The dashed line corresponds to a simulated non-porous Au +film with 20 nm thickness. The plasmonic resonances of the nanopores give rise to the increased +transmission in the experimental data from 650 nm to 850 nm compared to the simulation of a +non-porous film. +for narrow slot-like pores with an excitation polarized perpendicular to the pores’ long axis, +see Supporting Information and Ref. [25] for an extended discussion. +In Fig. 1 (c), we compare the optical transmission of freestanding 20 nm PAuM to the +6 + +10/6/2017 +dwell +HV +HFW +mag 只 +WD +1 μm +10:37:01 AM +10 μs +5.00 kV +5.08 μm +25 000 x +4.4 mm +G17 after H202transmission of a simulated gold film without nanopores of the same thickness. The general +shape of the optical transmission is in good agreement with the simulated results (dashed +line). However, deviations occur from 650 nm to 850 nm with an increased transmission +probability. Such increased transmission is expected when the nanopores of PAuM are res- +onantly excited and act as nano-antennas [27]. The colored lines correspond to individual +transmission measurements at different sample positions, and their varying deviations from +the simulated trend indicate the varying geometries of the nanopores. We conclude that Ra- +man enhancement can be expected in the spectral range from 650 nm to 850 nm in agreement +with our previous study [25]. The random nature of pore geometries in both - dimension +and orientation - provides a substantial number of pores that will resonantly couple to an +incident laser light with arbitrary polarization and wavelength. +Measurement of Surface Enhancement Using Graphene Probes +Graphene is ideal as a model material to test and quantify surface-sensitive Raman scat- +tering as it can mimic a surface or thin film due to its two-dimensional nature [28–30]. +Furthermore, the Raman spectrum of graphene is well understood and the main Raman fea- +tures are intense and independent of excitation wavelength as well as polarization [31]. Any +dependence of the Raman features of graphene interfaced with our porous Au membrane +can hence be attributed entirely to nanopore interaction. +To probe the interaction and enhancement of PAuM with a surface, we place a graphene +sheet, acting as an test surface, on a flat Si/SiO2 substrate, see Supporting Information S2. +A PAuM is transferred on top of this model system (see Methods) to cover the graphene +sheet partially as sketched in Fig. 2(a). +In this way, we can probe the two effects that +contribute to surface sensitive Raman scattering: First, the surface enhancement by the +plasmonic nanopores of the PAuM via the graphene Raman signal and second, the bulk- +signal suppression by monitoring the Raman signal of the Si substrate with and without the +membrane. Figure 2(f) shows a light microscope image of the PAuM-graphene-stack on a +substrate. The PAuM appears as the yellow region, and monolayer graphene flake is marked +by the dotted line. As can be seen in the image, the graphene flake is partially covered by +the PAuM. +First, we compare individual Raman spectra of PAuM-covered and uncovered graphene +7 + +500 +550 +0 +2 +4 +Raman Intensity (a.u.) +Raman shift (cm-1) +Si +1600 +2600 +0 +2 +4 +6 + PAuM + Ref. +Raman intensity (a.u.) +Raman shift (cm-1) +G +2D +1600 +2600 +0 +4 +8 + PAuM + Ref. +Raman intensity (a.u.) +Raman shift (cm-1) +x20 +1600 +2600 +0 +2 +4 +6 + PAuM + Ref. +Raman intensity (a.u.) +Raman shift (cm-1) +x50 +0.0 +2.5 +I(2D) +# +0 +25 50 75 +I(2D) +0 +75 +150 +I(2D) +0 +5 +10 15 +0 +1 +Si Raman Intensity +x- position (µm) +PAuM +SiO2 +graphene +Si +FIG2 +SiO2 +graphene +PAuM +a +e +Si +200nm +Top +5um +a +b +c +d +e +f +g +h +i +j +graphene +PAuM on Graphene +PAuM +x +# +532nm +660nm +785nm +FIG. 2. Graphene as model surface to probe surface-sensitive Raman scattering by +nanoporous Au membranes a) Sample schematic where a graphene flake on a Si/SiO2 substrate +is partially covered by a PAuM. b-d) Raman spectra of the bare graphene (black, reference) com- +pared to spectra from graphene plus PAuM for (b, green) 532 nm, (c, red) 660 nm, and 785 nm (d, +purple) excitation. The background in (b-d) stems from gold luminescence with an modulation due +to etaloning for 532 nm. e) 1st order Silicon Raman peak with PAuM for 532 nm (green), 660 nm +(red), and 785 nm (purple) normalized and compared to the corresponding reference Raman spec- +tra (black) without PAuM. The spectra for each wavelength are offset for clarity. f) Microscope +image of graphene flake (dashed turquoise) partially covered by PAuM (yellow). × (with PAuM) +and # (reference) mark the locations of the spectra shown (b)-(e). g-i) Spatial Raman maps of +the graphene 2D mode for (g) 532 nm, (h) 660 nm, and (i) 785 nm excitation. For each excitation +wavelength, the intensity is normalized to the 2D intensity of bare graphene reference with interfer- +ence effect taken into account. j) Spatial profile of the normalized silicon Raman intensity along a +horizontal line through the locations x and # as indicated in (f) for all three excitation wavelengths. +The scale bar in (g-j) is 2 µm. +for three excitation wavelengths in Fig. 2(b-d). The uncovered graphene serves as refer- +ence and the corresponding reference spectra are shown in black for each excitation wave- +8 + +length, where × and # in Fig. 2(f) mark the measurement positions. For 532 nm excita- +tion, Fig. 2(b), the G- and 2D Raman modes of graphene show comparable intensities for +PAuM-covered (green) and uncovered graphene (black). This behavior is expected since the +nanopores in the membrane are not in resonance with the 532 nm laser excitation [25]. The +2D-to-G intensity ratio further confirms that the graphene in these measurements is indeed +a monolayer [32, 33]. +For 660 nm excitation, the Raman spectrum for PAuM-covered graphene (red ) is sub- +stantially enhanced when compared with uncovered graphene (black) in Fig. 2(c). +The +enhancement is even more pronounced for 785 nm excitation, Fig. 2(d). In this wavelength +range, plasmonic enhancement from the pores comes into play, in good agreement with our +transmission data, Fig. 1(c), and previous works [25]. For a quantitative analysis of the en- +hancement, we take care of potential reflection effects at the graphene-Si/SiO2 interface [34]. +After excluding interference effects (Supporting Information), we find enhancement factors +between 33 and 165 for the G and 2D modes (Table I). Note that the 2D-mode intensity +for bare graphene and 785 nm is below the noise level. We therefore use the noise level to +approximate the 2D-mode intensity. +The spatial Raman maps shown in Fig. 2(g-i) trace the intensity of the 2D Raman mode +of graphene normalized to the uncovered graphene reference for 532 nm, 660 nm and 785 nm +excitation, respectively. +Interference effects are accounted for in all maps. +The Raman +map in Fig. 2(g) confirms the negligible effect of the PAuM on the Raman response for +532 nm excitation. In contrast, the Raman maps for 660 nm and 785 nm excitation show the +striking enhancement by the PAuM. Clearly, the enhancement only occurs in areas of PAuM- +covered graphene. Local variations in the enhancements reflect the random distribution of +the plasmonic nanopores in the membrane with respect to geometry and orientation. +Second, we demonstrate the suppression of the bulk/substrate Raman signal by the +TABLE I. Enhancement factors, bulk suppression and Surface-enhancement × bulk-suppression as +figure of merit for the overall increase for 660 nm and 765 nm excitations. +λ +Enh (G) Enh (2D) Bulk suppression Surface-enhancement × bulk-suppression +660 nm +72 +69 +10 +690 to 720 +785 nm +33 +165 +6.6 +220 to 1100 +9 + +PAuM. To do so, we make use of the silicon substrate 300 nm below the PAuM, where +the silicon Raman signals are not enhanced. We compare the 1st order Raman peak of +silicon at 521 cm−1 with (colored) and without a PAuM (black) for our three excitation +wavelengths in Fig. 2(e). The spectra indicate a clear suppression of the silicon Raman +signal with PAuM for all wavelengths. This agrees with the attenuation expected for the +incoming laser light and the Raman scattered light for a non-porous membrane of compa- +rable thickness, see Fig. 2(c). A line scan across the edge of the PAuM (Fig. 2(j)), reveals a +constant silicon Raman signal for all wavelengths on either side of the edge. The experimen- +tally observed suppression by the PAuM amounts to a factor of 10 for 532 nm and 660 nm, +and 6.6 for 785 nm see Table I. Since the Si Raman signal without PAuM used as reference +is affected by interference, the bulk Raman signal suppression a factor 5 to 8 higher then +the experimentally obtained values, see Supporting Information S4. This brings the sup- +pression closer to the values expected from our transmission experiments, see Fig. 1(c). The +exact magnitude of bulk Raman signal suppression depends on the exact geometry of the +sample investigated. We therefore consider it most adequate to provide the experimental +values as a lower bound for bulk Raman signal suppression by our PAuM. The product of +surface Raman enhancement and bulk-suppression, which is the primary figure of merit for +surface-sensitive Raman scattering as suggested here, then amounts to values between 220 +and 1100, see Table I. +Depth Dependence of Raman Enhancement by PAuM +Next, we investigate the effective Raman enhancement of the material below the PAuM as +function of depth by simulation and experiment. Since nanopores act as slot antennas [25], +we simulate the Raman enhancement by a prototypical plasmonic nanoslot (10 nm × 68 nm) +similar in shape and size to nanopores found in the PAuM using a finite element solver +JCMsuite (version 5.2.0) for Maxwell’s equations. We assume a wavelength of 660 nm for +excitation and 800 nm for the Raman scattered light of the graphene 2D-mode (see methods +and Supporting Information S5). Figure 3(a) depicts the simulated enhancement in the xz- +plane along the nanoslot’s short axis (x-direction). We then extract the average enhancement +in the entire xy-plane from our simulation as function of depth z. To obtain the average, we +do not only consider the areas directly under the nanoslots but also the area around it. This +10 + +FIG3 - Depth +0 +5 +10 +15 +20 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 + sim. + exp +Norm. 2D mode enhancement +spacer thickness (nm) +-30 0 30 +-200 +-100 +0 +z-direction (nm) +x-direction (nm) +-1 +0 +1 +2 +3 +4 +log E4 enhancement +PAuM +SiO2 spacer +graphene +SiO2 +a +b +PAuM +SiO2 spacer +graphene +SiO2 +a +b +PAuM +SiO2 spacer +graphene +SiO2 +a +b +PAuM +SiO2 spacer +graphene +SiO2 +a +b +0 +-5 +-10 +-15 +-20 +0 +5 +10 +15 +20 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 + simulation + experimental +spacer thickness (nm) +Norm. 2D mode enhancement +z-direction +-30 0 30 +-300 +-200 +-100 +0 +z-direction (nm) +x-direction (nm) +-1 +0 +1 +2 +3 +4 +log E4 enhancement +PAuM +SiO2 spacer +graphene +SiO2 +a +b +0 +-5 +-10 +-15 +-20 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 + simulation + experimental +Norm. 2D mode enhancement +z-direction +-30 0 30 +-300 +-200 +-100 +0 +z-direction (nm) +x-direction (nm) +-1 +0 +1 +2 +3 +4 +log E4 enhancement +0 +-5 +-10 +-15 +-20 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 + simulation + experimental +Norm. 2D mode enhancement +z-direction +-30 0 30 +-300 +-200 +-100 +0 +z-direction (nm) +x-direction (nm) +-1 +0 +1 +2 +3 +4 +log E4 enhancement +V7 +FIG. 3. Enhancement of Raman signal as a function of distance from the surface a) +Simulated E4 enhancement (log scale) in the xz plane for a 22 nm thick gold membrane with a +10x68 nm slot on SiO2. b) The experimental (triangle) and simulated (circles) enhancement for the +graphene 2D Raman mode are shown as a function of SiO2 spacer thickness between the porous Au +membrane and graphene. The black line is an exponential fit to the simulated values. The dashed +area marks the volume within which 90 % of the total Raman enhancement occurs. The simulated +enhancement is normalized to the value at z = −0.35 nm, which is equivalent to the thickness of +single layer graphene. +includes an area 15 times larger than the area of the slot and accounts for the fact that each +nanopores is surrounded by continuous gold membrane segment, see Fig. 1(b). We plot the +average Raman enhancement of the graphene 2D-mode versus z in Fig. 3(b) and find that +the enhancement drops sharply with an increasing distance from the nanoslot. The decay +is described by an exponential function ez/τ with τ = 1.1 nm. This means that 90% of the +total Raman enhancement occurs within the first 2.5 nm below the nanoslot. +In the next step, we probe the field enhancement as function of distance from our PAuM +experimentally. To do so, we sputter 5 nm and 15 nm SiO2 spacers on graphene and subse- +quently transfer PAuM on top as sketched in Fig. 3(b), see Supporting Information S6-S8. +We plot the enhancement of the graphene 2D-mode with and without the two different +spacers together with the simulation data in Fig. 3(b). We find that the experimental en- +hancement is in excellent agreement with the simulation. This finding confirms that the +PAuM-enhanced Raman spectroscopy allows for truly surface-sensitive Raman scattering, +11 + +with an effective enhancement depth of less than 5 nm. +The good agreement between our simulation and the experimental data for graphene +allows us to infer more general properties of the surface enhancement provided by our PAuM. +Extended simulations indicate an equally fast decay with distance from the PAuM for the +entire relevant Raman frequency range of few cm−1 to 2500 cm−1 (Supporting Information +S5). Furthermore, we find the strongest enhancement and the sharpest enhancement decay +for nanopores with plasmonic resonances having high quality-factors. High quality-factors +are inherent to narrow pores, i.e., gap features < 10 nm [25]. Wider pores with lower aspect +ratios feature lower quality-factors, which leads to a reduced enhancement and greater decay +constants τ ∼ 5 nm, equivalent to 90% of the total Raman enhancement within the first +11.5 nm. The approach to achieve the best surface-to-bulk enhancement ratio of the Raman +signal is therefore to identify the highest Raman enhancement within a spatial map (see for +example Fig. 2) and evaluate its spectrum. +Application of surface-sensitive Raman scattering by PAuM: Probing the structural +properties of an oxide thin-film surface +In the next step, we further elucidate the use of the porous gold membranes for surface +investigations. We choose a complex oxide, a LaNiO3 thin film on LaAlO3, as showcase +material. +Complex oxides are particularly suitable because their physical properties are +highly sensitive to structural distortions [35–37]. LaNiO3 illustrates this well, as it is one of +the few conducting perovskite-type materials and therefore an important electrode material +for perovskite-type heterostructures [38]. However, its conductivity is thickness-dependent, +where below a thickness of 3 unit cells, LaNiO3 even exhibits insulating behavior. These +conductivity changes are attributed to structural inhomogeneities in the thin films. More +specifically, the bulk and the surface of a LaNiO3 film show different degrees of structural +distortions [26]. A previous Raman study of LaNiO3 thin films confirmed the structural +changes with film thickness [16]. However, these Raman spectroscopic measurements would +only give information about the average structures and did not distinguish between surface, +bulk or heterointerface. Here, using PAuM-enhanced Raman spectroscopy, we specifically +target the surface structure of LaNiO3 thin films to observe structural distortions directly. +A 20 nm-LaNiO3 thin film has been epitaxially grown on a (100)pc-oriented LaAlO3 sub- +12 + +FIG. 4. Surface Raman scattering of a LaNiO3 thin film. a) Sketch of the perovskite-type +LaNiO3 thin film on a (100)pc-oriented LaAlO3 substrate. The compound is partially covered by +a PAuM. × and # indicate the measurement positions of the spectra in (b) with and without +PAuM, respectively. b) The Raman spectrum of a LaNiO3 thin film on LaAlO3 with PAuM and +without PAuM are depicted. Important Raman bands have been highlighted for visual guidance. +The formation of a shoulder peak at 389 cm−1 to the main peak at 413 cm−1 is indicative of the +distinct structural difference of the surface layer compared to the bulk of the LaNiO3 thin film. +strate by pulsed laser deposition, see Methods. Subsequently, a PAuM is transferred onto +the thin film sample. As bulk, LaNiO3 and LaAlO3 crystallize in a rhombohedral perovskite- +type structure with the space group characterized by anti-phase rotations of the octahedra, +a−a−a− in Glazer’s notation (see Fig. 4a). The structure gives rise to five Raman-active +vibrational modes ΓRaman = A1g + 4 Eg [39]. Epitaxially strained LaNiO3 on LaAlO3 stabi- +lizes a monoclinic structure with the space group C2/c [16]. However, for simplicity reasons +and its close shape of the Raman spectrum, we retain the notations of the rhombohedral +bulk symmetry. For our Raman spectroscopy experiment, we use a excitation wavelength +of 785 nm, as it shows the best performance with PAuM, see Table I and Ref. [25] (see +Supporting Information S10 for the spectra under 660 nm excitation). Figure 4b shows the +Raman spectra of LaNiO3 on LaAlO3 with (top) and without PAuM (bottom). The light- +blue and gray boxes correspond to regions with vibrational bands of LaAlO3 and LaNiO3, +respectively. The Raman spectrum measured without PAuM is in perfect agreement with +13 + +b +a +without +with PAuM: +PAuM: +X +20 nm LaNiO3 thin film on LaAIO3 +PAuM +# +laser wavelength - 785 nm +LNO +Intensity (arb. units) +E. +Eg (LNO) +LaNiO, +LNO +mode splitting +withPAuM +X +M +M +without +LaAIO3 +LAC +PAuM +# +LAO +LAO +100 +200 +300 +400 +500 +600 +· Oxygen +OLanthanum +Wavenumber (cm-1)literature data [16, 40]. Yet, the comparison of the spectra with and without PAuM reveals +a number of striking differences: With the PAuM, the signal of the LaAlO3 substrate is +barely visible and can only be approximated as shoulder-like features in the region between +100 cm−1 and 160 cm−1. Furthermore, the LaNiO3-Eg mode around 400 cm−1 exhibits a +prominent difference. In the spectrum without PAuM, we find a single, albeit asymmetric, +peak at 412 cm−1. In contrast, the spectrum with PAuM has two distinct features cen- +tered at 389 cm−1 and 413 cm−1. (For a high resolution Raman spectrum of this region see +Supporting Information S10) The A1g mode of LaNiO3, on the other hand, shows only a +minor shift of ∼ 3 cm−1 from 213 cm−1 without PAuM to 216 cm−1 with PAuM. Note, at low +frequencies, the spectrum with PAuM is characterized by an intensity increase. We assign +this increase to the onset of intense low frequency Eg-mode of LaNiO3 at 74 cm−1 below the +spectral cut-off of the measurement set-up. +Our results allow for a number of interesting conclusions: First, in the presence of the +PAuM, the thin film signal is strongly enhanced with respect to the LaAlO3 substrate +signal. Second, the Eg-mode splitting in the Raman spectrum demonstrates that the PAuM- +enhanced Raman signal does not represent an average of the vibrational signal of the entire +film. Rather, the PAuM-enhanced Raman signal stems primarily from the surface layer, +namely the first few nanometers where the the PAuM enhancement is most effective. +From the material perspective, we know that the crystal structure changes significantly +within these first few nanometers [26]. A look at the vibrational patterns reveals further +details of those surface distortions. +The A1g and Eg around 213 cm−1 and 403 cm−1 of +bulk-like LaNiO3 correspond to an octahedral tilt and bending vibration patterns, respec- +tively [39, 40]. Therefore, we suggest that deformations of the octahedra, permitted by the +C/2c symmetry, dominate the structural changes in the surface region over changes of the +octahedron tilt-angles. +Note that the PAuM plasmonic enhancement of the Raman spectra of oxide materials +differs from graphene, our model surface previously, where the primary scattering volume is +reduced. Hence, the Raman spectra with a PAuM has a lower total intensity and signal-to- +noise ratio than without the PAuM, despite the enhancement of surface Raman signals +(Fig. 4). +Overall, our findings demonstrate that enhancement of the Raman signal by +PAuM allows the effective extraction of the Raman response of an oxide surface. +This +novel spectroscopy-based access to the structure reveals major structural changes at the +14 + +film surface compared to the bulk of the film, in agreement with the monoclinic symmetry +of the film. +DISCUSSION +The improvement of the surface-to-bulk Raman intensity ratio by up to three orders of +magnitude is the primary quality of the PAuM. Moreover, PAuM can sustain excitation +power densities up to 106 W cm−2 for 785 nm excitation without structural damage. This +exceeds the threshold of classical plasmonic structures by two orders of magnitude [25] +and further increases the detectable Raman signal from a surface. Furthermore, surface- +sensitive Raman measurements with PAuM equally function at cryogenic temperatures (see +Supporting Information S9). Hence, PAuM enhanced Raman spectroscopy may give access +to temperature-driven phase transition of a surface layer[3]. +The geometrical randomness of the pores in our PAuM always provides pores with suit- +able enhancement for an arbitrary combination of excitation wavelength, polarization, and +refractive index of the material, see Supporting Information S5. We can further tune the +ratio between surface enhancement and bulk suppression by altering the PAuM thickness. +Thicker PAuM lead to fewer nanopores and reduced transmittance, thinner PAuM to more +nanopores and increased transmittance. The alteration of the nanopore geometries as a +function of film thickness, furthermore, allows to adjust its resonance. +CONCLUSIONS +We have demonstrated that transferable nanoporous gold membranes (PAuM) enable +surface-sensitive Raman spectroscopy. Nanopores in the membrane act as plasmonic hot- +spots for enhanced Raman scattering. Simultaneously, the membrane nature of the PAuM +suppresses bulk Raman signals. +Using graphene as a model surface, we have shown an +increase of the Raman surface-to-bulk ratio of a factor 1100. Simulations combined with +Raman measurements on buried graphene samples showed that the enhancement drops +exponentially with distance from the PAuM. 90% of the enhancement occur within the top +2.5 nm of the probed material. To demonstrate the ultility of our approach, we applied it to +an open scientific question – the spectroscopic analysis of the surface structure of LaNiO3. +15 + +By PAuM-enabled surface-sensitive Raman scattering of a LaNiO3 thin film on LaAlO3, +we found major structural changes at the surface of LaNiO3, which had not been observed +by Raman spectroscopy to date. Surface-sensitive Raman scattering, as introduced in this +work, therefore extends the use of Raman spectroscopy as a surface analytical technique. +Our approach is not limited to crystalline surfaces, but may also be employed to monitor +surface-bound chemical reactions or to characterize biological membranes. +METHODS +PAuM Manufacturing and transfer: A non-continuous gold film of 20 nm is evapo- +rated on a silicon/silicon dioxide (Si/SiO2) Wafer with an oxide thickness of 30 nm at a rate +of 0.2 nm/s. The PAuM is subsequently coated with poly(methyl metacrylate) (PMMA) in +Anisol (2 w%), followed by a floating etch in buffered hydrofluoric acid (BHF), releasing the +PAuM/PMMA from the substrate. The PAuM/PMMA is subsequently rinsed by floating +on deionised water for a total of 60 min, after which it is scooped with the target substrate +and let dry. Oxygen plasma (for non-organic samples, 400 W for 4 min) or acetone/isopropyl +alcohol baths (for organic samples) are used to remove the PMMA. (See Supporting Section +S1 for details). Freestanding PAuM for pore size characterization and transmission mea- +surement are obtained using a pre-patterned Si/Si3N4 chip with arrays of 64 x 4 µm holes +as described by Celebi et al. [41]. +Optical Transmission of PAuM: Transmission measurements were performed using a +self-built setup by focussing white light from broadband supercontinuum laser (NKT) with +an objective (NA 0.9) on the freely suspended PAuM from the top. A long-distance objective +(NA 0.7) place below the PAuM collected the transmitted light, which was recorded with +a Princeton Instruments Acton spectrometer. Reference spectra taken without the PAuM +were used to substract the background and eliminate any wavelength-dependence of the +spectrometer. +Raman spectroscopy Raman measurements were performed on a Horiba LabRam Ra- +man spectrometer equipped with a motorized stage. Laser powers were kept below 3 mW +(100X Objective, NA 0.9) with integration times up to between 1s and 20s. +LaNiO3 thin film growth: The LaNiO3 film was grown on a single crystalline LaAlO3 +(001)pc substrate (CrysTec GmbH) by pulsed laser deposition using a 248 nm KrF excimer +16 + +laser (LPXpro, Coherent Ltd.). The film was grown at a substrate temperature of 700◦C +under an oxygen partial pressure of 0.1 mbar. The laser fluence was set to 1 J/cm2 with +a repetition rate of 2 Hz. X-ray diffraction measurements were performed on a four-circle +thin-film diffractometer (PanAlytical X’Pert3 MRD) with Cu Kα1 radiation (λ = 1.54 Å). +X-ray reflectometry was performed to quantify the LaNiO3 film thicknesses, see Supporting +Information S10. +Surface topography measurements were conducted using atomic force +microscopy in a Bruker Multimode 8 scanning probe microscope with Pt-coated Si tips +(MikroMasch, k = 5.4 N/m). +AUTHOR CONTRIBUTIONS +M.C.W., M.F., R.M.W, and S.H. conceived the project. R.M.W. and K.P.S. fabricated +the nanoporous membranes and graphene samples, and performed the transfers of PAuM +onto target substrates, supervised by J.V.. M.P. and S.H measured the optical transmission +of the PAuM. S.H. performed and analysed the wavelength-dependent Raman measure- +ments. E.B. and S.H. performed the Raman measurements at cryogenic temperatures. G.K. +performed the numerical simulations. M.F.S. and M.T. fabricated the LaNiO3 thin film on +LaAlO3. M.C.W. performed and analysed the Raman measurements on the LaNiO3 thin +films with support from S.H.. L.H., G.M., M.P., M.F., and L.N. assisted in the Raman mea- +surements and numerical simulations. R.M.W., G.K., M.F., M.C.W. and S.H. interpreted +the results and co-wrote the manuscript with input from all authors. S.H. coordinated and +supervised the project. +ACKNOWLEDGMENTS +R.M.W. thanks the Binnig and Rohrer Nanotechnology Center in Rueschlikon/Switzerland +and ETH Zurich Department of Materials. G.K. was funded by the Deutsche Forschungsge- +meinschaft (DFG, German Research Foundation) - Project-ID 182087777 - SFB 951. S.H. +acknowledges funding from the Deutsche Forschungsgemeinschaft (DFG) under the Emmy +Noether Initiative (Project-ID 433878606) and from financial support by ETH Zürich Ca- +reer Seed Grant SEED-16 17-1. +M.T. acknowledges the financial support by the Swiss +National Science Foundation under project No. 200021_188414. M.F. and L.N. acknowl- +17 + +edges the financial support by the Swiss National Science Foundation under project No. +200020_192362. +M.C.W. is grateful for financial support by the Région des Pays de la +Loire under the Etoile Montante Initiative (2022_11808) and the PULAR Academy. The +authors express their gratitude to Marcela Giraldo for initiating the meeting that started +this project. The authors acknowledge the use of the facilities at the Scientific Center for +Optical and Electron Microscopy (ScopeM) at ETH Zuürich. +COMPETING FINANCIAL INTEREST +RMW, LH, GM and SH are planning to use porous gold membranes commercially. RMW +holds a patent on manufacturing porous gold membranes (Patent US17/294748) Otherwise, +there are not competing financial interests. +REFERENCES +∗ Correspondence email address: sebastian.heeg@physik.hu-berlin.de +[1] D. K. Pandey, H. L. Kagdada, P. Sanchora, D. K. Singh, In Modern Techniques of Spectroscopy, +145–184. Springer, 2021. +[2] R. S. Das, Y. 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Park, Science 2014, 344, 6181 289. +21 + diff --git a/QdE2T4oBgHgl3EQfsAgY/content/tmp_files/load_file.txt b/QdE2T4oBgHgl3EQfsAgY/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d058c7b524f6deb1bf7b252912f6192d2a13d8df --- /dev/null +++ b/QdE2T4oBgHgl3EQfsAgY/content/tmp_files/load_file.txt @@ -0,0 +1,879 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf,len=878 +page_content='Surface-Sensitive Raman Scattering by Transferable Nanoporous Plasmonic Membranes Roman M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Wyss,1, 2 Günter Kewes,1 Martin Frimmer,3 Karl-Philipp Schlichting,4 Markus Parzefall,3 Eric Bonvin,3 Martin F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Sarott,5 Morgan Trassin,5 Lala 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+page_content=' ETH Zürich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' 8093 Zürich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Switzerland 6Institut des Molécules et Matériaux du Mans,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' UMR 6283 CNRS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Le Mans Université,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' 72085 Le Mans,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' France (Dated: January 11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' 2023) 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='04054v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='mtrl-sci] 10 Jan 2023 Abstract Raman spectroscopy is a powerful technique to characterize materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' It probes non-destructively chemical composition, crystallinity, defects, strain and coupling phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' However, the Raman response of surfaces or thin films is often weak and obscured by dominant bulk signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Here we overcome this limitation by placing a transferable porous gold membrane (PAuM) on top of the surface of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Slot-like nanopores in the membrane act as plasmonic slot antennas and enhance the Raman response of the surface or thin film underneath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Simultaneously, the PAuM suppresses the penetration of the excitation laser into the bulk, efficiently blocking the bulk Raman signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Using graphene as a model surface, we show that these two simultaneous effects lead to an increase in the surface-to-bulk Raman signal ratio by three orders of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' We find that 90 % of the Raman enhancement occurs within the top 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='5 nm of the material, demonstrating truly surface-sensitive Raman scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' To validate our approach, we analyze the surface of a LaNiO3 thin film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' We observe a Raman mode splitting for the LaNiO3 surface-layer, which is spectroscopic evidence that the surface structure differs from the bulk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' This result underpins that PAuM give direct access to Raman signatures of surfaces and their structural properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' KEYWORDS Raman spectroscopy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' surface;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' surface-sensitive Raman scattering;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' plasmonic nanopore;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' com- plex oxide thin film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' INTRODUCTION Raman spectroscopy, the inelastic scattering of light by vibrations or phonons, is a widespread analytical tool to study and characterize materials [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Owing to its versatil- ity, simplicity and specificity, Raman spectroscopy is used in material science [2], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' to study phase transitions [3, 4], catalysis [5] or novel 2D materials [6], or even in pharmaceu- tics and biomedical diagnostics [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' In principle, Raman spectroscopy is ideal to study the structure of surfaces, since their atomic registry differs from the bulk of the material and may additionally be modified by terminations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' This leads to changes in the frequency of the Raman active vibrations or to peak-splitting as a result of a change insymmetry [8–10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' However, the study of surfaces and thin films by Raman spectroscopy is notoriously diffi- 2 cult as light typically penetrates several micrometers into the material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' The overall Raman response is, therefore, dominated by the bulk, while the Raman signals of the surface are orders of magnitude weaker and mostly go undetected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Hence, obtaining Raman signals of thin films requires a minimal thickness of several tens to hundreds of nanometers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Raman signatures of surfaces are often not observed at all [11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' One way to address the issue of vanishing surface- or thin film Raman signals is ultravio- let (UV) Raman spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Here, a UV laser excites the sample instead of a laser in the visible or near infrared range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' UV-Raman takes advantage of the shallow penetration-depth of the UV-light into many materials [13, 14] and is not impeded by autofluorescence effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' However, the penetration of the laser in the material is still in the order of hundreds of nanometers, and can only be reduced further for materials with suitable band gaps [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Moreover, the low damage threshold of many materials to UV light limits the application of UV Raman [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Probing surface and thin-film Raman signals has also been addressed by mathematical decomposition of a large stack of spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' To do so, multiple spectra of a sam- ple are measured under varying conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' An example can be altering the laser focal point with respect the the sample surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Subsequently, a statistical analysis allows to decom- pose the spectra into substrate/bulk and surface/thin film contributions [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' However, this method requires an already detectable signal of the surface or the thin film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Furthermore, the measurement times are often beyond practical use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' A general strategy to enhance Raman signals is plasmon-enhanced Raman scattering (PERS) [17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Here, the enhancement in PERS occurs in the vicinity of metallic nanos- tructures and arises from the near-fields of localized surface plasmon resonances in the metal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Particularly striking enhancement occurs in a nanoscale gap between two metal nanostruc- tures, also referred to as plasmonic hotspot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' This configuration enables the detection of molecules adsorbed at the hotspot down to the single molecule level, embodying surface- enhanced Raman scattering (SERS) [19–22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Even though SERS can be realized with a large number of different nanoparticle geometries, its use to enhance the Raman signals of a surface or thin film is limited: there is no geometry that efficiently interfaces a plasmonic hotspot between two metallic structures with a flat and extended surface or film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Tip- enhanced Raman spectroscopy (TERS), where a plasmonic hotspot at the apex of a metal tip scans over a surface, partially solves this problem [23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' The enhancement, however, occurs only at one spot and is weaker than for gap type geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' The largest drawback 3 of TERS is that bulk Raman signals are recorded together with the TERS signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' On top, TERS remains a complex and challenging technique, such that its use to probe surfaces or thin films is rarely reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Overall, the ideal plasmonic structure to study surfaces and thin films with Raman spec- troscopy consists of a flat gap type plasmonic hotspot with simultaneous bulk Raman signal suppression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Plasmonic nanoslots, rectangular nanoscale voids in a thin metallic film (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' nanoporous membrane), fulfill these conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Upon resonant excitation, these slots act as plasmonic slot antennas that harbour localized and enhanced near-fields, which rapidly decay outside the pore within few nanometers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Placed on a material, the slot’s near-fields in- teract primarily with the material surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Away from the nanopore the metallic membrane reflects incident fields and bulk Raman signals, which effectively suppresses the bulk Raman signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Recently, transferable and easy-to-manufacture porous gold membranes (PAuM) with nanoscale pores acting as plasmonic slot antennas were introduced [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' It was shown that individual pores feature local Raman enhancement factors up to 104 to 105 and sustain high excitation powers (106 W cm−2), which makes them the ideal plasmonic structure to study surfaces and thin films with Raman spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Here, we use porous gold membranes to enhance the surface Raman signal and to simul- taneously suppress the bulk signal of the sample under investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Using wavelength- dependent Raman spectroscopy, we show that PAuM enhance the surface-to-bulk Raman signal ratio by up to three orders of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Combining experiment and simulation, we reveal that the enhancement decays exponentially in the material such that 90 % of the enhanced signal occurs within the top 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='5 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Hence, our approach enables highly surface- sensitive Raman spectroscopy for weak or bulk-obscured Raman signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' We directly apply this technique to study the surface of a 20 nm LaNiO3 thin film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' We find Raman signatures of the surface that differ from the bulk of the 20 nm film, in line with theoretical predictions and experimental observations using scanning tunneling microscope (STM) [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Our work, therefore, underscores the power of PAuM-supported Raman spectroscopy of surfaces and thin films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' 4 RESULTS Our paper is structured as follows: First, we introduce the PAuM manufacturing and its working principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Second, we demonstrate the surface enhancement and bulk suppression of Raman signals by a wavelength-dependent study with graphene as a model surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Third, we unravel the depth dependence of the Raman response in experiment and simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' To do so, we probe graphene sheets buried at various depths from the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Finally, we showcase the use of PAuM to enhance the Raman response of a 20 nm LaNiO3 thin film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Manufacturing and Working Principle of PAuM Figure 1 (a) illustrates the key idea of this work: Without PAuM, a laser penetrates several multiples of its wavelength into the material, limited by absorption and focal depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' The Raman scattered signal, therefore, originates primarily from within the bulk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' The surface Raman signal remains weak or non-detectable due to the vanishing small scattering volume of the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' In contrast, using PAuM, the surface Raman signal is drastically enhanced compared the bulk Raman signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' This results from two simultaneous effects: 1) local plasmonic enhancement within the metallic nanopores and 2) the suppression of the laser penetration into the bulk and of residual bulk Raman signal reaching the detection pathway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Our non-continuous membranes are formed by evaporation of a 20 nm gold film on a SiO2/Si wafer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' The membrane is subsequently transferred onto the sample [25] (see Methods and Supporting Information S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' To characterize PAuM, we transfer a 1 cm2-sized membrane on a Si/Si3N4 chip bearing arrays of 4 µm holes (Methods) as shown in the optical microscope image in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' 1(b), top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' The scanning electron microscope (SEM) images Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' 1(b), middle and lower panel, reveal that the PAuM spans over the circular hole as a freestanding, mechanically stable membrane [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' The pores in the PAuM are visible in the SEM images as dark spots and come in various shapes and sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' The majority of pores is round or slot-like and below 100 nm in size (See Supporting Information S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' A recent study demonstrated that the nanopores in the PAuM act as plasmonic nanoslot antennas [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' The nanopores harbour intense light fields upon excitation at their plasmonic resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' The energy of the plasmonic resonance depends one the shape and aspect ratio of the individual pores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' The highest field enhancement occurs 5 FIG1 a Surface Bulk Raman Shift Raman Intensity Surface Bulk Raman Shift Raman Intensity Bulk Surface PAuM 500 600 700 800 15 30 45 pores in PAuM Transmission (%) wavelength sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' no pores b c 1 cm 1 µm 100 nm FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Principle of surface-sensitive Raman scattering enabled by PAuM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' a) Nanopores in the gold membrane enhance the Raman signal of the surface while suppressing the bulk Raman signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' b) Photographic image (top) of a 20 nm PAuM transferred on a Si/Si3N4 partially sus- pended over 4 µm holes (See Methods), forming freestanding membrane-like structures, as shown in the Scanning electron microscope (SEM) image (middle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' The nanopores are visible as dark irregular, slot-like and circular features (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' c) Optical transmission of a freestanding PAuM measured as function of wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' The dashed line corresponds to a simulated non-porous Au film with 20 nm thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' The plasmonic resonances of the nanopores give rise to the increased transmission in the experimental data from 650 nm to 850 nm compared to the simulation of a non-porous film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' for narrow slot-like pores with an excitation polarized perpendicular to the pores’ long axis, see Supporting Information and Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' [25] for an extended discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' 1 (c), we compare the optical transmission of freestanding 20 nm PAuM to the 6 10/6/2017 dwell HV HFW mag 只 WD 1 μm 10:37:01 AM 10 μs 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='00 kV 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='08 μm 25 000 x 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='4 mm G17 after H202transmission of a simulated gold film without nanopores of the same thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' The general shape of the optical transmission is in good agreement with the simulated results (dashed line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' However, deviations occur from 650 nm to 850 nm with an increased transmission probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Such increased transmission is expected when the nanopores of PAuM are res- onantly excited and act as nano-antennas [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' The colored lines correspond to individual transmission measurements at different sample positions, and their varying deviations from the simulated trend indicate the varying geometries of the nanopores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' We conclude that Ra- man enhancement can be expected in the spectral range from 650 nm to 850 nm in agreement with our previous study [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' The random nature of pore geometries in both - dimension and orientation - provides a substantial number of pores that will resonantly couple to an incident laser light with arbitrary polarization and wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Measurement of Surface Enhancement Using Graphene Probes Graphene is ideal as a model material to test and quantify surface-sensitive Raman scat- tering as it can mimic a surface or thin film due to its two-dimensional nature [28–30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Furthermore, the Raman spectrum of graphene is well understood and the main Raman fea- tures are intense and independent of excitation wavelength as well as polarization [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Any dependence of the Raman features of graphene interfaced with our porous Au membrane can hence be attributed entirely to nanopore interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' To probe the interaction and enhancement of PAuM with a surface, we place a graphene sheet, acting as an test surface, on a flat Si/SiO2 substrate, see Supporting Information S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' A PAuM is transferred on top of this model system (see Methods) to cover the graphene sheet partially as sketched in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' In this way, we can probe the two effects that contribute to surface sensitive Raman scattering: First, the surface enhancement by the plasmonic nanopores of the PAuM via the graphene Raman signal and second, the bulk- signal suppression by monitoring the Raman signal of the Si substrate with and without the membrane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Figure 2(f) shows a light microscope image of the PAuM-graphene-stack on a substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' The PAuM appears as the yellow region, and monolayer graphene flake is marked by the dotted line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' As can be seen in the image, the graphene flake is partially covered by the PAuM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' First, we compare individual Raman spectra of PAuM-covered and uncovered graphene 7 500 550 0 2 4 Raman Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=') Raman shift (cm-1) Si 1600 2600 0 2 4 6 PAuM Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Raman intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=') Raman shift (cm-1) G 2D 1600 2600 0 4 8 PAuM Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Raman intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=') Raman shift (cm-1) x20 1600 2600 0 2 4 6 PAuM Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Raman intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=') Raman shift (cm-1) x50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='5 I(2D) # 0 25 50 75 I(2D) 0 75 150 I(2D) 0 5 10 15 0 1 Si Raman Intensity x- position (µm) PAuM SiO2 graphene Si FIG2 SiO2 graphene PAuM a e Si 200nm Top 5um a b c d e f g h i j graphene PAuM on Graphene PAuM x # 532nm 660nm 785nm FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Graphene as model surface to probe surface-sensitive Raman scattering by nanoporous Au membranes a) Sample schematic where a graphene flake on a Si/SiO2 substrate is partially covered by a PAuM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' b-d) Raman spectra of the bare graphene (black, reference) com- pared to spectra from graphene plus PAuM for (b, green) 532 nm, (c, red) 660 nm, and 785 nm (d, purple) excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' The background in (b-d) stems from gold luminescence with an modulation due to etaloning for 532 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' e) 1st order Silicon Raman peak with PAuM for 532 nm (green), 660 nm (red), and 785 nm (purple) normalized and compared to the corresponding reference Raman spec- tra (black) without PAuM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' The spectra for each wavelength are offset for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' f) Microscope image of graphene flake (dashed turquoise) partially covered by PAuM (yellow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' × (with PAuM) and # (reference) mark the locations of the spectra shown (b)-(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' g-i) Spatial Raman maps of the graphene 2D mode for (g) 532 nm, (h) 660 nm, and (i) 785 nm excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' For each excitation wavelength, the intensity is normalized to the 2D intensity of bare graphene reference with interfer- ence effect taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' j) Spatial profile of the normalized silicon Raman intensity along a horizontal line through the locations x and # as indicated in (f) for all three excitation wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' The scale bar in (g-j) is 2 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' for three excitation wavelengths in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' 2(b-d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' The uncovered graphene serves as refer- ence and the corresponding reference spectra are shown in black for each excitation wave- 8 length, where × and # in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' 2(f) mark the measurement positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' For 532 nm excita- tion, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' 2(b), the G- and 2D Raman modes of graphene show comparable intensities for PAuM-covered (green) and uncovered graphene (black).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' This behavior is expected since the nanopores in the membrane are not in resonance with the 532 nm laser excitation [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' The 2D-to-G intensity ratio further confirms that the graphene in these measurements is indeed a monolayer [32, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' For 660 nm excitation, the Raman spectrum for PAuM-covered graphene (red ) is sub- stantially enhanced when compared with uncovered graphene (black) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' 2(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' The enhancement is even more pronounced for 785 nm excitation, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' 2(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' In this wavelength range, plasmonic enhancement from the pores comes into play, in good agreement with our transmission data, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' 1(c), and previous works [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' For a quantitative analysis of the en- hancement, we take care of potential reflection effects at the graphene-Si/SiO2 interface [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' After excluding interference effects (Supporting Information), we find enhancement factors between 33 and 165 for the G and 2D modes (Table I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Note that the 2D-mode intensity for bare graphene and 785 nm is below the noise level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' We therefore use the noise level to approximate the 2D-mode intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' The spatial Raman maps shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' 2(g-i) trace the intensity of the 2D Raman mode of graphene normalized to the uncovered graphene reference for 532 nm, 660 nm and 785 nm excitation, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Interference effects are accounted for in all maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' The Raman map in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' 2(g) confirms the negligible effect of the PAuM on the Raman response for 532 nm excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' In contrast, the Raman maps for 660 nm and 785 nm excitation show the striking enhancement by the PAuM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Clearly, the enhancement only occurs in areas of PAuM- covered graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Local variations in the enhancements reflect the random distribution of the plasmonic nanopores in the membrane with respect to geometry and orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Second, we demonstrate the suppression of the bulk/substrate Raman signal by the TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Enhancement factors, bulk suppression and Surface-enhancement × bulk-suppression as figure of merit for the overall increase for 660 nm and 765 nm excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' λ Enh (G) Enh (2D) Bulk suppression Surface-enhancement × bulk-suppression 660 nm 72 69 10 690 to 720 785 nm 33 165 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='6 220 to 1100 9 PAuM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' To do so, we make use of the silicon substrate 300 nm below the PAuM, where the silicon Raman signals are not enhanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' We compare the 1st order Raman peak of silicon at 521 cm−1 with (colored) and without a PAuM (black) for our three excitation wavelengths in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' 2(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' The spectra indicate a clear suppression of the silicon Raman signal with PAuM for all wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' This agrees with the attenuation expected for the incoming laser light and the Raman scattered light for a non-porous membrane of compa- rable thickness, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' 2(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' A line scan across the edge of the PAuM (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' 2(j)), reveals a constant silicon Raman signal for all wavelengths on either side of the edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' The experimen- tally observed suppression by the PAuM amounts to a factor of 10 for 532 nm and 660 nm, and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='6 for 785 nm see Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Since the Si Raman signal without PAuM used as reference is affected by interference, the bulk Raman signal suppression a factor 5 to 8 higher then the experimentally obtained values, see Supporting Information S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' This brings the sup- pression closer to the values expected from our transmission experiments, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' 1(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' The exact magnitude of bulk Raman signal suppression depends on the exact geometry of the sample investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' We therefore consider it most adequate to provide the experimental values as a lower bound for bulk Raman signal suppression by our PAuM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' The product of surface Raman enhancement and bulk-suppression, which is the primary figure of merit for surface-sensitive Raman scattering as suggested here, then amounts to values between 220 and 1100, see Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Depth Dependence of Raman Enhancement by PAuM Next, we investigate the effective Raman enhancement of the material below the PAuM as function of depth by simulation and experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Since nanopores act as slot antennas [25], we simulate the Raman enhancement by a prototypical plasmonic nanoslot (10 nm × 68 nm) similar in shape and size to nanopores found in the PAuM using a finite element solver JCMsuite (version 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='0) for Maxwell’s equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' We assume a wavelength of 660 nm for excitation and 800 nm for the Raman scattered light of the graphene 2D-mode (see methods and Supporting Information S5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Figure 3(a) depicts the simulated enhancement in the xz- plane along the nanoslot’s short axis (x-direction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' We then extract the average enhancement in the entire xy-plane from our simulation as function of depth z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' To obtain the average, we do not only consider the areas directly under the nanoslots but also the area around it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' This 10 FIG3 - Depth 0 5 10 15 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='0 sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' exp Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' 2D mode enhancement spacer thickness (nm) 30 0 30 200 100 0 z-direction (nm) x-direction (nm) 1 0 1 2 3 4 log E4 enhancement PAuM SiO2 spacer graphene SiO2 a b PAuM SiO2 spacer graphene SiO2 a b PAuM SiO2 spacer graphene SiO2 a b PAuM SiO2 spacer graphene SiO2 a b 0 5 10 15 20 0 5 10 15 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='0 simulation experimental spacer thickness (nm) Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' 2D mode enhancement z-direction 30 0 30 300 200 100 0 z-direction (nm) x-direction (nm) 1 0 1 2 3 4 log E4 enhancement PAuM SiO2 spacer graphene SiO2 a b 0 5 10 15 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='0 simulation experimental Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' 2D mode enhancement z-direction 30 0 30 300 200 100 0 z-direction (nm) x-direction (nm) 1 0 1 2 3 4 log E4 enhancement 0 5 10 15 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='0 simulation experimental Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' 2D mode enhancement z-direction 30 0 30 300 200 100 0 z-direction (nm) x-direction (nm) 1 0 1 2 3 4 log E4 enhancement V7 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Enhancement of Raman signal as a function of distance from the surface a) Simulated E4 enhancement (log scale) in the xz plane for a 22 nm thick gold membrane with a 10x68 nm slot on SiO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' b) The experimental (triangle) and simulated (circles) enhancement for the graphene 2D Raman mode are shown as a function of SiO2 spacer thickness between the porous Au membrane and graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' The black line is an exponential fit to the simulated values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' The dashed area marks the volume within which 90 % of the total Raman enhancement occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' The simulated enhancement is normalized to the value at z = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='35 nm, which is equivalent to the thickness of single layer graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' includes an area 15 times larger than the area of the slot and accounts for the fact that each nanopores is surrounded by continuous gold membrane segment, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' We plot the average Raman enhancement of the graphene 2D-mode versus z in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' 3(b) and find that the enhancement drops sharply with an increasing distance from the nanoslot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' The decay is described by an exponential function ez/τ with τ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='1 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' This means that 90% of the total Raman enhancement occurs within the first 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='5 nm below the nanoslot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' In the next step, we probe the field enhancement as function of distance from our PAuM experimentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' To do so, we sputter 5 nm and 15 nm SiO2 spacers on graphene and subse- quently transfer PAuM on top as sketched in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' 3(b), see Supporting Information S6-S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' We plot the enhancement of the graphene 2D-mode with and without the two different spacers together with the simulation data in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' 3(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' We find that the experimental en- hancement is in excellent agreement with the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' This finding confirms that the PAuM-enhanced Raman spectroscopy allows for truly surface-sensitive Raman scattering, 11 with an effective enhancement depth of less than 5 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' The good agreement between our simulation and the experimental data for graphene allows us to infer more general properties of the surface enhancement provided by our PAuM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Extended simulations indicate an equally fast decay with distance from the PAuM for the entire relevant Raman frequency range of few cm−1 to 2500 cm−1 (Supporting Information S5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Furthermore, we find the strongest enhancement and the sharpest enhancement decay for nanopores with plasmonic resonances having high quality-factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' High quality-factors are inherent to narrow pores, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=', gap features < 10 nm [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Wider pores with lower aspect ratios feature lower quality-factors, which leads to a reduced enhancement and greater decay constants τ ∼ 5 nm, equivalent to 90% of the total Raman enhancement within the first 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='5 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' The approach to achieve the best surface-to-bulk enhancement ratio of the Raman signal is therefore to identify the highest Raman enhancement within a spatial map (see for example Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' 2) and evaluate its spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Application of surface-sensitive Raman scattering by PAuM: Probing the structural properties of an oxide thin-film surface In the next step, we further elucidate the use of the porous gold membranes for surface investigations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' We choose a complex oxide, a LaNiO3 thin film on LaAlO3, as showcase material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Complex oxides are particularly suitable because their physical properties are highly sensitive to structural distortions [35–37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' LaNiO3 illustrates this well, as it is one of the few conducting perovskite-type materials and therefore an important electrode material for perovskite-type heterostructures [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' However, its conductivity is thickness-dependent, where below a thickness of 3 unit cells, LaNiO3 even exhibits insulating behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' These conductivity changes are attributed to structural inhomogeneities in the thin films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' More specifically, the bulk and the surface of a LaNiO3 film show different degrees of structural distortions [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' A previous Raman study of LaNiO3 thin films confirmed the structural changes with film thickness [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' However, these Raman spectroscopic measurements would only give information about the average structures and did not distinguish between surface, bulk or heterointerface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Here, using PAuM-enhanced Raman spectroscopy, we specifically target the surface structure of LaNiO3 thin films to observe structural distortions directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' A 20 nm-LaNiO3 thin film has been epitaxially grown on a (100)pc-oriented LaAlO3 sub- 12 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Surface Raman scattering of a LaNiO3 thin film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' a) Sketch of the perovskite-type LaNiO3 thin film on a (100)pc-oriented LaAlO3 substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' The compound is partially covered by a PAuM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' × and # indicate the measurement positions of the spectra in (b) with and without PAuM, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' b) The Raman spectrum of a LaNiO3 thin film on LaAlO3 with PAuM and without PAuM are depicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Important Raman bands have been highlighted for visual guidance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' The formation of a shoulder peak at 389 cm−1 to the main peak at 413 cm−1 is indicative of the distinct structural difference of the surface layer compared to the bulk of the LaNiO3 thin film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' strate by pulsed laser deposition, see Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Subsequently, a PAuM is transferred onto the thin film sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' As bulk, LaNiO3 and LaAlO3 crystallize in a rhombohedral perovskite- type structure with the space group characterized by anti-phase rotations of the octahedra, a−a−a− in Glazer’s notation (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' 4a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' The structure gives rise to five Raman-active vibrational modes ΓRaman = A1g + 4 Eg [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Epitaxially strained LaNiO3 on LaAlO3 stabi- lizes a monoclinic structure with the space group C2/c [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' However, for simplicity reasons and its close shape of the Raman spectrum, we retain the notations of the rhombohedral bulk symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' For our Raman spectroscopy experiment, we use a excitation wavelength of 785 nm, as it shows the best performance with PAuM, see Table I and Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' [25] (see Supporting Information S10 for the spectra under 660 nm excitation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Figure 4b shows the Raman spectra of LaNiO3 on LaAlO3 with (top) and without PAuM (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' The light- blue and gray boxes correspond to regions with vibrational bands of LaAlO3 and LaNiO3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' The Raman spectrum measured without PAuM is in perfect agreement with 13 b a without with PAuM: PAuM: X 20 nm LaNiO3 thin film on LaAIO3 PAuM # laser wavelength - 785 nm LNO Intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' units) E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Eg (LNO) LaNiO, LNO mode splitting withPAuM X M M without LaAIO3 LAC PAuM # LAO LAO 100 200 300 400 500 600 Oxygen OLanthanum Wavenumber (cm-1)literature data [16, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Yet, the comparison of the spectra with and without PAuM reveals a number of striking differences: With the PAuM, the signal of the LaAlO3 substrate is barely visible and can only be approximated as shoulder-like features in the region between 100 cm−1 and 160 cm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Furthermore, the LaNiO3-Eg mode around 400 cm−1 exhibits a prominent difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' In the spectrum without PAuM, we find a single, albeit asymmetric, peak at 412 cm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' In contrast, the spectrum with PAuM has two distinct features cen- tered at 389 cm−1 and 413 cm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' (For a high resolution Raman spectrum of this region see Supporting Information S10) The A1g mode of LaNiO3, on the other hand, shows only a minor shift of ∼ 3 cm−1 from 213 cm−1 without PAuM to 216 cm−1 with PAuM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Note, at low frequencies, the spectrum with PAuM is characterized by an intensity increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' We assign this increase to the onset of intense low frequency Eg-mode of LaNiO3 at 74 cm−1 below the spectral cut-off of the measurement set-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Our results allow for a number of interesting conclusions: First, in the presence of the PAuM, the thin film signal is strongly enhanced with respect to the LaAlO3 substrate signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Second, the Eg-mode splitting in the Raman spectrum demonstrates that the PAuM- enhanced Raman signal does not represent an average of the vibrational signal of the entire film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Rather, the PAuM-enhanced Raman signal stems primarily from the surface layer, namely the first few nanometers where the the PAuM enhancement is most effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' From the material perspective, we know that the crystal structure changes significantly within these first few nanometers [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' A look at the vibrational patterns reveals further details of those surface distortions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' The A1g and Eg around 213 cm−1 and 403 cm−1 of bulk-like LaNiO3 correspond to an octahedral tilt and bending vibration patterns, respec- tively [39, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Therefore, we suggest that deformations of the octahedra, permitted by the C/2c symmetry, dominate the structural changes in the surface region over changes of the octahedron tilt-angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Note that the PAuM plasmonic enhancement of the Raman spectra of oxide materials differs from graphene, our model surface previously, where the primary scattering volume is reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Hence, the Raman spectra with a PAuM has a lower total intensity and signal-to- noise ratio than without the PAuM, despite the enhancement of surface Raman signals (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Overall, our findings demonstrate that enhancement of the Raman signal by PAuM allows the effective extraction of the Raman response of an oxide surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' This novel spectroscopy-based access to the structure reveals major structural changes at the 14 film surface compared to the bulk of the film, in agreement with the monoclinic symmetry of the film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' DISCUSSION The improvement of the surface-to-bulk Raman intensity ratio by up to three orders of magnitude is the primary quality of the PAuM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Moreover, PAuM can sustain excitation power densities up to 106 W cm−2 for 785 nm excitation without structural damage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' This exceeds the threshold of classical plasmonic structures by two orders of magnitude [25] and further increases the detectable Raman signal from a surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Furthermore, surface- sensitive Raman measurements with PAuM equally function at cryogenic temperatures (see Supporting Information S9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Hence, PAuM enhanced Raman spectroscopy may give access to temperature-driven phase transition of a surface layer[3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' The geometrical randomness of the pores in our PAuM always provides pores with suit- able enhancement for an arbitrary combination of excitation wavelength, polarization, and refractive index of the material, see Supporting Information S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' We can further tune the ratio between surface enhancement and bulk suppression by altering the PAuM thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Thicker PAuM lead to fewer nanopores and reduced transmittance, thinner PAuM to more nanopores and increased transmittance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' The alteration of the nanopore geometries as a function of film thickness, furthermore, allows to adjust its resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' CONCLUSIONS We have demonstrated that transferable nanoporous gold membranes (PAuM) enable surface-sensitive Raman spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Nanopores in the membrane act as plasmonic hot- spots for enhanced Raman scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Simultaneously, the membrane nature of the PAuM suppresses bulk Raman signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Using graphene as a model surface, we have shown an increase of the Raman surface-to-bulk ratio of a factor 1100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Simulations combined with Raman measurements on buried graphene samples showed that the enhancement drops exponentially with distance from the PAuM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' 90% of the enhancement occur within the top 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='5 nm of the probed material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' To demonstrate the ultility of our approach, we applied it to an open scientific question – the spectroscopic analysis of the surface structure of LaNiO3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' 15 By PAuM-enabled surface-sensitive Raman scattering of a LaNiO3 thin film on LaAlO3, we found major structural changes at the surface of LaNiO3, which had not been observed by Raman spectroscopy to date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Surface-sensitive Raman scattering, as introduced in this work, therefore extends the use of Raman spectroscopy as a surface analytical technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Our approach is not limited to crystalline surfaces, but may also be employed to monitor surface-bound chemical reactions or to characterize biological membranes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' METHODS PAuM Manufacturing and transfer: A non-continuous gold film of 20 nm is evapo- rated on a silicon/silicon dioxide (Si/SiO2) Wafer with an oxide thickness of 30 nm at a rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='2 nm/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' The PAuM is subsequently coated with poly(methyl metacrylate) (PMMA) in Anisol (2 w%), followed by a floating etch in buffered hydrofluoric acid (BHF), releasing the PAuM/PMMA from the substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' The PAuM/PMMA is subsequently rinsed by floating on deionised water for a total of 60 min, after which it is scooped with the target substrate and let dry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Oxygen plasma (for non-organic samples, 400 W for 4 min) or acetone/isopropyl alcohol baths (for organic samples) are used to remove the PMMA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' (See Supporting Section S1 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Freestanding PAuM for pore size characterization and transmission mea- surement are obtained using a pre-patterned Si/Si3N4 chip with arrays of 64 x 4 µm holes as described by Celebi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Optical Transmission of PAuM: Transmission measurements were performed using a self-built setup by focussing white light from broadband supercontinuum laser (NKT) with an objective (NA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='9) on the freely suspended PAuM from the top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' A long-distance objective (NA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='7) place below the PAuM collected the transmitted light, which was recorded with a Princeton Instruments Acton spectrometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Reference spectra taken without the PAuM were used to substract the background and eliminate any wavelength-dependence of the spectrometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Raman spectroscopy Raman measurements were performed on a Horiba LabRam Ra- man spectrometer equipped with a motorized stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Laser powers were kept below 3 mW (100X Objective, NA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='9) with integration times up to between 1s and 20s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' LaNiO3 thin film growth: The LaNiO3 film was grown on a single crystalline LaAlO3 (001)pc substrate (CrysTec GmbH) by pulsed laser deposition using a 248 nm KrF excimer 16 laser (LPXpro, Coherent Ltd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' The film was grown at a substrate temperature of 700◦C under an oxygen partial pressure of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='1 mbar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' The laser fluence was set to 1 J/cm2 with a repetition rate of 2 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' X-ray diffraction measurements were performed on a four-circle thin-film diffractometer (PanAlytical X’Pert3 MRD) with Cu Kα1 radiation (λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='54 Å).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' X-ray reflectometry was performed to quantify the LaNiO3 film thicknesses, see Supporting Information S10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Surface topography measurements were conducted using atomic force microscopy in a Bruker Multimode 8 scanning probe microscope with Pt-coated Si tips (MikroMasch, k = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='4 N/m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' AUTHOR CONTRIBUTIONS M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=', M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=', R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='W, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' conceived the project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' fabricated the nanoporous membranes and graphene samples, and performed the transfers of PAuM onto target substrates, supervised by J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='. M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='H measured the optical transmission of the PAuM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' performed and analysed the wavelength-dependent Raman measure- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' performed the Raman measurements at cryogenic temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' performed the numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' fabricated the LaNiO3 thin film on LaAlO3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' performed and analysed the Raman measurements on the LaNiO3 thin films with support from S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='. L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=', G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=', M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=', M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=', and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' assisted in the Raman mea- surements and numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=', G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=', M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=', M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' interpreted the results and co-wrote the manuscript with input from all authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' coordinated and supervised the project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' ACKNOWLEDGMENTS R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' thanks the Binnig and Rohrer Nanotechnology Center in Rueschlikon/Switzerland and ETH Zurich Department of Materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' was funded by the Deutsche Forschungsge- meinschaft (DFG, German Research Foundation) - Project-ID 182087777 - SFB 951.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' acknowledges funding from the Deutsche Forschungsgemeinschaft (DFG) under the Emmy Noether Initiative (Project-ID 433878606) and from financial support by ETH Zürich Ca- reer Seed Grant SEED-16 17-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' acknowledges the financial support by the Swiss National Science Foundation under project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' 200021_188414.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' acknowl- 17 edges the financial support by the Swiss National Science Foundation under project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' 200020_192362.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' is grateful for financial support by the Région des Pays de la Loire under the Etoile Montante Initiative (2022_11808) and the PULAR Academy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' The authors express their gratitude to Marcela Giraldo for initiating the meeting that started this project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' The authors acknowledge the use of the facilities at the Scientific Center for Optical and Electron Microscopy (ScopeM) at ETH Zuürich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' COMPETING FINANCIAL INTEREST RMW, LH, GM and SH are planning to use porous gold membranes commercially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' RMW holds a patent on manufacturing porous gold membranes (Patent US17/294748) Otherwise, there are not competing financial interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' REFERENCES ∗ Correspondence email address: sebastian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='heeg@physik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='hu-berlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content='de [1] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfsAgY/content/2301.04054v1.pdf'} +page_content=' Pandey, H.' metadata={'source': 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index 0000000000000000000000000000000000000000..b2afe9a73a0f6defbdabc89511103bc683326334 --- /dev/null +++ b/RtFPT4oBgHgl3EQfqDWD/content/tmp_files/2301.13140v1.pdf.txt @@ -0,0 +1,877 @@ +Generalized phase conjugation for incoherent light in complex media +YoonSeok Baek,1, ∗ Hilton B. de Aguiar,1 and Sylvain Gigan1 +1Laboratoire Kastler Brossel, ENS–Universite PSL, CNRS, Sorbonne Université, +Collège de France, 24 Rue Lhomond, F-75005 Paris, France +Shaping light deep inside complex media, such as biological tissue, is critical to many research +fields. Although the coherent control of scattered light via wavefront shaping has made significant +advances in addressing this challenge, controlling light over extended or multiple targets without +physical access inside a medium remains elusive. Here we present a generalized phase conjugation +method for incoherent light, which enables the non-invasive light control based on incoherent emis- +sion from multiple target positions. Our method characterizes the scattering responses of hidden +sources by retrieving mutually incoherent scattered fields from speckle patterns. By time-reversing +scattered fluorescence with digital phase conjugation, we experimentally demonstrate focusing of +light on individual and multiple targets. We also demonstrate maximum energy delivery to an ex- +tended target through a scattering medium by exploiting transmission eigenchannels. This paves +the way to control light propagation in complex media using incoherent contrasts mechanisms. +I. +Main +Delivering optical energy and transmitting information +through complex media remains an important challenge +in many fields of studies, including optical manipulation +[1], deep-tissue imaging [2, 3] and optogenetics [4, 5]. In +recent years, it has been shown that the coherent control +of scattered light can manipulate spatial, spectral and +temporal distributions of light in scattering media [6– +10]. However, such capabilities are greatly limited with- +out physical access inside a medium because the scat- +tering response to target position is difficult to charac- +terize. +As a result, non-invasive light control over ex- +tended or multiple targets remains elusive despite being +crucial for real-world applications. Optimizing incident +wavefront based on a feedback signal [11–18] is mostly +limited to focusing on a single isolated target, and even +then it has limitations that require numerous changes of +the wavefront. While time-reversal or phase conjugation +techniques [19–30] allow for effective light delivery to an +optical or virtual source, they cannot individually control +light on multiple targets. +Here we address these challenges by generalizing phase +conjugation for incoherent light. Our approach utilizes +incoherent emission from multiple targets. We first char- +acterize scattering responses of these hidden sources by +retrieving mutually incoherent fields from spatially mod- +ulated speckle patterns. The retrieved fields are related +to the field transmission matrix [31], and their phase con- +jugation enables light control over the desired positions. +We demonstrate this experimentally by focusing light +on individual and multiple fluorescent targets through +a scattering medium. Finally, we show that transmission +eigenchannels can be identified by decomposing the inco- +herent fields and demonstrate maximum energy delivery +to a hidden extended target. +∗ yoonseok.baek@lkb.ens.fr +II. +Results +A schematic of the experiment is illustrated in Fig. 1. +We consider a scenario where multiple fluorescent targets +are hidden by a scattering medium and act as guidestars. +Fluorescence emitted by these guidestars is scattered, re- +sulting in an incoherent addition of speckle patterns on +the camera. Under this condition, we aim to deliver light +back to each of the guidestars by time-reversing the scat- +tered fluorescence. Our approach consists of retrieving +multiple incoherent fields that compose the fluorescence, +and using them to generate phase conjugated beams. To +this end, we introduce wavefront modulation of the scat- +tered fluorescence with a spatial light modulator (SLM). +This modulation induces changes in the measured inco- +herent speckle patterns, providing information to retrieve +the scattered fields that will later be used for phase con- +jugation. To explain the retrieval process in detail, we +introduce a partial field transmission matrix T whose in- +put and output are fields at the SLM plane and guidestar +positions, respectively, Eguidestar = T ESLM. Accord- +ing to the time-reversal symmetry, the scattering of flu- +orescence emitted by N guidestars can be expressed by +the rows of the transmission matrix, T = [t1, . . . , tN]⊤, +where tn represents the scattered field that corresponds +to an individual guidestar. We note that the brightness +of each guidestar is assumed to be the same, and it is +generally represented by the amplitude of tn. Then, the +incoherent speckle pattern measured by the camera is ex- +pressed as I0 = diag +� +(T F )†T F +� +, where F represents the +discrete Fourier transform between the SLM and camera +plane. If we apply M different SLM modulations and ex- +press the mth modulation as a diagonal matrix Sm, the +incoherent speckle pattern after the modulation is +Im = diag +� +(T SmF )† T SmF +� +. +(1) +A set of scattered fields that satisfy Eq. 1 can be found +by iteratively minimizing the error between the measured +and predicted Im (see Supplementary Information Sec- +tion 1 for more information). We note that finding T is +arXiv:2301.13140v1 [physics.optics] 30 Jan 2023 + +2 +Fluorescent +guidestars +Scattering +medium +Scattered fields +Mixed-state +phase retrieval +Wavefront +synthesis +SLM +Lens +Camera images +1 +1 +2 +M +2 +M +... +1 2 +N +... +... +(a.u.) +0 +1 +0 +π +(rad) +A +θ +Figure 1. Schematic of phase conjugation with incoher- +ent fluorescence: Multiple fluorescent guidestars are hidden +behind a scattering medium. The scattered fluorescence, com- +posed of mutually incoherent speckle fields, is modulated by +an SLM. The modulated fluorescence is Fourier transformed +by a lens and results in the incoherently added speckle pat- +terns on the camera. Then mixed-state phase retrieval recov- +ers a set of scattered fields, whose phase conjugation enables +targeted light control. +equivalent to mixed-state reconstruction [32] with inher- +ent ambiguity, where T and its unitary transformation +are indistinguishable by intensity. This can be confirmed +by replacing T with UT in Eq. 1. For this reason, the +scattered fields are retrieved as a mixture of tn: +H = UT , +(2) +where H is a set of retrieved fields hn, H = [h1, ..., hN]⊤, +and U is an arbitrary unitary matrix. Despite this am- +biguity in the reconstruction, the retrieved fields offer +unique capabilities for phase conjugation, as we show be- +low. +Incoherent phase conjugation +The time-reversal of scattered fluorescence will regen- +erate light at hidden sources, creating foci on the entire +targets. One way to accomplish this is to phase conjugate +the incoherent scattered fields from individual guidestars, +tn. Alternatively, we chose to use hn since it gives the +identical phase conjugation result (see Supplementary In- +formation Section 2). +To demonstrate this incoherent phase conjugation, we +introduced several 1 µm fluorescent beads as guidestars +behind the scattering medium. We retrieved the multiple +scattered fields according to the number of the guidestars. +In our experiments, each 1 µm bead was considered as +an individual guidestar as the speckle grain size at the +target plane was ∼0.9 µm. The incoherent phase con- +jugation was implemented by sequentially generating N +phase-conjugated fields of hn using the SLM, and by mea- +suring time-averaged responses (see Methods for phase +conjugation). +To evaluate the performance of the phase conjugation, +we first conducted an experiment with a single fluores- +cent bead (Fig. 2a). By phase-conjugating the scattered +field, we observed a strong focus on the bead (Fig. 2b). +This is in clear contrast with the random speckle gener- +ated by a beam with a random wavefront (Fig. 2c). The +enhancement factor, defined as the ratio between the op- +timized focus intensity and mean background intensity, +was ∼4,400. We then placed multiple fluorescent beads +(Fig. 2d–f) behind the scattering medium. By incoher- +ently phase conjugating the scattered fields, we success- +fully generated foci at every guidestar positions (Fig. 2g– +i). Despite the minimal spectral memory effect [33], we +were also able to excite the bead through the scattering +medium by generating a phase conjugation beam at the +excitation wavelength (Fig. S1). +Selective focusing on individual targets +In order to selectively focus on individual targets, it +is necessary to demix the individual fields tn from their +mixture hn. We note that tn is not strictly orthogonal, +and thus the orthogonalization of hn can not be a solu- +tion. Our solution was to directly invert Eq. 2 by finding +U. To this end, we utilized the memory effect [34, 35], +where neighboring guidestars generate correlated speckle +patterns. Specifically, we iteratively applied a random +unitary transformation to the retrieved fields H, such +that the correlation between the transformed speckle pat- +terns is maximized (see Supplementary Information Sec- +tion 3). +Figure 3 shows the experimental result with 5 fluores- +cent beads. +When the scattered fields hn are directly +used for phase conjugation, each phase conjugation gen- +erated foci on several guidestars with different intensities +(Fig. 3b). The demixed fields, on the other hand, gen- +erated a focus on a single guidestar, showing that tn is +successfully recovered (Fig. 3c). The memory effect range +in this experiment, defined as the full width at half max- +imum of speckle cross-correlation, was 5 µm, which is +much smaller than the spatial extent of the guidestars. +This result shows that the selective focusing via demix- +ing is possible as long as a pair of guidestars lies within +the memory effect range. +Targeted energy delivery +Maximum energy delivery through scattering media re- +quires an eigenchannel of T [36, 37]. The transmission +eigenchannels correspond to the singular vectors of T , +and the first singular vector with the largest singular + +3 +5 μm +d +f +g +h +i +b +a +c +5 μm +5 μm +5 μm +0 +(a.u.) +1 +0 +(a.u.) +1 +Figure 2. Incoherent phase conjugation for multiple targets: a, The fluorescence image of an 1 µm bead taken from the +side without a scattering medium. b, Intensity at the target plane with the phase conjugation of scattered fluorescence. (Inset) +The phase of the scattered field shown in the HSV colormap. The central highlighted part is used for the phase conjugation. c, +Intensity at the target plane generated by a random wavefront. d–f, The fluorescence images of multiple beads hidden behind +the scattering medium. g–i, Intensity at the target plane with incoherent phase conjugation of scattered fields. +a +b +Before demixing +h1 +h2 +h3 +h4 +h5 +c +After demixing +t1 +t2 +t3 +t4 +t5 +5 μm +Figure 3. Selective focusing on individual targets: a, The fluorescence image of a target comprised of 1 µm beads. b,c, +Intensity at the target plane after the phase conjugation of individual scattered fields before (b) and after (c) the demixing +process. (Insets) The phase of the scattered fields used for phase conjugation. +value, delivers the maximum energy to the target. Al- +though the direct access to T is not always possible, the +transmission eigenchannels of T can be found using H. +This is because the eigenchannels of T and H are iden- +tical because H†H = T †T according to Eq. 2. Thus, +we can deliver the maximum energy to extended targets +using the first singular vector of H. +To demonstrate the targeted energy delivery, we placed +a 5 µm fluorescent ink droplet behind the scattering +medium (Fig. 4a). Based on the size of the target, we +estimated the number of incoherent fields and retrieved +23 scattered fields. We then performed the singular value +decomposition of the retrieved fields H. Finally, we in- +jected fields that corresponds to the singular vectors vn +and observed the energy delivered to the target. When a +random wavefront is injected to the scattering medium, a +speckle pattern is generated at the target plane (Fig. 4b). +In contrast, the singular vectors produce intensity distri- +butions highly concentrated on the target (Fig. 4c). By +summing the results of all the singular vectors, we con- +firmed that the energy is delivered only to the target area +(Fig. 4d). The first singular vector v1 shows a 174-times +increase in the energy on the target, compared to random +wavefronts. The enhancement decays with the singular +vector index [Fig. 4(e)]. We observed that the values are +not perfectly sorted in a descending order, which we be- +lieve is due to the numerical error in the retrieved fields +and to the use of the phase-only SLM. +III. +Discussion and conclusion +We have presented an approach to control light in scat- +tering media without the physical access to a target plane +by extending phase conjugation to incoherent light. We +have demonstrated focusing and maximum energy deliv- + +五年4 +d +e +1 +23 +0 +100 +200 +Singular vector index +Energy enhancement +a +5 μm +0 +1 +(a.u.) +0 +1 +(a.u.) +c +b +v3 +v1 +v2 +Figure 4. Targeted energy delivery: a, The fluorescence image of an extended target hidden behind a scattering medium. +The image was taken from the side without the scattering medium. b, Intensity at the target plane when a random phase +pattern is displayed on the SLM. c, Intensity at the target plane using the first 3 singular vectors of H. The images are +normalized for the result of the first singular vector. d, The sum of all the results using 23 singular vectors. e, Enhancement +of the energy delivered to the target compared to random realizations. Dash circles in (c) and (d) indicate the boundary of +the target. +ery to extended fluorescent targets through a scattering +medium. Our approach differs from the existing phase +conjugation techniques by addressing all the mutually in- +coherent fields of scattered fluorescence. Another impor- +tant aspect is that it does not require the precise align- +ment between the camera and SLM [38], since the scat- +tered fields are retrieved at the SLM plane. Its principle +of characterizing the scattering response is entirely pas- +sive, as it does not alter the emission of guidestars, as +opposed to to techniques that modulates the excitation +wavefront (e.g. [14]). +For the proposed method, it is important to estimate +the number of mutually incoherent waves, N. +This is +because the underestimation of N results in imperfect +reconstruction of the scattered fields. We note that the +overestimation is allowed because it results in redundant +reconstruction (see Fig. S2). Nevertheless, it is recom- +mended to use the smallest possible value of N for the +minimal measurements and computation time. There are +different methods to estimate N. The contrast of fluores- +cence speckle is an useful indicator for N, as it decreases +as +√ +N [39]. It is also possible to find N by analyzing the +error in the mixed-state phase retrieval or the singular +value distribution [40] for different values of N. In our +demonstrations, we did not consider the spectral degrees +of freedom because narrow spectral responses were mea- +sured by using interference filters. If the detection band- +width is greater than the spectral memory effect range, +different spectral components should be considered in es- +timating N. +Another important consideration is the number of +modulation M required for the field retrieval, which +scales linearly with N. +In experiments, reliable phase +conjugation results were obtained when M ≥ 6N (see +Supplementary Information Section 1). +This linearity +can be attributed to the multiplexed information in the +intensity of multiple incoherent fields. We emphasize that +M scales with the number of incoherent waves N not with +the number of controlled input modes of the SLM. Re- +cent advances in phase retrieval [41, 42] show that few +measurements are sufficient in retrieving a coherent field +(N = 1). In this regard, we believe that in principle even +fewer M may be used for our method. +In our proof-of-principle experiments, we used the sim- +ple algorithms for the retrieval and demixing of the scat- +tered fields. The performance of the algorithms can be +enhanced by incorporating constraints, convex optimiza- +tion [42] or the generalized memory effect [43]. The total +measurement time can be reduced by designing a setup +with minimal energy loss, and by using a sensitive de- +tector, such as an EMCCD. The incoherent phase conju- +gation can be improved by employing a high-speed SLM +or possibly by shaping partially coherent light [44]. Such +improvements will benefit applications that require light +to be delivered to multiple targets simultaneously. +In conclusion, our method enables versatile light con- +trol over extended or multiple targets using incoher- +ent contrast mechanisms. The concept can be applied +to different incoherent emissions, such as spontaneous +Raman scattering [17, 18], and a wide range of pho- +toluminescence [45]. +Furthermore, it enables the pas- +sive characterization of a transmission matrix, open- +ing up the possibility to generalized light control using +transmission-matrix-based operators [10]. +We envision +that the proposed approach will enable targeted light de- +livery through thick biological tissue, facilitating biomed- +ical applications, such as optogenetic stimulation and +phototherapy. + +5 +IV. +Methods +Experimental setup +The experimental setup is shown in Fig. S3. A laser +diode (λ = 488 nm, LP488-SF20G, Thorlabs) was used +to excite fluorescence. The excitation beam was deliv- +ered to guidestars by a lens (L1, f = 100 mm) and an +objective lens (Plan N 20× 0.4, Olympus). +To moni- +tor the guidestars and phase conjugation, a dichroic mir- +ror (DMLP490R, Thorlabs), a lens (L2, f = 200 mm), +a bandpass filter (FL532-10, Thorlabs), and a camera +(acA5472-17um, Basler) were placed on the side without +a scattering medium. +The guidestars were fluorescent +ink mixed with UV glue (NOA 68, Norland), and fluo- +rescent beads (F8803, Invitrogen) immersed in glycerol. +A scattering medium was a 220-grit ground glass diffuser, +placed approximately 170 µm away from the guidestars. +On the detection side, the scattered fluorescence was col- +lected by an objective lens (MPlan N 50× 0.75, Olympus) +and two lenses (L3, f = 75 mm; L4, f = 150 mm). An +SLM (X10468-04, Hamamtsu) and a linear polarizer were +used to modulate the fluorescence. The modulated flu- +orescence is Fourier transformed by lenses (L5, f = 100 +mm; L6, f = 200 mm; L7, f = 250 mm) and then cap- +ture by an sCMOS camera (PCO.edge 5.5, PCO) with +bandpass filters (BP2, FL532-3 and FBH520-40, Thor- +labs). An iris was placed between L6 and L7 to adjust +the speckle grain size at the camera. For phase conjuga- +tion, a laser (λ = 532 nm, Compass 215M-50, Coherent) +was collimated using a 5 µm pinhole and a lens (L8, f = 6 +mm). The collimated laser was then shaped by the SLM +to generate a phase-conjugated beam, which propagated +back to the medium. A flip mirror was used to switch be- +tween the fluorescence detection and phase conjugation. +Phase-conjugated beam generation +A phase-conjugated beam is generated using the colli- +mated laser beam and the SLM. The collimated beam is +shaped to the phase conjugate of a given scattered field +Escattered by displaying a phase pattern that corresponds +to −arg (Escattered) on the SLM. The resultant phase +conjugated beam propagates back through the scattering +medium, retracing the scattering paths of fluorescence. +Acknowledgement +This research was funded by the FET-Open (Dynamic- +863203) and the European Research Council under Grant +Agreement No. +724473 (SMARTIES). Y.B. acknowl- +edges the support from Basic Science Research Pro- +gram through the National Research Foundation of +Korea (NRF) funded by the Ministry of Education +(2022R1A6A3A03072108). +References +[1] Tomáš Čižmár, Michael Mazilu, and Kishan Dholakia, In +situ wavefront correction and its application to microma- +nipulation, Nature Photonics 4, 388 (2010). +[2] Vasilis Ntziachristos, Going deeper than microscopy: the +optical imaging frontier in biology, Nature methods 7, 603 +(2010). +[3] Joel Kubby, Sylvain Gigan, +and Meng Cui, Wavefront +shaping for biomedical imaging (Cambridge University +Press, 2019). +[4] Edward S Boyden, Feng Zhang, Ernst Bamberg, Georg +Nagel, +and Karl Deisseroth, Millisecond-timescale, ge- +netically targeted optical control of neural activity, Nature +neuroscience 8, 1263 (2005). +[5] Jonghee Yoon, Minji Lee, KyeoReh Lee, Nury Kim, +Jin Man Kim, Jongchan Park, Hyeonseung Yu, Chulhee +Choi, Won Do Heo, +and YongKeun Park, Optogenetic +control of cell signaling pathway through scattering skull +using wavefront shaping, Scientific reports 5, 1 (2015). +[6] Ivo M Vellekoop and AP Mosk, Focusing coherent light +through opaque strongly scattering media, Optics letters +32, 2309 (2007). +[7] Allard P Mosk, Ad Lagendijk, Geoffroy Lerosey, +and +Mathias Fink, Controlling waves in space and time for +imaging and focusing in complex media, Nature photonics +6, 283 (2012). +[8] Roarke Horstmeyer, Haowen Ruan, and Changhuei Yang, +Guidestar-assisted wavefront-shaping methods for focusing +light into biological tissue, Nature photonics 9, 563 (2015). +[9] Stefan Rotter and Sylvain Gigan, Light fields in complex +media: Mesoscopic scattering meets wave control, Reviews +of Modern Physics 89, 015005 (2017). +[10] Hui Cao, Allard Pieter Mosk, and Stefan Rotter, Shaping +the propagation of light in complex media, Nature Physics +18, 994 (2022). +[11] Ori Katz, Eran Small, Yefeng Guan, and Yaron Silber- +berg, Noninvasive nonlinear focusing and imaging through +strongly scattering turbid layers, Optica 1, 170 (2014). +[12] Anat Daniel, Dan Oron, +and Yaron Silberberg, Light +focusing through scattering media via linear fluorescence +variance maximization, and its application for fluorescence +imaging, Optics Express 27, 21778 (2019). +[13] Antoine Boniface, Baptiste Blochet, Jonathan Dong, +and Sylvain Gigan, Noninvasive light focusing in scatter- +ing media using speckle variance optimization, Optica 6, +1381 (2019). +[14] Antoine Boniface, Jonathan Dong, +and Sylvain Gi- +gan, Non-invasive focusing and imaging in scattering me- +dia with a fluorescence-based transmission matrix, Nature +communications 11, 1 (2020). +[15] Dayan Li, Sujit Kumar Sahoo, Huy Quoc Lam, Dong +Wang, +and Cuong Dang, Non-invasive optical focusing +inside strongly scattering media with linear fluorescence, +Applied Physics Letters 116, 241104 (2020). +[16] Bernhard Rauer, Hilton B de Aguiar, Laurent Bourdieu, +and Sylvain Gigan, Scattering correcting wavefront shap- +ing for three-photon microscopy, Optics Letters 47, 6233 +(2022). +[17] Jonathan V +Thompson, +Graham A Throckmorton, +Brett H Hokr, and Vladislav V Yakovlev, Wavefront shap- +ing enhanced raman scattering in a turbid medium, Optics +letters 41, 1769 (2016). + +6 +[18] Bingxin Tian, Bernhard Rauer, Antoine Boniface, Jun +Han, Sylvain Gigan, +and Hilton B de Aguiar, Non- +invasive chemically selective energy delivery and focusing +inside a scattering medium guided by raman scattering, +Optics Letters 47, 2145 (2022). +[19] Zahid Yaqoob, Demetri Psaltis, Michael S Feld, +and +Changhuei Yang, Optical phase conjugation for turbidity +suppression in biological samples, Nature photonics 2, 110 +(2008). +[20] Meng Cui and Changhuei Yang, Implementation of a dig- +ital optical phase conjugation system and its application to +study the robustness of turbidity suppression by phase con- +jugation, Optics express 18, 3444 (2010). +[21] Chia-Lung Hsieh, Ye Pu, Rachel Grange, +and Demetri +Psaltis, Digital phase conjugation of second harmonic ra- +diation emitted by nanoparticles in turbid media, Optics +express 18, 12283 (2010). +[22] Xiao Xu, Honglin Liu, +and Lihong V Wang, Time- +reversed ultrasonically encoded optical focusing into scat- +tering media, Nature photonics 5, 154 (2011). +[23] Ivo M. 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We initialize the algorithm by letting the scattered fields hn as N complex Gaussian random fields. +Then we apply the SLM modulation Sm to the scattered fields hn. We note that Sm is a diagonal matrix whose +diagonal elements correspond to the field modulation given by the SLM. In the experiments, SLM was divided into +macro-pixels (composed of 40 × 40 pixels) having random phase values (0 or π). After the SLM modulation Sm, the +field at the camera plane is expressed as ˜h(m) +n += (SmF )⊤ hn. Then we conduct Fourier magnitude projection using +an auxiliary function ψ: +ψ(m) +n +(k) = +� +Im(k) +� +n +���˜h(m) +n +(k) +��� +2 +�γ +˜h(m) +n +(k), +(S1) +where k is a coordinate in the spatial frequency domain, ˜h(m) +n +(k) is the modulated field at the camera plane, corre- +sponding to ˜h(m) +n +, and γ is a constant parameter. Then the fields are updated by compensating the SLM modulation: +hn = +� +(SmF )⊤�−1 +ψ(m) +n +, +(S2) +where ψ(m) +n +is the vector representation of ψ(m) +n +(k). The update through Eq. S1–S2 is continued for the entire M +modulations. The whole process is repeated for several times to obtain consistent hn. With γ = 1/2, this method +can be interpreted as the maximum likelihood reconstruction and retrieves the incoherent fields [1]. However, we +observed that few initial iterations with γ = 1 accelerates the convergence greatly [2], in both numerical simulations +and experiments. Thus, in our experiments, we used γ = 1 for the first 20 iterations and γ = 1/2 for the rest of the +iterations. We confirmed numerically that the algorithm retrieves a correct set of fields for M ≥ 4N in the absence +of measurement noise. The size of macro-pixel had almost no effect on the reconstruction, except when its size is +comparable to the SLM. We observed that the minimum value of M required for correct reconstruction increases +depending on the noise level. For the experimental results shown in the main text, we used 6–8N modulations. +2. +Incoherent phase conjugation +When all the mutually incoherent components of fluorescence are time-reversed, the intensity at the nth guidestar +can be expressed by multiplying the nth row of the transmission matrix, t⊤ +n , and phase-conjugated field t∗ +m: +� +m +��t⊤ +n t∗ +m +��2 = t⊤ +n T †T t∗ +n, +(S3) +where ∗ denotes the complex conjugate. For a medium with negligible reflection and absorption, T †T ≈ 1. Thus +Eq. S3 is simplified to t⊤ +n t∗ +n, which remains more or less constant regardless of n. As a result, the incoherent phase +conjugation of tn generates foci on the entire guidestars with roughly the same intensity. We note that as long as the +loss of a medium is minimal, the phase conjugation well approximates the lossless case, producing high-contrast foci +on the entire guidestars [3]. Similarly, the intensity for the incoherent phase conjugation of hn is expressed as, +� +m +��t⊤ +n h∗ +m +��2 = t⊤ +n H†Ht∗ +n. +(S4) +Since H†H = T †T according to Eq. 2, Eq. S4 is identical to Eq. S3. Thus, the incoherent phase conjugation of hn +and tn are identical. +3. +Demixing incoherent fields +The scattered fields tn can be recovered by reversing Eq. 2. To this end, we developed an algorithm inspired by +the simulated annealing [4] that exploits the memory effect between the speckle patterns at the camera plane. The + +2 +algorithm starts by applying a random unitary transformation to the retrieved fields H and expressing the result at +the camera plane: +H = Uj +fj ˆUHF , +(S5) +where j is the iteration index, Uj is a random unitary matrix, fj is an exponent that decays to 0 over the iteration, +and ˆU is the current best estimate of U −1. The decaying behavior of fj makes Uj induce smaller changes as the +iteration progresses: limj→∞ Uj +fj = 1. Then the correlation metric C is calculated at every iteration, +C = +N +� +n=1 +� +max +� +|hn|2 ⋆ |h(n mod N)+1|2��2 +, +(S6) +where hn(k) is the unitary-transformed speckle field at the camera plane, corresponding to the nth row of H, and ⋆ +is the cross-correlation. If the value of C is greater than the previous values, ˆU is updated to Uj +fj ˆU. If not, ˆU is +unchanged. At the end of the iteration, the algorithm results in ˆU ≈ U −1, and the scattered fields are recovered by +inverting Eq. 2: T = ˆUH. We initialized the iteration with ˆU = 1 and fj = 100/(j + 1), and the maximum iteration +number of 104. We note that the experimental noise in H can lead to an incorrect demixing. To mitigate this issue, +we calculate C only using speckles brighter than the average, |hn|2 > µ, where µ is the mean value of every |hn|2. + +3 +a +b +0 +1 +(a.u.) +a +0 +0.2 +(a.u.) +b +c +0 +0.2 +0.4 +0.6 +0.8 +1 +10 +-10 +-5 +5 +0 +x (μm) +Intensity (a.u.) +5 μm +Figure S1. Phase conjugation using an excitation beam: a Fluorescence image of a bead when an excitation beam +(λ = 475 µm) is shaped using the phase conjugation pattern of scattered fluorescence (λ = 532 µm). b Fluorescence image of +the same bead when the excitation beam is shaped by a random phase pattern. c Intensity profiles of (a) and (b) along the +horizontal line that crosses the center. + +4 +Target +5 μm +a +N = 2 +N = 3 +N = 4 +b +c +d +e +Retrieved field index +0 +1 +0 +1 +(a.u.) +Figure S2. Phase conjugation with different numbers of retrieved fields: a Fluorescence image of hidden targets. b +Results of incoherent phase conjugation when different numbers of scattered fields are retrieved. c–e Phase conjugation of +individual fields when two (c), three (d), and four (e) scattered fields are retrieved. +SLM +Laser diode +(488 nm) +BS +Obj. 2 +Obj. 1 +Dichroic +mirror +Iris +L5 +L3 +L4 +L7 +L6 +BP2 +sCMOS +Polarizer +L1 +Sample +Laser +(532 nm) +BP1 +CMOS +L8 +Pinhole +Flip mirror +L2 +Mirror +Figure S3. Experimental setup: L, lens; Obj., objective lens; BS, beam splitter; BP, banspass filter. + +5 +References +[1] Pierre Thibault and Andreas Menzel, Reconstructing state mixtures from diffraction measurements, Nature 494, 68 (2013). +[2] Fucai Zhang and JM Rodenburg, Phase retrieval based on wave-front relay and modulation, Physical Review B 82, 121104 +(2010). +[3] Mickaël Tanter, Jean-Louis Thomas, and Mathias Fink, Time reversal and the inverse filter, The Journal of the Acoustical +Society of America 108, 223 (2000). +[4] Scott Kirkpatrick, C Daniel Gelatt Jr, and Mario P Vecchi, Optimization by simulated annealing, science 220, 671 (1983). + diff --git a/RtFPT4oBgHgl3EQfqDWD/content/tmp_files/load_file.txt b/RtFPT4oBgHgl3EQfqDWD/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..adb097317386ff462112bc20aa41c1895f13ee22 --- /dev/null +++ b/RtFPT4oBgHgl3EQfqDWD/content/tmp_files/load_file.txt @@ -0,0 +1,339 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf,len=338 +page_content='Generalized phase conjugation for incoherent light in complex media YoonSeok Baek,1, ∗ Hilton B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' de Aguiar,1 and Sylvain Gigan1 1Laboratoire Kastler Brossel, ENS–Universite PSL, CNRS, Sorbonne Université, Collège de France, 24 Rue Lhomond, F-75005 Paris, France Shaping light deep inside complex media, such as biological tissue, is critical to many research fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' Although the coherent control of scattered light via wavefront shaping has made significant advances in addressing this challenge, controlling light over extended or multiple targets without physical access inside a medium remains elusive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' Here we present a generalized phase conjugation method for incoherent light, which enables the non-invasive light control based on incoherent emis- sion from multiple target positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' Our method characterizes the scattering responses of hidden sources by retrieving mutually incoherent scattered fields from speckle patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' By time-reversing scattered fluorescence with digital phase conjugation, we experimentally demonstrate focusing of light on individual and multiple targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' We also demonstrate maximum energy delivery to an ex- tended target through a scattering medium by exploiting transmission eigenchannels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' This paves the way to control light propagation in complex media using incoherent contrasts mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' Main Delivering optical energy and transmitting information through complex media remains an important challenge in many fields of studies, including optical manipulation [1], deep-tissue imaging [2, 3] and optogenetics [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' In recent years, it has been shown that the coherent control of scattered light can manipulate spatial, spectral and temporal distributions of light in scattering media [6– 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' However, such capabilities are greatly limited with- out physical access inside a medium because the scat- tering response to target position is difficult to charac- terize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' As a result, non-invasive light control over ex- tended or multiple targets remains elusive despite being crucial for real-world applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' Optimizing incident wavefront based on a feedback signal [11–18] is mostly limited to focusing on a single isolated target, and even then it has limitations that require numerous changes of the wavefront.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' While time-reversal or phase conjugation techniques [19–30] allow for effective light delivery to an optical or virtual source, they cannot individually control light on multiple targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' Here we address these challenges by generalizing phase conjugation for incoherent light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' Our approach utilizes incoherent emission from multiple targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' We first char- acterize scattering responses of these hidden sources by retrieving mutually incoherent fields from spatially mod- ulated speckle patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' The retrieved fields are related to the field transmission matrix [31], and their phase con- jugation enables light control over the desired positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' We demonstrate this experimentally by focusing light on individual and multiple fluorescent targets through a scattering medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' Finally, we show that transmission eigenchannels can be identified by decomposing the inco- herent fields and demonstrate maximum energy delivery to a hidden extended target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' ∗ yoonseok.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content='baek@lkb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content='ens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content='fr II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' Results A schematic of the experiment is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' We consider a scenario where multiple fluorescent targets are hidden by a scattering medium and act as guidestars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' Fluorescence emitted by these guidestars is scattered, re- sulting in an incoherent addition of speckle patterns on the camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' Under this condition, we aim to deliver light back to each of the guidestars by time-reversing the scat- tered fluorescence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' Our approach consists of retrieving multiple incoherent fields that compose the fluorescence, and using them to generate phase conjugated beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' To this end, we introduce wavefront modulation of the scat- tered fluorescence with a spatial light modulator (SLM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' This modulation induces changes in the measured inco- herent speckle patterns, providing information to retrieve the scattered fields that will later be used for phase con- jugation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' To explain the retrieval process in detail, we introduce a partial field transmission matrix T whose in- put and output are fields at the SLM plane and guidestar positions, respectively, Eguidestar = T ESLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' Accord- ing to the time-reversal symmetry, the scattering of flu- orescence emitted by N guidestars can be expressed by the rows of the transmission matrix, T = [t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' , tN]⊤, where tn represents the scattered field that corresponds to an individual guidestar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' We note that the brightness of each guidestar is assumed to be the same, and it is generally represented by the amplitude of tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' Then, the incoherent speckle pattern measured by the camera is ex- pressed as I0 = diag � (T F )†T F � , where F represents the discrete Fourier transform between the SLM and camera plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' If we apply M different SLM modulations and ex- press the mth modulation as a diagonal matrix Sm, the incoherent speckle pattern after the modulation is Im = diag � (T SmF )† T SmF � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' (1) A set of scattered fields that satisfy Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' 1 can be found by iteratively minimizing the error between the measured and predicted Im (see Supplementary Information Sec- tion 1 for more information).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' We note that finding T is arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content='13140v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content='optics] 30 Jan 2023 2 Fluorescent guidestars Scattering medium Scattered fields Mixed-state phase retrieval Wavefront synthesis SLM Lens Camera images 1 1 2 M 2 M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' 1 2 N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=') 0 1 0 π (rad) A θ Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' Schematic of phase conjugation with incoher- ent fluorescence: Multiple fluorescent guidestars are hidden behind a scattering medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' The scattered fluorescence, com- posed of mutually incoherent speckle fields, is modulated by an SLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' The modulated fluorescence is Fourier transformed by a lens and results in the incoherently added speckle pat- terns on the camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' Then mixed-state phase retrieval recov- ers a set of scattered fields, whose phase conjugation enables targeted light control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' equivalent to mixed-state reconstruction [32] with inher- ent ambiguity, where T and its unitary transformation are indistinguishable by intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' This can be confirmed by replacing T with UT in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' For this reason, the scattered fields are retrieved as a mixture of tn: H = UT , (2) where H is a set of retrieved fields hn, H = [h1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=', hN]⊤, and U is an arbitrary unitary matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' Despite this am- biguity in the reconstruction, the retrieved fields offer unique capabilities for phase conjugation, as we show be- low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' Incoherent phase conjugation The time-reversal of scattered fluorescence will regen- erate light at hidden sources, creating foci on the entire targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' One way to accomplish this is to phase conjugate the incoherent scattered fields from individual guidestars, tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' Alternatively, we chose to use hn since it gives the identical phase conjugation result (see Supplementary In- formation Section 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' To demonstrate this incoherent phase conjugation, we introduced several 1 µm fluorescent beads as guidestars behind the scattering medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' We retrieved the multiple scattered fields according to the number of the guidestars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' In our experiments, each 1 µm bead was considered as an individual guidestar as the speckle grain size at the target plane was ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content='9 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' The incoherent phase con- jugation was implemented by sequentially generating N phase-conjugated fields of hn using the SLM, and by mea- suring time-averaged responses (see Methods for phase conjugation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' To evaluate the performance of the phase conjugation, we first conducted an experiment with a single fluores- cent bead (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' 2a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' By phase-conjugating the scattered field, we observed a strong focus on the bead (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' 2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' This is in clear contrast with the random speckle gener- ated by a beam with a random wavefront (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' 2c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' The enhancement factor, defined as the ratio between the op- timized focus intensity and mean background intensity, was ∼4,400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' We then placed multiple fluorescent beads (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' 2d–f) behind the scattering medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' By incoher- ently phase conjugating the scattered fields, we success- fully generated foci at every guidestar positions (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' 2g– i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' Despite the minimal spectral memory effect [33], we were also able to excite the bead through the scattering medium by generating a phase conjugation beam at the excitation wavelength (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' Selective focusing on individual targets In order to selectively focus on individual targets, it is necessary to demix the individual fields tn from their mixture hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' We note that tn is not strictly orthogonal, and thus the orthogonalization of hn can not be a solu- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' Our solution was to directly invert Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' 2 by finding U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' To this end, we utilized the memory effect [34, 35], where neighboring guidestars generate correlated speckle patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' Specifically, we iteratively applied a random unitary transformation to the retrieved fields H, such that the correlation between the transformed speckle pat- terns is maximized (see Supplementary Information Sec- tion 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' Figure 3 shows the experimental result with 5 fluores- cent beads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' When the scattered fields hn are directly used for phase conjugation, each phase conjugation gen- erated foci on several guidestars with different intensities (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' 3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' The demixed fields, on the other hand, gen- erated a focus on a single guidestar, showing that tn is successfully recovered (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' 3c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' The memory effect range in this experiment, defined as the full width at half max- imum of speckle cross-correlation, was 5 µm, which is much smaller than the spatial extent of the guidestars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' This result shows that the selective focusing via demix- ing is possible as long as a pair of guidestars lies within the memory effect range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' Targeted energy delivery Maximum energy delivery through scattering media re- quires an eigenchannel of T [36, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' The transmission eigenchannels correspond to the singular vectors of T , and the first singular vector with the largest singular 3 5 μm d f g h i b a c 5 μm 5 μm 5 μm 0 (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=') 1 0 (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=') 1 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' Incoherent phase conjugation for multiple targets: a, The fluorescence image of an 1 µm bead taken from the side without a scattering medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' b, Intensity at the target plane with the phase conjugation of scattered fluorescence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' (Inset) The phase of the scattered field shown in the HSV colormap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' The central highlighted part is used for the phase conjugation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' c, Intensity at the target plane generated by a random wavefront.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' d–f, The fluorescence images of multiple beads hidden behind the scattering medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' g–i, Intensity at the target plane with incoherent phase conjugation of scattered fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' a b Before demixing h1 h2 h3 h4 h5 c After demixing t1 t2 t3 t4 t5 5 μm Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' Selective focusing on individual targets: a, The fluorescence image of a target comprised of 1 µm beads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' b,c, Intensity at the target plane after the phase conjugation of individual scattered fields before (b) and after (c) the demixing process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' (Insets) The phase of the scattered fields used for phase conjugation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' value, delivers the maximum energy to the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' Al- though the direct access to T is not always possible, the transmission eigenchannels of T can be found using H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' This is because the eigenchannels of T and H are iden- tical because H†H = T †T according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' Thus, we can deliver the maximum energy to extended targets using the first singular vector of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' To demonstrate the targeted energy delivery, we placed a 5 µm fluorescent ink droplet behind the scattering medium (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' 4a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' Based on the size of the target, we estimated the number of incoherent fields and retrieved 23 scattered fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' We then performed the singular value decomposition of the retrieved fields H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' Finally, we in- jected fields that corresponds to the singular vectors vn and observed the energy delivered to the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' When a random wavefront is injected to the scattering medium, a speckle pattern is generated at the target plane (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' 4b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' In contrast, the singular vectors produce intensity distri- butions highly concentrated on the target (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' 4c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' By summing the results of all the singular vectors, we con- firmed that the energy is delivered only to the target area (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' 4d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' The first singular vector v1 shows a 174-times increase in the energy on the target, compared to random wavefronts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' The enhancement decays with the singular vector index [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' 4(e)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' We observed that the values are not perfectly sorted in a descending order, which we be- lieve is due to the numerical error in the retrieved fields and to the use of the phase-only SLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' Discussion and conclusion We have presented an approach to control light in scat- tering media without the physical access to a target plane by extending phase conjugation to incoherent light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' We have demonstrated focusing and maximum energy deliv- 五年4 d e 1 23 0 100 200 Singular vector index Energy enhancement a 5 μm 0 1 (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=') 0 1 (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=') c b v3 v1 v2 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' Targeted energy delivery: a, The fluorescence image of an extended target hidden behind a scattering medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' The image was taken from the side without the scattering medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' b, Intensity at the target plane when a random phase pattern is displayed on the SLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' c, Intensity at the target plane using the first 3 singular vectors of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' The images are normalized for the result of the first singular vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' d, The sum of all the results using 23 singular vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' e, Enhancement of the energy delivered to the target compared to random realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' Dash circles in (c) and (d) indicate the boundary of the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' ery to extended fluorescent targets through a scattering medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' Our approach differs from the existing phase conjugation techniques by addressing all the mutually in- coherent fields of scattered fluorescence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' Another impor- tant aspect is that it does not require the precise align- ment between the camera and SLM [38], since the scat- tered fields are retrieved at the SLM plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' Its principle of characterizing the scattering response is entirely pas- sive, as it does not alter the emission of guidestars, as opposed to to techniques that modulates the excitation wavefront (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' [14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' For the proposed method, it is important to estimate the number of mutually incoherent waves, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' This is because the underestimation of N results in imperfect reconstruction of the scattered fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' We note that the overestimation is allowed because it results in redundant reconstruction (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' Nevertheless, it is recom- mended to use the smallest possible value of N for the minimal measurements and computation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' There are different methods to estimate N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' The contrast of fluores- cence speckle is an useful indicator for N, as it decreases as √ N [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' It is also possible to find N by analyzing the error in the mixed-state phase retrieval or the singular value distribution [40] for different values of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' In our demonstrations, we did not consider the spectral degrees of freedom because narrow spectral responses were mea- sured by using interference filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' If the detection band- width is greater than the spectral memory effect range, different spectral components should be considered in es- timating N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' Another important consideration is the number of modulation M required for the field retrieval, which scales linearly with N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' In experiments, reliable phase conjugation results were obtained when M ≥ 6N (see Supplementary Information Section 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' This linearity can be attributed to the multiplexed information in the intensity of multiple incoherent fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' We emphasize that M scales with the number of incoherent waves N not with the number of controlled input modes of the SLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' Re- cent advances in phase retrieval [41, 42] show that few measurements are sufficient in retrieving a coherent field (N = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' In this regard, we believe that in principle even fewer M may be used for our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' In our proof-of-principle experiments, we used the sim- ple algorithms for the retrieval and demixing of the scat- tered fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' The performance of the algorithms can be enhanced by incorporating constraints, convex optimiza- tion [42] or the generalized memory effect [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' The total measurement time can be reduced by designing a setup with minimal energy loss, and by using a sensitive de- tector, such as an EMCCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' The incoherent phase conju- gation can be improved by employing a high-speed SLM or possibly by shaping partially coherent light [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' Such improvements will benefit applications that require light to be delivered to multiple targets simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' In conclusion, our method enables versatile light con- trol over extended or multiple targets using incoher- ent contrast mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' The concept can be applied to different incoherent emissions, such as spontaneous Raman scattering [17, 18], and a wide range of pho- toluminescence [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' Furthermore, it enables the pas- sive characterization of a transmission matrix, open- ing up the possibility to generalized light control using transmission-matrix-based operators [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' We envision that the proposed approach will enable targeted light de- livery through thick biological tissue, facilitating biomed- ical applications, such as optogenetic stimulation and phototherapy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' 5 IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' Methods Experimental setup The experimental setup is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' A laser diode (λ = 488 nm, LP488-SF20G, Thorlabs) was used to excite fluorescence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' The excitation beam was deliv- ered to guidestars by a lens (L1, f = 100 mm) and an objective lens (Plan N 20× 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content='4, Olympus).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' To moni- tor the guidestars and phase conjugation, a dichroic mir- ror (DMLP490R, Thorlabs), a lens (L2, f = 200 mm), a bandpass filter (FL532-10, Thorlabs), and a camera (acA5472-17um, Basler) were placed on the side without a scattering medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' The guidestars were fluorescent ink mixed with UV glue (NOA 68, Norland), and fluo- rescent beads (F8803, Invitrogen) immersed in glycerol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' A scattering medium was a 220-grit ground glass diffuser, placed approximately 170 µm away from the guidestars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' On the detection side, the scattered fluorescence was col- lected by an objective lens (MPlan N 50× 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content='75, Olympus) and two lenses (L3, f = 75 mm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' L4, f = 150 mm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' An SLM (X10468-04, Hamamtsu) and a linear polarizer were used to modulate the fluorescence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' The modulated flu- orescence is Fourier transformed by lenses (L5, f = 100 mm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' L6, f = 200 mm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' L7, f = 250 mm) and then cap- ture by an sCMOS camera (PCO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content='edge 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content='5, PCO) with bandpass filters (BP2, FL532-3 and FBH520-40, Thor- labs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' An iris was placed between L6 and L7 to adjust the speckle grain size at the camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' For phase conjuga- tion, a laser (λ = 532 nm, Compass 215M-50, Coherent) was collimated using a 5 µm pinhole and a lens (L8, f = 6 mm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' The collimated laser was then shaped by the SLM to generate a phase-conjugated beam, which propagated back to the medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' A flip mirror was used to switch be- tween the fluorescence detection and phase conjugation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' Phase-conjugated beam generation A phase-conjugated beam is generated using the colli- mated laser beam and the SLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' The collimated beam is shaped to the phase conjugate of a given scattered field Escattered by displaying a phase pattern that corresponds to −arg (Escattered) on the SLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' The resultant phase conjugated beam propagates back through the scattering medium, retracing the scattering paths of fluorescence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' Acknowledgement This research was funded by the FET-Open (Dynamic- 863203) and the European Research Council under Grant Agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' 724473 (SMARTIES).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' acknowl- edges the support from Basic Science Research Pro- gram through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2022R1A6A3A03072108).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' References [1] Tomáš Čižmár, Michael Mazilu, and Kishan Dholakia, In situ wavefront correction and its application to microma- nipulation, Nature Photonics 4, 388 (2010).' metadata={'source': 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the SLM plane from M modulated intensity images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' We initialize the algorithm by letting the scattered fields hn as N complex Gaussian random fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' Then we apply the SLM modulation Sm to the scattered fields hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' We note that Sm is a diagonal matrix whose diagonal elements correspond to the field modulation given by the SLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' In the experiments, SLM was divided into macro-pixels (composed of 40 × 40 pixels) having random phase values (0 or π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' After the SLM modulation Sm, the field at the camera plane is expressed as ˜h(m) n = (SmF )⊤ hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' Then we conduct Fourier magnitude projection using an auxiliary function ψ: ψ(m) n (k) = � Im(k) � n ���˜h(m) n (k) ��� 2 �γ ˜h(m) n (k), (S1) where k is a coordinate in the spatial frequency domain, ˜h(m) n (k) is the modulated field at the camera plane, corre- sponding to ˜h(m) n , and γ is a constant parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' Then the fields are updated by compensating the SLM modulation: hn = � (SmF )⊤�−1 ψ(m) n , (S2) where ψ(m) n is the vector representation of ψ(m) n (k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' The update through Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' S1–S2 is continued for the entire M modulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' The whole process is repeated for several times to obtain consistent hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' With γ = 1/2, this method can be interpreted as the maximum likelihood reconstruction and retrieves the incoherent fields [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' However, we observed that few initial iterations with γ = 1 accelerates the convergence greatly [2], in both numerical simulations and experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' Thus, in our experiments, we used γ = 1 for the first 20 iterations and γ = 1/2 for the rest of the iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' We confirmed numerically that the algorithm retrieves a correct set of fields for M ≥ 4N in the absence of measurement noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' The size of macro-pixel had almost no effect on the reconstruction, except when its size is comparable to the SLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' We observed that the minimum value of M required for correct reconstruction increases depending on the noise level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' For the experimental results shown in the main text, we used 6–8N modulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' Incoherent phase conjugation When all the mutually incoherent components of fluorescence are time-reversed, the intensity at the nth guidestar can be expressed by multiplying the nth row of the transmission matrix, t⊤ n , and phase-conjugated field t∗ m: � m ��t⊤ n t∗ m ��2 = t⊤ n T †T t∗ n, (S3) where ∗ denotes the complex conjugate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' For a medium with negligible reflection and absorption, T †T ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' Thus Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' S3 is simplified to t⊤ n t∗ n, which remains more or less constant regardless of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' As a result, the incoherent phase conjugation of tn generates foci on the entire guidestars with roughly the same intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' We note that as long as the loss of a medium is minimal, the phase conjugation well approximates the lossless case, producing high-contrast foci on the entire guidestars [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' Similarly, the intensity for the incoherent phase conjugation of hn is expressed as, � m ��t⊤ n h∗ m ��2 = t⊤ n H†Ht∗ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' (S4) Since H†H = T †T according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' 2, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' S4 is identical to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' Thus, the incoherent phase conjugation of hn and tn are identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' Demixing incoherent fields The scattered fields tn can be recovered by reversing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' To this end, we developed an algorithm inspired by the simulated annealing [4] that exploits the memory effect between the speckle patterns at the camera plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' The 2 algorithm starts by applying a random unitary transformation to the retrieved fields H and expressing the result at the camera plane: H = Uj fj ˆUHF , (S5) where j is the iteration index, Uj is a random unitary matrix, fj is an exponent that decays to 0 over the iteration, and ˆU is the current best estimate of U −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' The decaying behavior of fj makes Uj induce smaller changes as the iteration progresses: limj→∞ Uj fj = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' Then the correlation metric C is calculated at every iteration, C = N � n=1 � max � |hn|2 ⋆ |h(n mod N)+1|2��2 , (S6) where hn(k) is the unitary-transformed speckle field at the camera plane, corresponding to the nth row of H, and ⋆ is the cross-correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' If the value of C is greater than the previous values, ˆU is updated to Uj fj ˆU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' If not, ˆU is unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' At the end of the iteration, the algorithm results in ˆU ≈ U −1, and the scattered fields are recovered by inverting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' 2: T = ˆUH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' We initialized the iteration with ˆU = 1 and fj = 100/(j + 1), and the maximum iteration number of 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' We note that the experimental noise in H can lead to an incorrect demixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' To mitigate this issue, we calculate C only using speckles brighter than the average, |hn|2 > µ, where µ is the mean value of every |hn|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' 3 a b 0 1 (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=') a 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content='2 (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=') b c 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content='8 1 10 10 5 5 0 x (μm) Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=') 5 μm Figure S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' Phase conjugation using an excitation beam: a Fluorescence image of a bead when an excitation beam (λ = 475 µm) is shaped using the phase conjugation pattern of scattered fluorescence (λ = 532 µm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' b Fluorescence image of the same bead when the excitation beam is shaped by a random phase pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' c Intensity profiles of (a) and (b) along the horizontal line that crosses the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' 4 Target 5 μm a N = 2 N = 3 N = 4 b c d e Retrieved field index 0 1 0 1 (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=') Figure S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' Phase conjugation with different numbers of retrieved fields: a Fluorescence image of hidden targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' b Results of incoherent phase conjugation when different numbers of scattered fields are retrieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' c–e Phase conjugation of individual fields when two (c), three (d), and four (e) scattered fields are retrieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' SLM Laser diode (488 nm) BS Obj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' 2 Obj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' 1 Dichroic mirror Iris L5 L3 L4 L7 L6 BP2 sCMOS Polarizer L1 Sample Laser (532 nm) BP1 CMOS L8 Pinhole Flip mirror L2 Mirror Figure S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' Experimental setup: L, lens;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' Obj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=', objective lens;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' BS, beam splitter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' BP, banspass filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' 5 References [1] Pierre Thibault and Andreas Menzel, Reconstructing state mixtures from diffraction measurements, Nature 494, 68 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' [2] Fucai Zhang and JM Rodenburg, Phase retrieval based on wave-front relay and modulation, Physical Review B 82, 121104 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' [3] Mickaël Tanter, Jean-Louis Thomas, and Mathias Fink, Time reversal and the inverse filter, The Journal of the Acoustical Society of America 108, 223 (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} +page_content=' [4] Scott Kirkpatrick, C Daniel Gelatt Jr, and Mario P Vecchi, Optimization by simulated annealing, science 220, 671 (1983).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFPT4oBgHgl3EQfqDWD/content/2301.13140v1.pdf'} diff --git a/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf 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Kucera,2 and C. Richard DeVore2 +1Department of Physics +Catholic University of America +Washington, DC 20064, USA +2Heliophysics Science Division +NASA Goddard Space Flight Center +Greenbelt, MD 20771, USA∗ +(Accepted January 6, 2023) +Submitted to ApJ +ABSTRACT +We investigate the properties of nonlinear fast magnetosonic (NFM) waves in a solar prominence, mo- +tivated by recent high-resolution and high-cadence Hinode/SOT observations of small-scale oscillations +in a prominence pillar. As an example, we analyze the details of the 2012 February 14 Hinode/SOT +observations of quasi-periodic propagating features consistent with NFM waves, imaged in emission in +Ca II and in the far blue wing of Hα. We perform wavelet analysis and find oscillations in the 1-3 min +period range. Guided by these observations, we model the NFM waves with a three-dimensional mag- +netohydrodynamics (3D MHD) model, extending previous 2.5D MHD studies. The new model includes +the structure of the high-density, low-temperature material of the prominence pillar embedded in the +hot corona, in both potential and non-force-free sheared magnetic field configurations. The nonlinear +model demonstrates the effects of mode coupling and the propagating density compressions associated +with linear and NFM waves. The guided fast magnetosonic waves, together with density compressions +and currents, are reproduced in the 3D pillar structure. We demonstrate or the first time the dynamic +effects of the Lorentz force due to the magnetic shear in the non-force-free field on the pillar structure +and on the propagation of the waves. The insights gained from the 3D MHD modeling are useful for +improving coronal seismology of prominence structures that exhibit fast MHD wave activity. +1. INTRODUCTION +Solar prominences (also called filaments; e.g. Tandberg-Hanssen 1995) are highly complex magnetic structures that +extend from the photosphere up into the corona, where they support material that is much denser (n ∼ 1010−12 +cm−3) and cooler (T ∼ 1×104 K) than the surrounding plasma (n ∼ 108−9 cm−3, T ∼ 1-2×106 K). High-resolution +observations of prominences have been available in H I Balmer (Hα) and Ca II emission for decades from ground-based +telescopes and, more recently, in various ion emission bands from satellite-borne instruments. These observations show +that the prominence material is highly dynamic, exhibiting persistent flows, waves, and other oscillations, as well as +MHD instabilities that can lead to its violent eruption (for reviews, see Labrosse et al. 2010; Parenti 2014; Arregui +et al. 2018). Idealized models of quiescent prominences often assume an equilibrium magnetic structure that supports +the cool material statically within the hot corona. The observations indicate that both the magnetic and thermal +structures of prominences are often out of equilibrium, highly dynamic at small scales and gradually evolving at large +scales. +Small-scale propagating and oscillating features in cool prominence threads and low-lying coronal loops have been +studied from space for many years using high-resolution and high-cadence spectral observations. Hinode’s Solar Optical +Corresponding author: Leon Ofman +ofman@cua.edu +∗ Visiting, Department of Geosciences, Tel Aviv University, Tel Aviv, Israel +arXiv:2301.04503v1 [astro-ph.SR] 11 Jan 2023 + +2 +Ofman et al. +Telescope (Hinode/SOT; Kosugi et al. 2007) has observed such phenomena in Hα and Ca II emission (Okamoto et al. +2007; Ofman & Wang 2008; Schmieder et al. 2013; Ofman et al. 2015; Kucera et al. 2018), as has the Interface Region +Imaging Spectrograph (IRIS; De Pontieu et al. 2014) in chromospheric Mg II emission as spectral lines and slit jaw +images (Kucera et al. 2018). High-resolution prominence observations by Hinode/SOT show that the prominences +material exhibits constant down-flows, lateral flows, upflows, and dynamic evolution with the observed velocities in +the range 1 − 100 km s−1 consistent with the effects of magneto-fluid instabilities (e.g., Berger et al. 2017). Recent +ground-based high-resolution observations using the New Vacuum Solar Telescope (NVST; Liu et al. 2014) report +evidence of small-scale oscillations and waves detected in Hα in quiescent prominences (e.g., Li et al. 2018, 2022). +Advanced high-resolution resistive 3D MHD modeling of prominence structure evolution shows that the nonlinear +development of the magnetic Rayleigh–Taylor instability produces small scale structures in the prominence material +(Jenkins & Keppens 2022), and possibly can provide an alternative (to waves) interpretation of some of the observed +small-scale oscillating structures. +The (quasi-) periodic, small-scale, oscillating features, with typical time scales of minutes in prominence threads +and pillars, have been identified and modeled previously as linear fast magnetosonic waves (e.g., Schmieder et al. +2013). Because the nonlinearity of these waves is evident in the observations in the form of steepening and asymmetric +density compressions, the models were later extended to nonlinear fast magnetosonic (NFM) waves using an MHD +model with two-dimensional spatial variations and three-dimensional vector fields (2.5D MHD; see Ofman et al. 2015; +Ofman & Kucera 2020). The observed waves can be used to deduce the magnetic structure of prominences by applying +techniques of coronal seismology (e.g., Nakariakov & Verwichte 2005; Anfinogentov et al. 2022). These indirect methods +are invaluable, as the coronal magnetic field is very difficult to measure directly using spectroscopic or other methods, +while force-free extrapolation methods have limited applicability in realistic coronal structures. Coronal seismology +remains based primarily on linear MHD wave theory. However, nonlinearity may significantly affect the wave structure, +phase speed, wave dissipation, and couplings. Thus, interpreting observations of nonlinear waves requires the use of +nonlinear wave theory or nonlinear MHD modeling for improved accuracy of the analysis. +Plasma flows, in addition to waves, are often observed in cool prominence threads in emission lines such as Hα +and Ca II (Okamoto et al. 2007; Alexander et al. 2013; Kucera et al. 2014; Parenti 2014; Diercke et al. 2018). These +flows may affect the oscillations through, for example, changes in the density that affect the phase speed of the waves +and Doppler shifts of the oscillation frequencies. Recently, Kucera et al. (2018) and Ofman & Kucera (2020) used +Hinode/SOT Ca II spectral lines to study small-scale motions in prominences. The observed propagating fluctuations +were identified as NFM waves using a combination of data analysis and modeling. The observed NFM waves had +typical periods ∼ 5-11 minutes and wavelengths ∼ 2000 km, while the flows had typical speeds ∼ 15-50 km s−1. The +main properties of the observed NFM waves, combined with the effects of mass flows in prominence threads, were +replicated by the model (Ofman & Kucera 2020). The magnetic field strengths in the prominence were estimated to +lie in the range 5-17 G. +In the present study, we extend the previous studies of propagating waves in prominences with data analysis and +modeling of a prominence pillar observed on 2012 February 14 from Hinode/SOT. We employ a new, fully three +dimensional (3D) MHD model of small-scale NFM waves in an idealized prominence pillar with more realistic structure +than in the previous studies. The new model allows us to investigate more complex fast magnetosonic wave generation, +propagation, and interaction than in the previous 2.5D configurations, for example, by including the effects of magnetic +shear, and for the first time study the effects of non-force free shear magnetic field. The results are useful for interpreting +high-resolution Hinode/SOT observations of prominence small-scale oscillations and for making further advancements +in the coronal seismology of solar prominences using MHD waves. +Our paper is organized as follows. In §2 we present new observations of propagating features in a prominence pillar. +In §3 we describe the new 3D MHD model, along with the initial and boundary conditions used in the calculations. In +§4, we present the numerical results and compare them with observations. Finally, the discussion and our conclusions +are given in §5. +2. OBSERVATIONS AND DATA ANALYSIS +The studied prominence was observed by Hinode/SOT on 2014 February 14 from 10:48 - 13:15 UT as part of Hinode +Operation Plan (HOP) 114. The data consisted of measurements from both the Broadband Filter Instrument (BFI) +and the Narrowband Filter Instrument (NFI) (Kosugi et al. 2007; Tsuneta et al. 2008). The BFI was used to observe +the Ca II H line at 3969 ˚A and the NFI was used to observe the Hα line at 6563.2 ˚A, both with cadences of 22.4 s. + +Nonlinear Fast Waves in a Prominence Pillar +3 +(a) NSO/GONG H−alpha 11−Feb−2012 11:00:54 UT +450 +500 +550 +600 +650 +700 +Solar−X (arcsec) +500 +550 +600 +650 +Solar−Y (arcsec) +(b) Hinode SOT Ca II + 14−Feb−2012 12:46:49 UT +840 +860 +880 +900 +Solar−X (arcsec) +440 +460 +480 +500 +520 +Solar−Y (arcsec) +(c) Hinode SOT Hα far blue wing +14−Feb−2012 12:46:58 UT +840 +860 +880 +900 +Solar−X (arcsec) +440 +460 +480 +500 +520 +Solar−Y (arcsec) +Figure 1. (a) GONG Hα image showing the prominence on 11 Feb 2012, three days before the prominence was observed on +the limb. The box shows the approximate field of view of the images in (b) and (c). (b) Hinode Ca II image showing the +prominence on the limb; the box is the field of view shown in Figure 2. An animation that corresponds to this panel is available +online. The video shows the Hinode Ca II emission observed on 14-Feb-2012 in the time interval 10:51:06-13:13:45 UT in an +accelerated time of 16 s. (c) Hα far blue-wing image; the box is the field of view shown in Figure 3. +The Hα line positions for this data set were not well calibrated, but appear to be from near line center and in the +blue wing of the line (about 416 m˚A from line center), making them not useful for Doppler measurements. The image +field of view is about 112′′ square, and the spatial resolution is 0.2-0.3′′. Maps were processed with the fg prep.pro +routine provided by to the Solar Soft library (https://www.lmsal.com/solarsoft/, Freeland & Handy (1998)) by the +Hinode team, including dark-current subtraction and flat-field removal. Drift and jitter were corrected using an image +cross-correlation (fg rigidalign.pro) routine. +For context, we inspected observations from the Global Oscillation Network Group to image the on-disk structure +of the prominence in the days preceding its appearance at the limb. GONG Hα images are provided by a network of +six stations around the globe (Harvey et al. 1996) with a pixel size of about 1′′. +The features observed on the limb were part of a long prominence that extended more or less East-West above the +northern active-region belt and curved equator-ward on the western end. A portion of that prominence seen against the +solar disk three days before the observations we analyzed is shown in Figure 1a. The most evident prominence features +seen in Hα are a series of barbs connected by fainter spine flows. These barbs evolve over time, and it is difficult +to identify individual barbs near the limb. However, the appearance of the region at the limb from Hinode/SOT, +Figures 1b (Ca II) and 1c (Hα), is consistent with associating the pillars with barbs that are oriented mostly along +the line of sight. +Figure 2 shows time-distance diagrams for the small-scale propagating features (i.e., ‘pulses’) observed in the Ca II +images in two locations. The pulses were measured along a 5-pixel-wide area centered on the solid red and green lines +shown in panels (a) and (d), respectively. Panel (b) shows a series of pulses with plane-of-sky velocities 12-16 km s−1 +determined from the slopes of the dashed red lines, which were visually fit to the intensity peaks. The peaks are about +1 min apart, and the distances between pulses are in the range 1330-2030 km. Panel (e) shows another set of pulses +corresponding to the location shown in panel (d). These pulses have peaks 1.5-2.3 min apart, velocities 8-11 km s−1 +obtained from the slopes of the dashed green lines, and distance between pulses in the range 800-1600 km. Panels (c) +and (f) show the intensities along the horizontal lines shown in panels (b) and (e) respectively. The variations between +the maximum and minimum intensities of the individual features are about 10% of the total intensity. +Figure 3 shows time-distance diagrams for moving features seen in the Hα blue wing. Shown are a series of pulses +with plane-of-sky velocities 12-16 km s−1, peaks 1-5 min apart with sharp non-sinusoidal peaks indicative of nonlinear +steepening, and distances between pulses of 1000-3000 km. The plane-of-the-sky propagation speed is likely reduced +compared to the ‘true’ phase speed due to projection effects, and the value is in qualitative agreement with possible +fast magnetosonic speeds in cool prominence material of the order ∼ 20 km s−1 (see, e.g., Schmieder et al. 2013). The +variations between the maximum and minimum intensities of the individual features are about 30-60% of the total +intensity. +We have performed a wavelet analysis (Torrence & Compo 1998) of the oscillations in Ca II and the far blue wing of +Hα cuts shown in Figures 2 and 3 using the Morlet wavelet. In Figure 4 we show the results of the analysis with evident + +4 +Ofman et al. +(a) Hinode SOT Ca II 14−Feb−2012 12:28:55 UT +845 +850 +855 +860 +865 +870 +Solar−X (arcsec) +485 +490 +495 +500 +505 +Solar−Y (arcsec) +(b) Hinode SOT Ca II +0 +2 +4 +6 +8 +10 +12 +14 +Time after 14-Feb-12 12:20:00 UT + (minutes) +0 +2 +4 +6 +8 +Distance (arcsec) +0 +2 +4 +6 +8 +10 +12 +14 +0 +2 +4 +6 +8 +(c) Ca II at D=5 arcsec +2 +4 +6 +8 +10 +12 +14 +Time after 14−Feb−12 12:20:00 UT +400 +450 +500 +550 +600 +Ca II (DN) +(d) Hinode SOT Ca II 14−Feb−2012 12:46:49 UT +845 +850 +855 +860 +865 +870 +Solar−X (arcsec) +485 +490 +495 +500 +505 +Solar−Y (arcsec) +(e) Hinode SOT Ca II +0 +5 +10 +15 +20 +Time after 14-Feb-12 12:36:00 UT + (minutes) +0 +2 +4 +6 +8 +Distance (arcsec) +0 +5 +10 +15 +20 +0 +2 +4 +6 +8 +(f) Ca II at D=5 arcsec +5 +10 +15 +Time after 14−Feb−12 12:36:00 UT +500 +550 +600 +650 +700 +Ca II (DN) +Figure 2. (a) Image of the Ca II emission obtained with Hinode/SOT on 14-Feb-2012 12:28:56UT. The solid red line indicates +the data location for the time-distance diagram. (b) Time-distance diagram showing the propagation of the features along the +solid red line in (a). (c) Plots of intensity as a function of time at the locations shown with the blue horizontal line in (b). +(d)-(f) The same for a different set of pulses obtained along the solid green line in (d). The slopes of the dashed red (b) and +green (e) lines indicate the propagation speed of the pulses in the plane of the sky. A video showing the field of view in panels +(a) and (d) is included online. The video shows the Hinode SOT Ca II intensity observed on 14-Feb-2012 in the time interval +12:18:06 UT to 15:56:59 UT in an accelerated time of 4 s. +highest confidence level for the wavelet magnitude greater than 85%. The cones of influence indicate the regions that +may be affected by the boundaries. The results show the global wavelet power integrated inside the cone of influence, +indicating significant power in ∼ 1 − 3 min period oscillations, consistent with the temporal evolution at the indicated +temporal cuts, and in the animations of the observed oscillations included online. The wavelet analysis and the global +wavelets provide unbiased quantification of the observed oscillations and their statistical significance. +Thus, we observe multiple cases of a short series of oscillatory features propagating in a direction roughly away from +the limb in the plane of the sky, separated by ∼ 1 min. Each individual feature is slightly elongated perpendicular +to the direction of motion, hence is similar to features described previously by others (Schmieder et al. 2013; Ofman +et al. 2015; Kucera et al. 2018). Because the pillars are likely to be elongated structures along the line of sight, these +moving features may be related to motions observed in different (perpendicular) line of sight in extended prominence +structures as transverse oscillations combined with flows of cool material (Ofman & Wang 2008; Okamoto et al. 2016; +Ofman & Kucera 2020). +3. NUMERICAL 3D MHD MODEL, BOUNDARY CONDITIONS, AND PARAMETERS +In order to model the NFM waves in a prominence pillar we solve the resistive 3D MHD equations using our code +NLRAT described in detail in previous papers (Ofman & Thompson 2002; Provornikova et al. 2018; Ofman & Liu +2018; Ofman & Wang 2022). The normalized resistive MHD equations with gravity, using standard notation for the + +Nonlinear Fast Waves in a Prominence Pillar +5 +(a) Hinode SOT Hα far blue wing +14−Feb−2012 11:33:52 UT +850 +855 +860 +865 +870 +875 +Solar−X (arcsec) +465 +470 +475 +480 +485 +490 +495 +Solar−Y (arcsec) +(b) Hinode SOT H-alpha far blue wing +0 +2 +4 +6 +8 +10 +Time after 14-Feb-12 11:30:00 UT + (minutes) +0 +1 +2 +3 +4 +5 +Distance (arcsec) +0 +2 +4 +6 +8 +10 +0 +1 +2 +3 +4 +5 +(c) Hα at D=3.5 arcsec +2 +4 +6 +8 +Time after 14−Feb−12 11:30:00 UT +600 +700 +800 +900 +1000 +1100 +Hα far blue wing (DN) +(d) Hα at D=2.5 arcsec +2 +4 +6 +8 +Time after 14−Feb−12 11:30:00 UT +500 +600 +700 +800 +900 +1000 +1100 +Hα far blue wing (DN) +(e) Hα at D=2 arcsec +2 +4 +6 +8 +Time after 14−Feb−12 11:30:00 UT +500 +600 +700 +800 +900 +1000 +Hα far blue wing (DN) +Figure 3. (a) Image in the far blue wing of Hα obtained with Hinode/SOT on 14-Feb-2012 at 11:33:52 UT. The solid purple +line shows the location of the data for the time-distance diagram. (b) Time-distance diagram showing the propagation of the +pulses along the solid purple line indicated in (a). (c)-(d) plots of intensity as a function of time at the locations shown with +the blue horizontal lines in (b). The slopes of the dashed purple lines in (b) indicate the propagation speed of the pulses in the +plane of the sky. A video that corresponds to the field of view in panel (a) is included online. The observed Hinode SOT Hα +far blue wing field of view on 14-Feb-2021 in the time interval 11:25:17-11:42:26UT is shown in an accelerated time of 2 s. +variables, are +∂ρ +∂t + ∇ · (ρV) = 0, +(1) +∂(ρV) +∂t ++ ∇ · +� +ρVV + +� +Eup + B · B +2 +� +I − BB +� += − 1 +Fr +ρFg, +(2) +∂B +∂t − ∇ × (V × B) = 1 +S ∇2B, +(3) +∂(ρE) +∂t ++ ∇ · +� +V +� +ρE + Eup + B · B +2 +� +− B(B · V) + 1 +S (∇ × B) × B +� += − 1 +Fr +ρFg · V. +(4) +With our normalization Eu = β/2 is the magnetic Euler number (ratio of thermal pressure to Alfv´en-wave pressure), +Fr = V 2 +ARs/(GMs) is the magnetic Froude number (ratio of magnetic force to gravitational force), where G is the +gravitational constant, Ms is the solar mass, Rs is the solar radius, and S is the Lundquist number (ratio of resistive +diffusion time to Alfv´en time). The details of the normalization of the variables can be found in Ofman & Liu (2018). +The gravitational force, +Fg = +a2 +0 +(Rs + z − zmin)2ˆz, +(5) +is modeled with the assumption of small height of the prominence compared to the solar radius Rs, where a0 = 0.1Rs +about 70 Mm is the normalization length scale of the coordinates, and zmin is the height of the lower boundary in +the model. We note that in the present model we have excluded radiative losses and thermal conduction, and the +prominence pillar structure is provided as an initial state, rather than produced self-consistently by the model. The +total energy density is given by +ρE = +Eup +(γ − 1) + ρV 2 +2 ++ B2 +2 . +(6) +In the present model, we neglect radiative cooling and thermal conduction because these losses are small on the typical +time scales of the NFM waves. For coronal temperature T = 1 × 106 K, density n = 109 cm−3, and magnetic field +magnitude B = 10 G we obtain the Alfv´en speed VA = 690 km s−1, the Alfv´en time τA = 101 s, the plasma β ≈ 0.07, + +6 +Ofman et al. +(c) +(d) +(a) +(b) +Figure 4. The results of the wavelet analysis of the oscillations shown in Figures (a) 2c; (b) 2f (c) 3d; (d) 3e; respectively. The +Morlet wavelet was used, and the 85% confidence level contour is indicated on the wavelet power. The cones of influence where +boundary effects may affect the results are indicated with the red curve on each wavelet panel. The global wavelets in the cone +of influence for each case are shown in the corresponding right panels. + +Nonlinear Fast Waves in a Prominence Pillar +7 +the Froude number Fr = 0.25, and the Euler number Eu = 3.47 × 10−2 (for the case with B0 = 20 G, Eu is reduced +by a factor of four, to Eu = 8.67 × 10−3). Note that, for uniform magnetic field, the value of β is identical to the +coronal value all across the prominence pillar, due to the uniform thermal pressure along the magnetic field lines that +cross the pillar. For computational stability purposes, the effect of gravity in the model is reduced by a factor of 10 by +correspondingly increasing Fr, in order to slow the gravitational settling of the cool material in the prominence pillar. +The reduced gravity does not affect the results significantly, since the dominant restoring force of the oscillations is +the Lorentz force (i.e., magnetic field-line ‘tension’). In the above equations we have neglected viscosity, radiative +losses, and thermal conduction. The resistive terms are used with the Lundquist number set to S = 105, which does +not affect the results significantly on the NFM time scales. An empirical value of the nearly isothermal polytropic +index, γ = 1.05, is used that accounts for coronal heating. These modeling parameters improve the stability of the +background prominence pillar structure on the time scale of MHD wave propagation, without affecting significantly +the NFM wave dynamics. +0.15 +0.10 +0.05 +0.00 +0.05 +0.10 +0.15 +0.01 +0.1 +1 +10 +100 +- +- +- +n , p , T + 0 0 0 +x + +p0 +n0 +T0 +Figure 5. The x dependence (across the prominence pillar) of normalized initial temperature, T0 (red), density, n0 (blue), and +thermal pressure p0 (green) in the model prominence and surrounding corona. The magnitudes of the variables are shown on +Log scale. +The initial x-dependent temperature T0 and density n0 structures are given by +T0(x) = Tmax − (Tmax − Tmin)e−[(x−x0)/w]2q, +(7) +n0(x) = p0/T(x), +(8) +where the coronal temperature is Tmax, the prominence temperature is Tmin, the exponent q = 2 defines the sharpness +of the temperature transition between the corona and the prominence pillar, w = 0.05 is the half-width of the +prominence pillar, and x0 = 0 is the center position of the pillar. The initial temperature and density dependencies +on the x coordinate across the model prominence pillar are shown in Figure 5. The normalized thermal pressure, +p0 = n0T0 = 1, is uniform. In the present model we use Tmin/Tmax = 0.01, consistent with the typical ratios of the +prominence to corona temperatures. It is evident in Figure 5 that T0 decreases from its coronal value by two orders of +magnitude, while n0 increases correspondingly by the two orders of magnitude in the prominence pillar. Note, that the +fine-scale structuring of the background density in the direction of wave propagation, i.e., with height, may introduce +dispersion, enhanced damping, and small deceleration of the fast magnetosonic waves (e.g., Murawski et al. 2001). In +the present model the fast magnetosonic speed is lower by a factor of 10 in the pillar compared to the surrounding +corona, and the speed could be even lower in higher density cool prominence structures. The prominence-corona +transition region (PCTR) (see the review, Parenti 2014) along the magnetic field is evident in the model, with the +length computed as the difference between the half width at half maximum (HWHM) of the prominence pillar and +the half width at 10% of peak density as ∼ 1 Mm in physical units. This value of PCTR thickness is consistent +with previous studies (e.g., Chuideri Drago et al. 1992; Gun´ar et al. 2011). In normalized units the mass density +is equal to the number density ρ0 = n0. The initial state is in equilibrium when the magnetic field is uniform in +the x direction without gravity, which was first used to study prominence oscillations by Joarder & Roberts (1992) +and later in 2.5D MHD models of NFM waves in prominences (Ofman et al. 2015; Ofman & Kucera 2020). Here we +adopt this initial state in the 3D MHD model, as well as study additional magnetic configuration that depart from + +8 +Ofman et al. +equilibrium. Since we consider the effects of reduced solar gravity the initial state is not strictly in equilibrium. The +initial nonequilibrium leads to formation of gradients in the initially uniform magnetic field that produce a Lorentz +force balancing gravity. However, the departure from equilibrium is small in the low-β prominence model, as shown +below (see, Case 0 in Section 4). While the initial state of the density is uniform in the y and z directions, transverse +variability is introduced by the effects of the source of the waves (i.e., boundary conditions), in addition to the effects +of gravity. Since the source of the waves at the lower boundary of the prominence pillar depends on x and y and on +time (see, Equation 12 below), the compressional fast magnetosonic wave pressure introduces structure primarily in +the density and magnetic field in x, y, and z directions inside the pillar. +Realistic three-dimensional force-free extrapolations show that magnetic field of dips in quiescent prominences is +mostly horizontal (e.g., Dud´ık et al. 2012). Observed prominence structure shows evidence of magnetic shear and +flows (see, e.g., Antiochos et al. (1994) and the recent review by Gibson (2018)). Our aim is to investigate the effects +of uniform as well as sheared magnetic field on the propagation of nonlinear fast magnetosonic waves in the prominence +pillar. There are many past observations of flows in prominence pillars (e.g., Ofman & Kucera 2020, and references +within). While there could be several possible sources for the observed flows in prominences of jet-like or large-scale +flows, here, we model the effects of an unbalanced Lorentz force (i.e., non-force free magnetic configuration) with small +shear as the driver of the large-scale flows in the prominence foot, Cases 4-8. While in some observations of Polarity +Inversion Lines (PIL) in prominences the magnetic shear could be large and the magnetic field possibly force-free, +modeled with linear force-free field magnetic field (e.g., Aulanier & Demoulin 1998), or nonlinear force-free field (e.g., +Jiang et al. 2014), our model investigates for the first time the effect of non-force-free field on the formation of large- +scale flows and on the propagation of fast magnetosonic waves in the prominence pillar self-consistently. Our model +reproduces the main properties of such sheared magnetic configurations by introducing the x-dependent By component +that changes sign in the center of the prominence pillar at x=0, as modeled by Equation 9, +B0 = Bx0ˆx + By0tanh(x/w)ˆy, +(9) +where Bx0 and By0 are given in Table 1 for the eight cases studied, and w = 0.05 is the fixed half-width of the +prominence pillar. When By0 = 0, the magnetic field is potential, and the initial state given by Equations 7 and 8 is +in equilibrium. In order to study initial states that depart from equilibrium and contain currents (non-force-free), we +use small values of By0 ≪ Bx0 in the initial state. The corresponding current density j0 and Lorentz force L0 in the +x-y plane are given by +j0 = ∇ × B0 = By0 +w sech2(x/w)ˆz, +(10) +L0 = j0 × B0 = jz0(−By0 tanh(x/w)ˆx + Bx0ˆy). +(11) +The x dependence of j0 and L0 along with the corresponding magnetic field in the x-y plane are shown in Figure 6. +The present model produced the desired Lorentz force that points toward the center of the pillar. Note, that we have +also experimented with other forms of By, such as a centrally peaked profiles, and found similar results for the fast +magnetosonic waves, but with different forms of the Lorentz force and directions of the large-scale flows. The location +of the prominence pillar is depicted by the yellow-shaded area. +The adopted form of By is justified by the dynamics of the observed flows and allows exploring the fast magnetosonic +wave propagation effects in non-pontential and non-force-free magnetic field in a prominence. Moreover, this magnetic +configuration may correspond qualitatively to a section of a sigmoidal filament structure that is often unstable, leading +to eruption (e.g., Yan et al. 2012; Dai et al. 2021). +The vertical extent of the prominence pillar is evident in the observations in Figure 1 of about ∼ 40′′in the plane +of the sky (i.e., the lower limit). In the model we used ∆z = 0.4 for the height of the pillar, with the coordinates +normalization of 0.1Rs the vertical extent of the model prominence pillar is 0.04Rs = 28 Mm in agreement with +observations. Clearly, the extent of the observed low temperature prominence pillar material of ∼ 28 Mm is much +longer than the scale height for the 104 K prominence material of ∼ 0.6 Mm. Thus, one does not expect to see the +cool material in gravitational equilibrium at these heights in a field-free or in purely vertically directed magnetic field +region, and the prominence material must supported by a horizontal magnetic field component. +The boundary conditions at x = xmin and x = xmax are line tied, and the other boundary conditions are open +except for the lower boundary of the prominence pillar (z = zmin = 1). In order to launch the NFM waves at the + +Nonlinear Fast Waves in a Prominence Pillar +9 +-0.15 +-0.10 +-0.05 +-0.00 +0.05 +0.10 +0.15 +x +-1 +0 +1 +2 +3 +4 +By0, Jz0, Lx0 +Initial Magnetic Field Model +-0.15 +-0.10 +-0.05 +0.00 +0.05 +0.10 +0.15 +X +-0.15 +-0.10 +-0.05 +0.00 +0.05 +0.10 +0.15 +Y +Initial Magnetic Field Model +-0.15 +-0.10 +-0.05 +0.00 +0.05 +0.10 +0.15 +X +-0.15 +-0.10 +-0.05 +0.00 +0.05 +0.10 +0.15 +Y +Figure 6. The initial magnetic field with By0 = 0.2 (see Case 7, below). (a) The x dependencies of the y component of the +magnetic field B0 (black), the corresponding z component of the current density j0 (blue), and the resultant x component of +the Lorentz force L0 (red). (b) The initial magnetic field vectors in the x-y plane. The shaded area indicates schematically the +location of the prominence pillar. +Table 1. Parameters of the numerical 3D MHD models of prominence pillars with waves and flows. The velocity amplitude is +given in units of VA, the frequencies in τ −1 +A , and the magnetic field strength in Gauss. +Case # +Vd [VA] +ω [τ −1 +A ] +Bx,0 [G] +By,0/B0 +0 +0 +- +10 +0 +1 +0.01 +5.28 +10 +0 +2 +0.01 +12.56 +10 +0 +3 +0.02 +12.56 +10 +0 +4 +0.01 +5.28 +10 +0.1 +5 +0.01 +12.56 +10 +0.1 +6 +0.02 +12.56 +10 +0.1 +7 +0.02 +12.56 +10 +0.2 +8 +0.01 +6.28 +20 +0.1 +lower pillar boundary, the following time-dependent boundary conditions are applied on the Vz velocity component: +Vz(t, x, y, z = zmin) = Vd +2 [1 + cos(ωt)] e−[(x−x0)/sx]4−[(y−y0)/sy]2. +(12) +Motivated by the observed wave propagation primarily inside the pillar as evident in Figure 2 and the related anima- +tions, the source of the wave flux is set to be maximal in the center of the prominence pillar by using the parameters +x0 = y0 = 0.0, sx = 0.10, and sy = 0.15; the amplitude Vd controls the nonlinearity. The density and magnetic field +perturbations are computed by zero-order interpolation from the interior of the computational domain, whereas the +transverse velocity components are set to zero at the boundary. This results in periodic perturbations of the magnetic +field, density, and velocity Vz that inject NFM waves into the prominence pillar structure. In Table 1, we provide the +values of Vd, ω, Bx,0, and By,0 for the nine modeled cases in the present study. The results of Case 0 without waves +are provided for reference. +4. NUMERICAL RESULTS + +10 +Ofman et al. +Here we present the results of the 3D MHD modeling of the NFM waves in the prominence pillar for the parameters +given in Table 1. In order to explore the details of the waves, we first show in Figure 7 the results in the x-z plane cut +at y = 0 for Case 3 at t = 3.14τA. This prominence pillar is embedded in a uniform horizontal (potential) magnetic +field modeled as described in Equation 9 with By0. The NFM waves are launched by the time-dependent velocity +source (Vz, Eq. 12) at the lower boundary with amplitude Vd = 0.1VA and frequency ω = 12.56 localized at the center +of the prominence pillar. The waves propagate inside the pillar with nonlinear effects evident in non-modal structure +of the oscillations, i.e., asymmetric and sharp peaks in the variables. +The nonlinear wave pressure displaces the +magnetic field lines upwards, as is evident in this figure, affecting the temperature, magnetosonic speed, and plasma β +structure. The details of the perturbations due to the waves are particularly clear in the density contrast, ∆ρ/ρ0. The +prominence pillar acts as a leaky waveguide (Cally 1986) for the NFM waves, as the magnetosonic speed Vf is smaller +by an order of magnitude inside the prominence pillar compared to the outside (coronal) region. The small leakage +of the wave is most apparent in Figure 7e as the periodic density perturbations outside of the prominence pillar. The +squared magnitude of the current density, j2, shows the regions of enhanced currents that lead to Ohmic dissipation +(j2/S in normalized units) associated with the NFM waves. The velocity components in the x-z plane are shown +Figure 7f, where the arrows indicate the local direction (not magnitude) of the velocity vectors and the magnitude V +is color-shaded as indicated by the color bar. The corresponding magnetic field in the x-z plane is shown in Figure 7h. +The dominant Bx component is evident, along with the perturbations in the magnetic field magnitude B due to the +fast magnetosonic waves and the nonlinear wave pressure effects within the base of the prominence pillar. +(a) +(b) +(c) +(d) +(e) +(f) +(g) +(h) +Figure 7. The variables in the x-z plane at y = 0, t = 3.14 with Vd = 0.02, ω = 12.56 for Case 3. (a) ρ with overlaid magnetic +field lines. (b) The normalized fast magnetosonic speed Vf magnitude with several isocontours, (c) T, (d) β, (e) ∆ρ/ρ0, (f) V +magnitude with arrows indicating the local direction of the velocity vectors, (g) j2, (h) B magnitude with arrows indicating the +local magnetic field direction. An animation of panels (a) and (f) is available online. The video shows the density structure in +the x-z plane (left) with over-plotted field lines, and the corresponding velocity maps. The duration of the animation is 3.21 in +normalized time units in the 5 s video. +In Figure 8 we show the variables in a cut along the prominence pillar axis, i.e., in the y-z plane at x = 0 in the +high-density, low-temperature (relative to coronal values) region at time t = 3.14τA. The effects of the NFM waves +generated by the time-dependent boundary conditions in Vz are evident. In particular, the density perturbations are +in phase with the magnetic field perturbations, as seen by comparing the panels in Figure 8e and h, as expected for +the fast magnetosonic waves. The magnitude of the waves is largest in the center of the pillar due to the form of the +wave source in Equation 12, as well as to the waveguide trapping of the wave flux. The squared current magnitude j2 +is shown in Figure 8g, where the larger currents are associated with the wave fronts and are regions of higher Ohmic +dissipation (therefore also affecting the temperature). The directions of the perturbations in V and B in the y-z plane + +Nonlinear Fast Waves in a Prominence Pillar +11 +are shown in Figure 8f and h. The waves propagate in the z direction since Vf is nearly uniform in the y-z plane, with +very small perturbations due to the waves (note the intensity range in the color bar of Figure 8b). +(a) +(b) +(c) +(d) +(e) +(f) +(g) +(h) +Figure 8. The variables in the y − z plane at x = 0, t = 3.14 for Case 3. (a) ρ, (b) Vf, (c) T, (d) β, (e) ∆ρ/ρ0, (f) V with +arrows showing the direction of the velocity, (g) j2, (h) B with arrows showing the direction of the magnetic field. +The temporal evolution of the variables at a point in the center of the prominence pillar at x=0, y=0, z=1.2 for +Cases 0-3 is shown in Figure 9. The three components of the velocity and magnetic field perturbations (with respect to +B0) and the change in density and temperature are displayed. The difference between Case 1 and Case 2 is the increase +of wave frequency by a factor of 2.4, while in Case 3 the amplitude of the velocity at the boundary is increased by a +factor of 2 with respect to Cases 1 and 2. Evaluating the propagation speed of the NFM waves from the animations +for Cases 2 and 3, we find that they travel close and slightly above (5%-15%) the theoretical linear fast magnetosonic +speed. The speedup is larger for the higher amplitude waves suggesting nonlinearity effect (see, Ofman & Davila 1997, +and references therein). The nonlinearity of the fast magnetosonic waves is evident primarily in their non-sinusoidal +temporal structure, which shows evidence of steepening. This nonlinear effect is more evident in the low-frequency +waves and in the large-amplitude waves, where the wave peaks are sharper than the troughs due to steepening. The +magnetic field perturbations show secular growth of the amplitude, an indication of nonlinear wave pressure effects +on the background magnetic structure. The density perturbations show an oscillatory increasing trend, whereas the +temperature perturbations are small. +The case without without waves (Case 0) shows the evolution of the background state in the center of the pillar due +to the gravitational settling of the initial state. One can estimate the magnetic field change expected for the given +density increase of ∼5% due to the gravitational settling. This corresponds to a magnetic pressure change that is +5% of β, or about 0.4%, which equals to the value of ∆B/(2B0). Therefore, the estimated ∆B/B0 =0.8% = 0.008 +is consistent with the magnitude of the changes shown in Figure 9 in the field component plotted there for Case 0 +(green curve). It is evident that the small velocity Vz readjustment exhibits an initial oscillatory evolution due to +the effect of gravity, followed by a nearly constant downward velocity Vz ≈ −0.003VA corresponding in physical units +to about −2 km s−1. We investigated the effects of diffusion by repeating Case 0 with higher (S = 104) and lower +(S = 106) resistivity. In the latter case the spatial resolution was doubled in each direction (5143) compared to other +runs. We find that in the case with S = 104 the small down flow velocity increases by 30%. However, in the reduced +resistivity, high resolution run with S = 106, the asymptotic down flow speed remains nearly the same as in the case +with S = 105, where the density structure shows slight compression and broadening of the lower part of the pillar. +The diffusion of cold prominence material through the supporting magnetic field is expected in real prominences in + +12 +Ofman et al. +qualitative agreement with the present model for higher resistivity case, since the material is partially ionized (for +example, see Gilbert et al. 2002; Khomenko et al. 2014), and where the frozen in condition breaks down due to finite +resistivity (Low et al. 2012; Low & Egan 2014; Jenkins & Keppens 2021), see the review by Gibson (2018). While +in the MHD model the down flow velocity at lower resistivity is due to compressive effects, we find that this velocity is +small compared to the phase speed of the fast magnetosonic waves and therefore has no significant effect on the wave +propagation. +(a) +0 +1 +2 +3 +4 +time +-0.005 +0.000 +0.005 +0.010 +0.015 +Vx, Vy, Vz +(b) +0 +1 +2 +3 +4 +time +0.00 +0.05 +0.10 +0.15 +0.20 +DBx, DBy, DBz +(c) +0 +1 +2 +3 +4 +time +0.00 +0.05 +0.10 +0.15 +Dr/r0, DT/T0 +Figure 9. Temporal evolution of the variables in the center of the prominence pillar for Case 1 (blue) with Vd = 0.01, ω = 5.26, +Case 2 (red) with Vd = 0.01, ω = 12.56, and Case 3 (black) with Vd = 0.02, ω = 12.56. (a) Velocity components Vz (solid), +Vy (short dashes), Vx (long dashes). (b) Magnetic field component perturbations ∆Bx (solid), ∆By (short dashes), ∆Bz (long +dashes). (c) Changes in density ∆ρ/ρ0 (solid) and temperature ∆T/T0 (long dashes) normalized by the respective initial values +ρ0 and T0. The case without waves (Case 0) is shown (green) for reference. Times are in units of τA. +The structure of the magnetic field and density perturbations due to the NFM waves for Case 2 is shown in Figure 10. +In the present model the initial state was the result of Case 0 without waves, where slight dips form in the magnetic +configuration of the pillar due to the effects of reduced gravity. The figure and the animation show the magnetic field +lines and the density isocontours in the domain at t = 5.8τA. A small lifting of the magnetic field lines by the wave +pressure is evident mostly in the lower region of the pillar, while the small gravitational dipping of the field lines is +most evident in the upper part of the domain in the initial state, reduced at later times due to the effects of wave +pressure. Isocontours of density indicate the locations of the propagating compressions due to the guided NFM waves, +while further details of the wave propagation are exhibited in the animations provided. +The effects of non-potential, non-force-free magnetic fields on the propagation of the fast magnetosonic waves in the +prominence pillar are explored in Cases 4-8. The form of the background magnetic field is given by Equation 9. The +parameters of Cases 4-6 are the same as in Cases 1-3 except that By,0 = 0.1. In Case 7 we consider By = 0.2, with +the other parameters as in Case 3, and in Case 8 we consider B0 = 20 G, with the rest of the parameters the same as +in Case 6. These results are discussed below. +In Figure 11 we show the variables in the x-z plane at y = 0 for Case 6 with By0 = 0.1 at t = 3.03τA. The NFM wave +structure is most evident inside the prominence pillar in the relative density compressions, ∆ρ/ρ0, but also is seen in +the variability in ρ, β, j2, and the velocity and magnetic field magnitudes. Comparing ∆ρ/ρ0 to Case 3 (Fig. 7e), we +find that the relative magnitude of the leakage in the x direction is reduced. The effects of the x component of the +Lorentz force in compressing the prominence pillar density are seen by comparing the structure of ρ to the initial state + +Nonlinear Fast Waves in a Prominence Pillar +13 +Figure 10. The results of the 3D MHD model in Case 2. (a) Magnetic field lines and (b) density isocontours due to the +propagating NFM waves in the domain at t = 5.8τA. An animation of this figure is available online. The duration of the +animation is 3.6 in normalized time units that shown in 2 s video. +in Figure 6 and to ρ in Case 3 shown in Figure 7a. The apparent half-width is reduced by about 30% in the present +case. +(a) +(b) +(c) +(d) +(a) +(b) +(c) +(d) +Figure 11. The variables in the x-z plane cut at y = 0, t = 3.03τA for Vd = 0.02, ω = 12.56 (Case 6). (a) ρ with field lines +indicated with white lines. (b) Vf magnitude with several isocontours. (c) T. (d) β. (e) ∆ρ/ρ0. (f) V magnitude with arrows +showing the local direction of the velocity vectors. (g) j2. (h) B magnitude with arrows showing the local direction of the +magnetic field vectors (dominated by Bx). +Figure 12 shows the variables in the y-z plane at x = 0 for the case with By0 = 0.1 (Case 6) at t = 3.03τA. The +refraction of the wave fronts of the NFM waves due to the effect of the By0 magnetic field component is apparent +by comparison with Figure 8. The wave structure is evident in the density and magnetic field perturbations, as well +as in the corresponding current perturbations. In this magnetic configuration, the waves leak significantly out of the +prominence pillar through the side boundary at y = ymax, decreasing the wave energy flux in the center of the pillar, + +Time= 5.80e+00 +1.2 +0:10 +1005 +.00(b)14 +Ofman et al. +whereas in the uniform magnetic field case, the main leakage takes place through the top of the prominence pillar +(z = zmax) with open boundary condition. +(a) +(b) +(c) +(d) +(e) +(f) +(g) +(h) +Figure 12. Variables in the y-z plane at x = 0, t = 3.03 for Case 6. (a) ρ. (b) Vf. (c) T. (d) β. (e) ∆ρ/ρ0. (f) V . (g) j2. (h) +B. +The cut in the x-y plane (i.e., the solar ‘disk’ view) of the model shows the structure of the prominence pillar density, +temperature, fast magnetosonic speed, β, j2, B, and V at height z = 1.2 at t = 3.03τA for Case 6. The effects of +the upward propagating NFM waves are evident. In the x-y plane, the waves are most clear in ∆ρ/ρ0, j2, β, and the +magnitudes B and V . The y dependence of the wave structure is affected by both the driving source and the wave +refraction due to the By0 component of the background magnetic field. It is evident from ∆ρ/ρ0 that the leakage of the +wave is significant in the density compressions outside the prominence pillar region (i.e., |x| > 0.05). The Lorentz force +generates a compression of the density primarily in the x direction, with small magnetic field and density compression +in the y direction, as can be seen from the density and velocity structures in the x-y plane. +The temporal evolution of the variables for Cases 4-6 in the center of the prominence pillar at x=0, y=0, z=1.2 are +shown in Figure 14. Evidently, the non-force-free magnetic field introduces flows due to the Lorentz force that lead to +compression in the prominence pillar, and corresponding increases of magnetic field strength and density that disrupt +the initial gravitational equilibrium. In particular, it is evident that the Vy component has similar accelerated evolution +in Cases 4-6, with weak dependence on the properties of the fast magnetosonic waves. Thus, the effects of the Lorentz +force in the non-force-free field in introducing mass flows self-consistently becomes evident. The flow accelerates during +the simulated time, exceeding 10% of the Alfv´en speed (about 70 km s−1 with the present normalization) by the final +modeled time. +The effect of increased Lorentz force on the structure of the prominence pillar and on the NFM waves is demonstrated +in Case 7 with By0 = 0.2. As expected, the increased Lorentz force leads to more rapid and powerful compression of +the prominence pillar than in Cases 4-6. This affects the properties of the background density structure of the pillar +and also the NFM waves. In particular, the wave frequency has decreased due to the increased compression, mainly +due to the increase in density and corresponding decrease in Vf inside the prominence pillar. This also leads to the +decrease of the velocity amplitude associated with the NFM waves inside the pillar for the fixed wave source at the +boundary. +In Case 8 we investigate the effects of increased magnetic field strength on the NFM waves by doubling the assumed +magnitude of the magnetic field, B0 = 20 G. This change with respect to previous cases results in a doubling of VA +and a decrease of plasma β by a factor of four. Since the velocity amplitude Vd is in units VA, the magnitude of the +nonlinearity of the fast magnetosonic wave in Case 8 is essentially the same as in Case 4. Also, we note that τA is half +the value in Case 8 compared to other cases, and that the value of ω in Case 8 is equal to the value in Case 7 when + +Nonlinear Fast Waves in a Prominence Pillar +15 +(a) +(b) +(c) +(d) +(e) +(f) +(g) +(h) +Figure 13. Variables in the x-y plane at z = 1.2, t = 3.03 for Case 6. (a) ρ. (b) Vf with several isocontours. (c) T. (d) β. (e) +∆ρ/ρ0. (f) V magnitude with direction arrows. (g) j2. (h) B magnitude with direction arrows. +(a) +0 +1 +2 +3 +4 +time +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +0.12 +Vx, Vy, Vz +(b) +0 +1 +2 +3 +time +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +0.12 +0.14 +DBx, DBy, DBz +(c) +0 +1 +2 +3 +time +0.0 +0.1 +0.2 +0.3 +0.4 +Dr/r0, DT/T0 +Figure 14. The temporal evolution of the variables in the center of the prominence pillar for Case 4 (red) with Vd = 0.01, +ω = 5.28, Case 5 (blue) with Vd = 0.01, ω = 12.58, and Case 6 (black) with Vd = 0.02, ω = 12.58. (a) Velocity components +Vz (solid), Vy (short dashes), Vx (long dashes). (b) Magnetic field component perturbations ∆Bx (solid), ∆By (short dashes), +∆Bz (long dashes). (c) Changes in density ∆ρ/ρ0 (solid) and temperature ∆T/T0 (long dashes) normalized by the respective +initial values ρ0 and T0. Times are in units of τA. +converted to rad s−1. The main difference with respect to previous cases is the effect of the wave pressure, which is +now four times larger in Case 8 and leads to correspondingly stronger density compressions. +5. DISCUSSION AND CONCLUSIONS + +16 +Ofman et al. +Recent high spatial and temporal resolution observations of a prominence pillar from Hinode/SOT in Hα and Ca II +and IRIS in Mg II show evidence of small-scale oscillations and propagating features associated with flows (Kucera +et al. 2018). Analysis of Doppler shifts from Hinode/SOT Hα and IRIS show red-wing/blue-wing contrasts that are +consistent with propagating waves and flows on extended magnetic field lines (e.g., Ofman & Kucera 2020). Here, +we analyze additional observations of propagating small-scale oscillations in the Hinode/SOT Ca II line and the blue +wing of Hα in a prominence on 2012 February 14. Using space-time plots and wavelet analysis, we find oscillations +with typical periods of order minutes and wavelengths of order 1000-2000 km with sharp peaks indicative of nonlinear +steepening, and we identify the propagating features as signatures of nonlinear fast magnetosonic (NFM) waves. +Motivated by past and recent observations of the small-scale oscillations, we developed an idealized 3D MHD model +of a prominence pillar that focuses on the generation and propagation of NFM waves guided by observed properties. +The advantage of the simplified 3D model is tractability of the wave features when the line-of-sight projection effects +inherent to single-point plane-of-the-sky observations are removed. +The 3D model reproduces the main physical +properties of the prominence cool material embedded in background magnetic field and of the observed propagating +small-scale features. +The present model extends previous 2.5D MHD studies of the propagating NFM waves into more complex and +realistic prominence structures by allowing 3D wave propagation and couplings. There is evidence of nonlinear coupling +of the NFM waves to other wave modes in the animation of the density structure, which shows secondary density +compressions due to slow magnetosonic mode that appear to follow the compressions associated with NFM waves. +There is also evidence of small amplitude Alfv´enic oscillation in the temporal signatures of the variables. However, we +find that the main effects of nonlinearity of the waves are the steepening and the coupling between the NFM waves +and the background pillar structure in the low-β plasma. +We modeled eight cases and varied the main parameters of the waves in two types of magnetic field configurations +(uniform potential and for the first time non-force-free) to provide insights on the effects of the various parameters +on the generation and propagation of NFM waves. +Evidence of velocity and magnetic shear is often observed in +pre-eruptive prominences configurations (Gibson 2018). +Therefore, we modeled non-force-free field with magnetic +shear that introduces large-scale flows, corresponding (aperiodic) compressions of the prominence pillar, and dynamic +changes in wave propagation properties, all self-consistently. We find that the effects of the non-force-free sheared +magnetic field on the pillar structure and on the wave propagation are significant, even for relatively small magnitude +of the shear-produced Lorentz force, due to the low-β state of the prominence material. Thus, the effects of magnetic +field shear on the NFM waves may affect the application of coronal seismology in prominence pillars. +The modeling results show qualitative agreement with the observed propagating oscillations with nonlinear steep- +ening in the prominence pillar, as demonstrated in previous studies (Ofman et al. 2015; Ofman & Kucera 2020) and +the present observational analysis. The 3D MHD results confirm further the interpretations of the observed propa- +gating small-scale features in terms of NFM waves that are wave-guided in the cool material (low fast magnetosonic +speed) of the prominence pillar region. From the model we find that the wave nonlinearity leads to secular changes +in background magnetic field structure, density, and temperature due to the wave pressure, in addition to the wave +steeping effects that affect the small-scale compressive structures. The low-frequency wave source leads to higher +amplitude guided NFM waves than the high-frequency waves, due to lower leakage and dissipation compared to the +high-frequency waves. +Our study demonstrates the potential applications to the observed small-scale waves together with modeling for +magnetic seismology of the prominence structure. One can apply coronal seismology by using the properties of the +observed waves, such as wavelengths and periods to determine the phase speed of the waves. The relation between the +phase speed and the magnetic field can be obtained from linear theory for linear waves in simplified geometry (e.g., +Nakariakov & Verwichte 2005). For nonlinear waves in more complex geometry the phase speed can be obtained from +a 3D MHD model. . Finally, by comparing the theoretical/modeled phase speed with the observed phase speed and +with the density and temperature information, one can determine the magnetic field in the pillar (taking into account +possible plane of the sky (POS) projection effects). The details of magnetic geometry structure could be deduced from +the observed direction of wave propagation where a 3D MHD model helps alleviate the POS observational ambiguity. +The present model considers the nonlinearity in various idealized magnetic field geometry scenarios, and in the future +more realistic 3D MHD wave models will include more detailed magnetic and density structure based on specific +observations, thus improving the accuracy of coronal seismology method. + +Nonlinear Fast Waves in a Prominence Pillar +17 +LO acknowledges support by NASA Cooperative Agreement 80NSSC21M0180 to The Catholic University of America. +Resources supporting this work were provided by the NASA High-End Computing (HEC) Program through the NASA +Advanced Supercomputing (NAS) Division at Ames Research Center. TAK and CRD were supported by NASA’s H- +ISFM program at Goddard Space Flight Center. 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F., et al. 2012, ApJ, 754, +16. doi:10.1088/0004-637X/754/1/16 + diff --git a/U9AyT4oBgHgl3EQf8vpk/vector_store/index.faiss b/U9AyT4oBgHgl3EQf8vpk/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..e4147751b34c997282816eb82bfbdedaff3f2d5c --- /dev/null +++ b/U9AyT4oBgHgl3EQf8vpk/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:ffb011e01cc2bdfbffe1f00076dcbdfb60d03ee3d101a9857f7e423f03d2daf3 +size 3080237 diff --git a/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf b/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..2efa54a8e158575e7e5e7eec76105136b670a5e5 --- /dev/null +++ b/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:1299764442a7bdc44681e5b9ed921b8484858065d0c04b38144106bf98e607bf +size 217892 diff --git a/U9E_T4oBgHgl3EQfxhwl/vector_store/index.pkl b/U9E_T4oBgHgl3EQfxhwl/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..569c737540e248d7221d353039710b9fde3184fe --- /dev/null +++ b/U9E_T4oBgHgl3EQfxhwl/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:f8367bef83a2221f55c000679045f2ffef780486539204c1c5d4570fcdaed8fd +size 101815 diff --git a/UdE0T4oBgHgl3EQf2gJy/content/tmp_files/2301.02713v1.pdf.txt b/UdE0T4oBgHgl3EQf2gJy/content/tmp_files/2301.02713v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..0b00790c221620c2cdaecb3175cef793fd1e4d1b --- /dev/null +++ b/UdE0T4oBgHgl3EQf2gJy/content/tmp_files/2301.02713v1.pdf.txt @@ -0,0 +1,583 @@ +���������� +������� +Citation: Kauffmann, J.; Rajagopalan, +G.; Akiyama, K.; Fish, V.; Lonsdale, +C.; Matthews, L.D.; Pillai, T. The +Haystack Telescope as an +Astronomical Instrument. Preprints +2023, 1, 0. https://doi.org/ +Academic Editor: Lorenzo Amati +Publisher’s Note: MDPI stays neutral +with regard to jurisdictional claims in +published maps and institutional affil- +iations. +Copyright: +© 2022 by the authors. +Licensee MDPI, Basel, Switzerland. +This article is an open access article +distributed +under +the +terms +and +conditions of the Creative Commons +Attribution (CC BY) license (https:// +creativecommons.org/licenses/by/ +4.0/). +Article +The Haystack Telescope as an Astronomical Instrument +Jens Kauffmann* +, Ganesh Rajagopalan +, Kazunori Akiyama +, Vincent Fish +, Colin Lonsdale +, +Lynn D. Matthews +, and Thushara G.S. Pillai +Haystack Observatory, Massachusetts Institute of Technology, 99 Millstone Rd., Westford, MA 01886, USA; +ganesanr@mit.edu (G.R.); kakiyama@mit.edu (K.A.); vfish@mit.edu (V.F.); cjl@mit.edu (C.L.); +lmatthew@mit.edu (L.D.M.); thushara@mit.edu (T.P.) +* Correspondence: jens.kauffmann@mit.edu +Abstract: The Haystack Telescope is an antenna with a diameter of 37 m and an elevation-dependent +surface accuracy of ≤100 µm that is capable of millimeter-wave observations. The radome-enclosed +instrument serves as a radar sensor for space situational awareness, with about one-third of the time +available for research by MIT Haystack Observatory. Ongoing testing with the K-band (18–26 GHz) and +W-band receivers (currently 85–93 GHz) is preparing the inclusion of the telescope into the Event Horizon +Telescope (EHT) array and the use as a single-dish research telescope. Given its geographic location, the +addition of the Haystack Telescope to current and future versions of the EHT array would substantially +improve the image quality. +Keywords: Very Long Baseline Interferometry; radio astronomy; millimeter astronomy; radio telescopes; +high angular resolution; astronomical instrumentation +1. Introduction: Astronomy Observations with the Haystack Telescope +MIT Haystack Observatory has been a home to a radome-enclosed telescope of 37 m +diameter since 1964 [1]1. Figure 1 illustrates the siting of the instrument, while Figure 2 presents +an overview of the dish. The original system was primarily conceived as a space radar and as +a platform for telecommunications experiments to support work by MIT Lincoln Laboratory. +Ownership was transferred to the Northeast Radio Observatory Corporation2 (NEROC) in +1970, with the goal to enable millimeter-wave observations for the astronomy community in the +Northeast US, while still being available as a radar sensor to MIT Lincoln Laboratory. The site +is known as MIT Haystack Observatory since that time. The telescope has undergone several +upgrades since its original dedication. Some of this work focused on improving the surface +accuracy of the dish, which was improved from an initial root-mean-square (RMS) error of +∼900 µm to ∼200 µm after 1992 [2]. +The Haystack Telescope has enabled key scientific discoveries, as summarized by Whitney +et al. [3]. Radar observations delivered key intelligence on the Apollo landing sites, and joint +observations with the Westford Telescope—a dish of 18 m diameter located about a mile from +the Haystack Telescope—produced the first radar maps of Venus’ surface that cleanly separated +radar echoes from the planet’s northern and southern hemispheres. Radar observations of +Venus and Mercury also delivered stringent tests of General Relativity by constraining the +gravitational time delay caused by the presence of the Sun (i.e., Shapiro delay, [4]). Single-dish +spectroscopy observations with the Haystack Telescope were essential in establishing “dense +molecular cores” as the key star-forming sites in molecular clouds [5], and they showed how +dense cores build up density as they contract out of more diffuse cloud material [6]. MIT +Haystack Observatory led the way during the inception of Very Long Baseline Interferometry +(VLBI), and 8 of the 22 awardees of the 1971 Rumford Prize of the American Academy of +Arts and Sciences for the inception of VLBI were working at the observatory. The Haystack +Telescope critically supported this work. VLBI observations with the instrument, such as the +arXiv:2301.02713v1 [astro-ph.IM] 6 Jan 2023 + +BY2 of 11 +discovery of apparent superluminal motion in quasars [7], shaped our understanding of the +universe at high angular resolution. +Figure 1. Aerial view of MIT Haystack Observatory. The Haystack Telescope, a dish of 37 m diam- +eter, is located in the large radome dominating the foreground. (Used with permission, courtesy of +Mark Derome). + +3 of 11 + +VOLUME 21, NUMBER 1, 2014 � LINCOLN LABORATORY JOURNAL +47 +NIKOLAS T. WAGGENER +FIGURE 3. A computer-aided design rendering of the HUSIR antenna shows its key features. +× 65–foot temporary fabric and steel building that would +become the assembly area for the new backstructure, +permitting work to proceed year-round and providing +the controlled environment necessary for the alumi- +num welding work. Several smaller site improvements +included assembly areas for the quadrapod (the support +structure for the subreflector) and the transition structure +(the steel backup structure that includes the elevation +counterweights, drive gears, and bearings). +To minimize the impact of adverse weather during the +critical lifts of the major subassemblies, the engineering +team targeted spring 2010 for the start of the integration +period. Because of the scope of the work and the relative +inexperience of Lincoln Laboratory with large construction +projects, the construction management firm Bond Broth- +ers of Everett, Massachusetts, was hired to supervise daily +operations at the site. Bond has a background in coordinat- +ing large civil engineering projects and a familiarity with +the various skilled trades required to successfully com- +plete this type of project. Keystone Engineering, formerly +of Georgetown, Massachusetts, was contracted to provide +much of the labor and equipment for the construction, and +assembled a crew of highly skilled workers, primarily from +the Local 7 ironworkers union. Hallamore Corporation of +Holbrook, Massachusetts, supplied the 400-foot-tall Mani- +towoc 18000 MAX-ER crawler crane, which handled the +Massachusetts, the engineering team considered various +design options, including actively controlled positioners +for the primary surface panels, before settling on the final +strategy. The details of the final design* of the antenna are +included in the appendix to this article. The heart of the +upgraded antenna would be a stifness-optimized primary +reflector (backstructure). The backstructure was designed +to deform such that it remains a paraboloid under gravity +or temperature-induced loading, albeit with a diferent +focal length that can be corrected by adjusting the posi- +tion of the secondary reflector (subreflector). The rest of +the new antenna structure was developed to support the +optimized backstructure, while utilizing as much of the +existing structure as possible (Figure 3). +By 2007, most of the design work was completed and +fabrication of many subassemblies had begun. Working +directly with subcontractors, the Lincoln Laboratory team +oversaw the completion of the new antenna hardware. +In 2008, construction began of the onsite facilities that +would be used to perform the final assembly and integra- +tion of the various subassemblies, many of which would +be too large to ship to the site fully assembled. The largest +of these site projects was the construction of a 140 × 160 +Transition structure +120 ft +diameter +dish +Backstructure +200 tons above elevation axis +20 ft radius sector gears +85 ft +340 tons above azimuth axis +Hydrostatic +azimuth bearing +and bull gear +42 ft +Concrete pedestal +Yoke +RF box +Subreflector +Quadrapod +Primary reflector +*The design of the antenna was originally proposed by Apostle +Cardiasmenos of L-3 Communications ESSCO. +Figure 2. Overview of the Haystack Telescope, with the radome removed [8]. The receiver equipment is +installed in the “RF box”, a container that is brought down to ground level during “box-down” periods to +enable major engineering activities. (Reprinted with permission, courtesy of MIT Lincoln Laboratory, +Lexington, Massachusetts). +A major upgrade, completed in 2014, improved the surface accuracy to ≤100 µm, depend- +ing on elevation. This work, executed by MIT Lincoln Laboratory under sponsorship by the +Defense Advanced Research Projects Agency (DARPA) and the U.S. Air Force, was part of the +upgrade delivering the Haystack Ultrawideband Satellite Imaging Radar (HUSIR). The HUSIR +system is designed around a W-band radar covering a substantial bandwidth of 92–100 GHz, +and it also includes an X-band radar operating at 9.5–10.5 GHz. The outstanding bandwidth +of ∆ν = 8 GHz enables the W-band radar to directly resolve structures of c/(2 · ∆ν) = 1.9 cm +size in range [9], with advanced image processing techniques delivering an effective resolution +well below this scale. In 2014, HUSIR’s W-band radar delivered the finest spatial resolution +of any imaging radar, while the X-band radar constituted the only system for imaging out to +geosynchronous orbits [9]. The systems have been upgraded since, and HUSIR continues to be +an essential contributing sensor for space situational awareness. +Today, NEROC has access to about one-third of the time available on the Haystack Tele- +scope. This time can be used to conduct experiments in astronomy and other fields of funda- +mental research. The primary access windows are weekends, and night hours at 23:00–07:00 +local time on Mon.–Fri. Access to other periods, as for example needed for time-critical experi- +ments in astronomy, is coordinated with MIT Lincoln Laboratory. Such work can currently use +K-band (18–26 GHz) and W-band receivers (85–93 GHz) dedicated to astronomical observations +that are separate from the HUSIR systems. An existing Q-band system covering 36–50 GHz +will be brought online in the future. + +4 of 11 +MIT Haystack Observatory currently studies the expected performance of the telescope at +∼230 GHz. This is done as part of the ngEHT project (as described elsewhere in this special +issue; also see https://www.ngeht.org, accessed on 2022 Dec. 15), which seeks to deliver a +“next generation EHT” by adding new stations and other capabilities to the Event Horizon +Telescope (EHT; see [10] for a recent description of the system). Inclusion of the Haystack +Telescope into the EHT would enhance the imaging capabilities of the array, as described below. +The upgraded dish provides exciting opportunities for astronomy. Unfortunately, until re- +cently it was not possible to make use of the telescope’s capabilities, given the lack of substantial +and systematic funding for astronomical experiments. This has changed in the past few years, +thanks to a private donation and a grant from the National Science Foundation supporting the +ngEHT project (AST-1935980). The telescope is currently regularly used to conduct experiments +in support of system commissioning and initial experiments into astrophysical research and +education. This includes three VLBI runs at 86 GHz that have delivered fringe detections on +intercontinental baselines. +This paper is organized as follows. The telescope, its current and future instrumentation, +and the characteristics of the site are described in Section 2. The discussion in Section 3 outlines +the case for research, education, and technology development on the Haystack Telescope. The +connection of the telescope to the EHT and ngEHT projects is described in Section 4. The +material is summarized in Section 5. +2. Telescope, Instrumentation, and Site +2.1. Telescope and Site +Figure 2 summarizes the characteristics of the dish. The reflector of the Haystack Telescope +has a diameter of 120 ft, equivalent to 36.57 m. It is formed by 432 panels that each have an +RMS surface accuracy of about 28 µm [8]. The main reflector itself is rigged to achieve am +RMS surface accuracy of 75 µm at an elevation of 25◦, with larger deformations occurring at +higher or lower elevations [11]. The moving sections have a mass of 340 t, with 200 t of mass +moving in elevation. The dish is capable of slewing at speeds of 5◦ s−1 in azimuth and 2◦ s−1 +in elevation, and it achieves accelerations of 1.◦5 s−2 and 2◦ s−2, respectively. By requirement, +the pointing accuracy is < 3.′′6, with a tracking accuracy < 1.′′8 [11]. +The telescope is housed in a radome of 150 ft diameter that was originally designed for use +in extreme arctic environments and is capable of withstanding 130 mph winds (i.e., 210 km h−1 +or 60 m s−1) [8]. The radome is skinned with three-ply ESSCOLAM 10 membranes with a +hydrophobic coating, which are characterized by a transmission of about 95% at 90 GHz [8]. +The receivers are housed in a “box” that is installed about 85 ft above ground. It can be +brought down to the floor of the telescope building during dedicated “box-down” periods. The +box houses radar equipment as well as the astronomy receivers, and it is very tightly packed +with systems. As a consequence, major engineering actives can only be performed during a +box-down window, during which the interior of the box can be accessed easily from all sides. +The number and duration of box-down periods is minimized in support of high-priority radar +observations. +The observatory’s land is distributed over the Massachusetts towns of Groton, Tyngsbor- +ough, and Westford, with Westford being the administrative home of MIT Haystack Observa- +tory. This thickly forested community is about an hour’s drive away from downtown Boston +(MA). The telescope itself is located at 42.◦62 N vs. 71.◦49 W, at an altitude of 130 m. +Haystack Observatory experiences extended periods of cold and dry weather during the +winter, thus providing the weather conditions needed for observations at millimeter wavelengths. +Historical measurements of the precipitable water vapor (PWV) column are available from the +Suominet3 network for atmospheric research. Archived data give a median PWV column of +8.3 mm for the period November 1 to April 30. Assuming an outside temperature of 0 ◦C, + +5 of 11 +modeling of the atmosphere with the AM4 radiative transfer code gives a corresponding optical +depth of 0.12 at ∼86 GHz under such conditions, equivalent to an atmospheric transmission +of 84% at 45◦ elevation. More realistically, observations by systems like the EHT are triggered +in better-than-median atmospheric conditions. To give an example, the PWV column is below +5.3 mm for 25% of the winter period. Rich additional documentation about the telescope and the +site can be found in Brown and Pensa [1], Whitney et al. [3], Waggener [8], Czerwinski and Usoff +[9], Usoff et al. [11], MacDonald et al. [12], and Eshbaugh et al. [13]. +2.2. Current Instrumentation +The telescope is equipped with receivers operating in the K (18–26 GHz), Q +(36–50 GHz), and W bands (70–115 GHz). The cryogenic frontends operate at around 20 K in +independent dewars. These are +arranged roughly on a vertical line that is offset from the +central focal point. MIT Lincoln Laboratory operates on-axis X-band and W-band radars, so +that the three astronomy receivers are offset from boresight. The sub-reflector on a hexapod is +remotely controlled to choose between the three K, Q and W-band receivers. Current observing +projects make use of the K and W bands, and these receivers are therefore currently kept +operational by engineering activities. +The K-band frontend is shared between MIT Haystack Observatory and MIT Lincoln +Laboratory. One polarization is available for astronomical observations, while the other +polarization is used for holography observations. Astronomical observations can be conducted +anywhere in the frequency range of 18–26 GHz. +The W-band frontend is currently configured as a single-sideband receiver that senses +horizontal and vertical polarization. Data are taken in a sideband of 8 GHz width that is +set by an analog bandpass filter. The system is currently set up to observe at frequencies of +85–93 GHz. Modest upgrades to the hardware would allow to access the full frequency range +of 70–115 GHz. The receiver was recently improved via the installation of a new wideband low +-noise amplifier (LNA) and components for the sideband rejection scheme. These investments +were made possible by an NSF MSRI-1 grant to the ngEHT project (AST-1935980). +The backends are located at the ground level of the telescope building. A radiofrequency- +over-fiber (RFoF) system is used to transport the signals into this room. The RFoF infrastructure +is currently being upgraded for transport bandwidths of up to 20 GHz for two polarizations. +An up-down converter (UDC) is used to condition the signals for the backends. The single-dish +backend currently processes up to 500 MHz in one polarization. Further investments in hardware +and software would enable processing of larger bandwidths and of a second polarization. The +backend measures continuum signals, and it currently also produces spectra of up to 500 Hz +resolution. VLBI data are acquired using a ROACH2 digital backend (R2DBE) connected to a +Mark 6 VLBI recorder. A Rakon Oven Controlled Crystal Oscillator (OCXO) is used as a frequency +standard for ongoing engineering experiments in VLBI. The acquisition of the RFoF infrastructure, +and the ongoing acquisition of a new OCXO, are supported by an NSF MSRI-1 grant to the ngEHT +project (AST-1935980). +2.3. Current and Future Instrument Development +Current work on the Haystack Telescope focuses on evaluation of the newly upgraded +system (i.e., after installation of the W-band LNA, RFoF system, and VLBI equipment). While +characterization of the W-band system is the main activity, the K-band receiver is occasionally +used to deliver complementary data on telescope performance under less ideal weather con- +ditions. This program consists of single-dish observations of calibrators like planets, as well +as participation in observations by VLBI networks. In the area of interferometry the goal is to +enable future VLBI observations at ≲90 GHz, and to assess the feasibility of VLBI observations +at ∼230 GHz in support of the EHT. More generally, the observations seek to demonstrate the + +6 of 11 +capability of the Haystack Telescope to deliver exciting astrophysical research as a single-dish +telescope and VLBI station. +Current funding from an NSF MSRI-1 grant (AST-1935980) supports the design of a +receiver for VLBI observations with the Haystack Telescope at ∼230 GHz in the context of +the ngEHT project. This undertaking might evolve into the design for a multi-band receiver +enabling parallel observations at ∼86 GHz and ∼230 GHz. This depends on future decisions +by the EHT and ngEHT projects concerning the need for multi-band observations in support of +“frequency phase transfer” (FPT, [14]), i.e., the transfer of VLBI calibration information obtained +at one frequency to other bands. +The long-term plan for the telescope foresees to support single-dish and VLBI observations +at K, Q, and W band, as well as at ∼230 GHz. Ongoing investigations will clarify whether +some or all of these receivers need to be able to observe in parallel (e.g., to support FPT). +The installation of wideband (i.e., ≥8 GHz) receivers and backends for single-dish and VLBI +observations, as well as of a maser clock as a frequency standard for VLBI experiments, form +part of this long-term plan. The development of the telescope must be supported via dedicated +grants from funding agencies, as the Haystack Telescope receives no general-purpose funding +to advance the capabilities of the facility. +3. Astrophysical Research, Education, and Technology Development +Section 4 explains how the Haystack Telescope can add new, sensitive, and important +baselines to the EHT. In the same way, the Haystack Telescope can complement the Global +Millimeterwave VLBI Array (GMVA). The increased availability of multi-band receivers on +radio observatories, as pioneered by the Korean VLBI Network (KVN) [15], raises the exciting +prospect for the Haystack Telescope to join an intercontinental network building on FPT. +The outstanding scientific capabilities of large single-dish instruments at millimeter wave- +lengths are demonstrated by the high impact of current research on the IRAM 30m-telescope. +Recent work with that telescope includes the study of star formation physics in nearby galax- +ies [16], and investigations of molecular cloud structure and evolution in the Milky Way that +support the aforementioned extragalactic work [17,18]. Many of these studies employ the +EMIR receiver system that samples 16 GHz of bandwidth per polarization in the 70–115 GHz +range [19]. Installation of receiver and backend systems matching or exceeding this capability +would open up new and exciting capabilities for the US-based community. +The Haystack Telescope offers unique, important, and currently missing educational +opportunities in the US. The instrument is associated with the NEROC community of 13 +research-intensive educational institutions in the Northeast US (see footnote 5 on page 10). +Several of these are closely involved in the EHT, the Large Millimeter Telescope (LMT), and +the Submillimeter Array (SMA). The Haystack Telescope is within easy driving distance from +all these institutions, providing outstanding hands-on experiences for junior researchers within +NEROC. More generally, the telescope can serve as a destination for educational astronomical +daytrips that are independent from daytime, and only modestly dependent on the weather, +by schools, colleges, and universities in six states of the US (all of MA, RI, CT; most of NH; +parts of ME and NY)5. +Operations of the Haystack Telescope are exclusively funded by grants with specific +objectives and deliverables. MIT Haystack Observatory receives no general long-term funding +that could make the instrument broadly accessible by the community. Future operations +are expected to be supported via a mix of funding streams. This will include observations +for specific user groups that will pay for having their data taken on the Haystack Telescope. +The current engineering work undertaken on an NSF MSRI-1 grant supporting the ngEHT +project constitutes one example for such observations. Similarly, NSF AAG grants could fund +data collection for specific astrophysical research projects. It is the ambition of MIT Haystack + +7 of 11 +Observatory to also make the telescope broadly available to the entire US community. The +feasibility of such a program depends on the grants acquired by the observatory. +4. Connection to the Event Horizon Telescope: Current Work and Future Roles +Ongoing work on the Haystack Telescope is in part funded by an NSF MSRI-1 grant +(AST-1935980). This award forms part part of the ngEHT project, and its goal is to evaluate +the telescope for inclusion into the EHT at ∼230 GHz via quantitative modeling and VLBI test +observations at ∼86 GHz. A private donation to MIT Haystack Observatory enables parallel +activities that enhance the overall capabilities of the instrument. +Figure 3 shows that addition of the Haystack Telescope to future versions of the EHT +network would produce new and critical baselines. +This is specifically demonstrated here +for a network that includes the telescopes that are now available for future EHT observations, +but that also includes a set of additional antennas enhancing the EHT. This can be seen in +Figure 3 (top right), where dishes enhancing the current EHT constellation are marked by +green stars. The resulting significant improvements to the uv-coverage of the array can result +in reductions of the inner sidelobes of the synthesized beam by a factor 1.4 (K. Akiyama, priv. +comm.). This is indicated by imaging simulations assuming the reference array summarised +in Figure 3 (top left). +The simulation in particular demonstrates that the addition of the +Haystack Telescope to the EHT array would substantially improve the sampling of the uv- +domain at baselines of ≲4 × 109 λ (Figure 3 bottom), resulting in the aforementioned reduction +of sidelobes. Data from the Haystack Telescope can also improve other practical aspects +of VLBI observations. For example, the dish can deliver an important connection between +dishes in Europe and the Americas. The telescope can also add substantial sensitivity to VLBI +networks, in particular at low frequencies where the telescope efficiencies are higher. All +these factors aid in the overall calibration of VLBI networks, delivering advantages beyond +the fundamental improvement in uv-coverage. Importantly, the specific model shown here +demonstates that that the Haystack Telescope would still add substantial value to the EHT array +when considering network configurations that include more telescopes than those used today. +The ngEHT project is developing reference arrays for future quantitative array assessments by +the collaboration. Such evaluations will in future also include the Haystack Telescope. + +8 of 11 +ngEHT +ngEHT + Haystack +ngEHT +ngEHT + Haystack +Figure 3. Outline of the impact of observations with the Haystack Telescope on VLBI arrays (K. Akiyama, +priv. comm.). The upper left panel illustrates an ngEHT reference array used for imaging simulations, as +appropriate for a target at a declination of +10◦. Specifically, current EHT stations are marked by blue +stars, while green stars indicate potential new sites. The Haystack Telescope is marked by a yellow star. +This highlights the important role the Haystack Telescope can play in linking VLBI stations across the +Atlantic Ocean. The upper right panel characterizes how inclusion of the Haystack Telescope into the +adopted ngEHT array improves the synthesized beam. Inner sidelobes are reduced by a factor 1.4. The +bottom panel illustrates that adding the Haystack Telescope would in particular help to populate the +inner area of the uv-plane. +The Haystack Telescope would add substantial sensitivity to the EHT array. Consider +observations at 0 ◦C outside temperature and a better-than-median winter PWV column of +5.3 mm (Section 2.1). In that case the atmospheric transmission is 64% at 230 GHz and 45◦ +elevation. Preliminary performance modeling (J. Kauffmann, priv. comm.) further indicates an +aperture efficiency of 35% and a radome transmission of 73%. Multiplication of all these factors +shows that the telescope’s effective combined efficiency is 16%. This number is small—but the +effective aperture of this telescope is still equivalent to an ideal dish of 0.161/2 · 37 m = 15 m +diameter above Earth’s atmosphere. This equal to the median dish size of current EHT stations6, +and smaller telescope diameters are considered for some future EHT stations. This underlines +the role which the Haystack Telescope can play within the future EHT network. That said, this +calculation purely considers the transmission losses of the system: the impact of ground-pickup +and sky brightness on the system temperature are not included, given insufficient modeling +at this time, while impact of the atmospheric transmission in the calibration to the T∗ +A-scale +is taken into account. For reference, repetition of the analysis for 86 GHz frequency and 45◦ + +12 +Haystack +EHT? +10 +8 +二 +6 +4 +2 +0 +22021 +ngEHT +-0 8—9— +u(109入)-6 +8 +-10 +-12 +10 +8 +6 +4. +2 +0 +Baseline Length12 +EHT +10 +8 +6 +4 +22021 +ngEHTBase +B +-8 +-10 +-12 +12 +10 +8 +6 +4 +2 +0 +Baseline Length2 -4 -6 —8 -10-12 +u (109入)9 of 11 +elevation yields an equivalent diameter of 31 m for an ideal telescope above Earth’s atmosphere. +At this frequency the Haystack Telescope could serve as an “anchor station” that can be used +to improve the overall calibration of the network. In particular, the instrument could serve this +purpose at ∼90 GHz in support of FPT to smaller dishes (see Issaoun et al., this volume). +A key component of ongoing work is to validate the telescope’s abilities via participation +in VLBI runs conducted at ∼86 GHz frequency. The Haystack Telescope has joined three +such experiments since April 2022. This has already resulted in the detection of fringes on +intercontinental baselines. Ongoing analysis will quantitatively characterize the value of the +Haystack Telescope in VLBI arrays. +5. Summary +The reflector of the Haystack Telescope has been upgraded to a dish of 37 m diameter +that has a surface accuracy of ≤100 µm, depending on elevation (Section 1). The instrument +serves as a radar sensor for space situational awareness, with about one-third of the time +available for research by MIT Haystack Observatory. Current work funded by an NSF MSRI-1 +grant conducts astronomical single-dish and VLBI observations at frequencies of ∼20 GHz and +∼90 GHz to study the inclusion of the telescope into the EHT array. Parallel work enabled +via a private donation generally enhances the capabilities of the instrument for research and +education. The telescope is housed in a radome of 150 ft diameter that is designed to support +radar observations at high frequency (Section 2). Current data indicate a median precipitable +water vapor (PWV) column of about 8 mm during winter months (i.e., November 1 to April 30). +These characteristics enable the Haystack Telescope to provide the US-based community with +new and important capabilities for research, education, and technology development in radio +astronomy (Section 3). In particular, the instrument can add new transatlantic baselines to the +EHT network that would drastically improve the image quality with a frequency-dependent +dish sensitivity equivalent to an ideal telescope of 15 m to 31 m above Earth’s atmosphere +(Section 4). Initial VLBI experiments conducted in April 2022 have resulted in fringe detections +on intercontinental baselines. +Author Contributions: Conceptualization, J.K. and G.R.; writing—original draft preparation, J.K. and +G.R.; writing—review and editing, K.A., V.F., C.L., L.M., and T.P.; visualization, K.A.; funding acquisition, +L.M. and V.F. All authors have read and agreed to the published version of the manuscript. +Funding: This work was in part enabled by grants from the National Science Foundation, including +DUE-1503793 and AST-1935980. The development of the Haystack Telescope is further supported by a +private donation. +Data Availability Statement: Not applicable. +Conflicts of Interest: The authors declare no conflict of interest. +Abbreviations +The following abbreviations are used in this manuscript: +EHT +Event Horizon Telescope +LMT +Large Millimeter Telescope +ngEHT +next generation Event Horizon Telescope +SMA +Submillimeter Array +VLBI +Very Long Baseline Interferometry + +10 of 11 +Notes +1 +This article includes numerous references to the “Celebrating 50 Years of Haystack” Special Issue of the Lincoln Laboratory Journal, +which is available at https://www.ll.mit.edu/about/lincoln-laboratory-publications/lincoln-laboratory-journal/lincoln-laboratory- +journal-0 (accessed on 2022 Dec. 15). +2 +The current NEROC members are Boston College, Boston University, Brandeis University, Dartmouth College, Harvard University, +Harvard-Smithsonian Center for Astrophysics, Massachusetts Institute of Technology, Merrimack College, University of Massachusetts +at Amherst, University of Massachusetts at Lowell, University of New Hampshire, and Wellesley College. NEROC’s mission is to +further research, education, and scientific collaboration in the field of radio science. NEROC is headquartered at MIT Haystack +Observatory. Also see https://www.haystack.mit.edu/about/northeast-radio-observatory-corporation-neroc/ (accessed on 2022 +Dec. 15). +3 +https://www.cosmic.ucar.edu/what-we-do/suominet-weather-precipitation-data (accessed on 2022 Dec. 15) +4 +https://lweb.cfa.harvard.edu/~spaine/am/ (accessed on 2022 Dec. 15) +5 +Permitting a one-way drive time of ≤3 h, following https://www.smappen.com/app/ (accessed on 2022 Dec. 15). +6 +The median dish diameter of the EHT array available for future observation cycles is 15 m. This characterizes an array formed from +the phased ALMA dishes, with a collection area equivalent to an antenna of 91 m diameter, the phased NOEMA dishes, equivalent to +an antenna of 52 m, and the phased dishes of the SMA, equivalent to an antenna of 17 m. The array also includes the LMT of 50 m +diameter, the IRAM 30m-telescope, the JCMT of 15 m diameter, the APEX, Kitt Peak, and GLT dishes of 12 m diameter, and the SMT +and SPT dishes of 10 m diameter. +References +1. +Brown, W.M.; Pensa, A.F. History of Haystack. Linc. Lab. J. 2014, 21, 4–7. +2. +Barvainis, R.; Ball, J.A.; Ingalls, R.P.; Salah, J.E. The Haystack observatory lambda 3-mm upgrade. Publ. Astron. Soc. Pac. 1993, +105, 1334. https://doi.org/10.1086/133315. +3. +Whitney, A.R.; Lonsdale, C.J.; Fish, V.L. Insights into the Universe: Astronomy with Haystack’s Radio Telescope. Linc. Lab. J. 2014, +21, 8–27. +4. +Shapiro, I.I.; Ash, M.E.; Ingalls, R.P.; Smith, W.B.; Campbell, D.B.; Dyce, R.B.; Jurgens, R.F.; Pettengill, G.H. Fourth test of general +relativity: New radar result. Phys. Rev. Lett. 1971, 26, 1132–1135. https://doi.org/10.1103/PhysRevLett.26.1132. +5. +Myers, P.C.; Benson, P.J. Dense cores in dark clouds. II - NH3 observations and star formation. +Astrophys. 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='org/licenses/by/ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='0/).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Article The Haystack Telescope as an Astronomical Instrument Jens Kauffmann* , Ganesh Rajagopalan , Kazunori Akiyama , Vincent Fish , Colin Lonsdale , Lynn D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Matthews , and Thushara G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Pillai Haystack Observatory, Massachusetts Institute of Technology, 99 Millstone Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=', Westford, MA 01886, USA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' ganesanr@mit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='edu (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' kakiyama@mit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='edu (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' vfish@mit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='edu (V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' cjl@mit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='edu (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' lmatthew@mit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='edu (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' thushara@mit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='edu (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=') Correspondence: jens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='kauffmann@mit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='edu Abstract: The Haystack Telescope is an antenna with a diameter of 37 m and an elevation-dependent surface accuracy of ≤100 µm that is capable of millimeter-wave observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' The radome-enclosed instrument serves as a radar sensor for space situational awareness, with about one-third of the time available for research by MIT Haystack Observatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Ongoing testing with the K-band (18–26 GHz) and W-band receivers (currently 85–93 GHz) is preparing the inclusion of the telescope into the Event Horizon Telescope (EHT) array and the use as a single-dish research telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Given its geographic location, the addition of the Haystack Telescope to current and future versions of the EHT array would substantially improve the image quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Keywords: Very Long Baseline Interferometry;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' radio astronomy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' millimeter astronomy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' radio telescopes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' high angular resolution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' astronomical instrumentation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Introduction: Astronomy Observations with the Haystack Telescope MIT Haystack Observatory has been a home to a radome-enclosed telescope of 37 m diameter since 1964 [1]1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Figure 1 illustrates the siting of the instrument, while Figure 2 presents an overview of the dish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' The original system was primarily conceived as a space radar and as a platform for telecommunications experiments to support work by MIT Lincoln Laboratory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Ownership was transferred to the Northeast Radio Observatory Corporation2 (NEROC) in 1970, with the goal to enable millimeter-wave observations for the astronomy community in the Northeast US, while still being available as a radar sensor to MIT Lincoln Laboratory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' The site is known as MIT Haystack Observatory since that time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' The telescope has undergone several upgrades since its original dedication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Some of this work focused on improving the surface accuracy of the dish, which was improved from an initial root-mean-square (RMS) error of ∼900 µm to ∼200 µm after 1992 [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' The Haystack Telescope has enabled key scientific discoveries, as summarized by Whitney et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Radar observations delivered key intelligence on the Apollo landing sites, and joint observations with the Westford Telescope—a dish of 18 m diameter located about a mile from the Haystack Telescope—produced the first radar maps of Venus’ surface that cleanly separated radar echoes from the planet’s northern and southern hemispheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Radar observations of Venus and Mercury also delivered stringent tests of General Relativity by constraining the gravitational time delay caused by the presence of the Sun (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=', Shapiro delay, [4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Single-dish spectroscopy observations with the Haystack Telescope were essential in establishing “dense molecular cores” as the key star-forming sites in molecular clouds [5], and they showed how dense cores build up density as they contract out of more diffuse cloud material [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' MIT Haystack Observatory led the way during the inception of Very Long Baseline Interferometry (VLBI), and 8 of the 22 awardees of the 1971 Rumford Prize of the American Academy of Arts and Sciences for the inception of VLBI were working at the observatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' The Haystack Telescope critically supported this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' VLBI observations with the instrument, such as the arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='02713v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='IM] 6 Jan 2023 BY2 of 11 discovery of apparent superluminal motion in quasars [7], shaped our understanding of the universe at high angular resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Aerial view of MIT Haystack Observatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' The Haystack Telescope, a dish of 37 m diam- eter, is located in the large radome dominating the foreground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' (Used with permission, courtesy of Mark Derome).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' 3 of 11 VOLUME 21, NUMBER 1, 2014 � LINCOLN LABORATORY JOURNAL 47 NIKOLAS T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' WAGGENER FIGURE 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' A computer-aided design rendering of the HUSIR antenna shows its key features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' × 65–foot temporary fabric and steel building that would become the assembly area for the new backstructure, permitting work to proceed year-round and providing the controlled environment necessary for the alumi- num welding work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Several smaller site improvements included assembly areas for the quadrapod (the support structure for the subreflector) and the transition structure (the steel backup structure that includes the elevation counterweights, drive gears, and bearings).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' To minimize the impact of adverse weather during the critical lifts of the major subassemblies, the engineering team targeted spring 2010 for the start of the integration period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Because of the scope of the work and the relative inexperience of Lincoln Laboratory with large construction projects, the construction management firm Bond Broth- ers of Everett, Massachusetts, was hired to supervise daily operations at the site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Bond has a background in coordinat- ing large civil engineering projects and a familiarity with the various skilled trades required to successfully com- plete this type of project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Keystone Engineering, formerly of Georgetown, Massachusetts, was contracted to provide much of the labor and equipment for the construction, and assembled a crew of highly skilled workers, primarily from the Local 7 ironworkers union.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Hallamore Corporation of Holbrook, Massachusetts, supplied the 400-foot-tall Mani- towoc 18000 MAX-ER crawler crane, which handled the Massachusetts, the engineering team considered various design options, including actively controlled positioners for the primary surface panels, before settling on the final strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' The details of the final design* of the antenna are included in the appendix to this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' The heart of the upgraded antenna would be a stifness-optimized primary reflector (backstructure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' The backstructure was designed to deform such that it remains a paraboloid under gravity or temperature-induced loading, albeit with a diferent focal length that can be corrected by adjusting the posi- tion of the secondary reflector (subreflector).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' The rest of the new antenna structure was developed to support the optimized backstructure, while utilizing as much of the existing structure as possible (Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' By 2007, most of the design work was completed and fabrication of many subassemblies had begun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Working directly with subcontractors, the Lincoln Laboratory team oversaw the completion of the new antenna hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' In 2008, construction began of the onsite facilities that would be used to perform the final assembly and integra- tion of the various subassemblies, many of which would be too large to ship to the site fully assembled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' The largest of these site projects was the construction of a 140 × 160 Transition structure 120 ft diameter dish Backstructure 200 tons above elevation axis 20 ft radius sector gears 85 ft 340 tons above azimuth axis Hydrostatic azimuth bearing and bull gear 42 ft Concrete pedestal Yoke RF box Subreflector Quadrapod Primary reflector The design of the antenna was originally proposed by Apostle Cardiasmenos of L-3 Communications ESSCO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Overview of the Haystack Telescope, with the radome removed [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' The receiver equipment is installed in the “RF box”, a container that is brought down to ground level during “box-down” periods to enable major engineering activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' (Reprinted with permission, courtesy of MIT Lincoln Laboratory, Lexington, Massachusetts).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' A major upgrade, completed in 2014, improved the surface accuracy to ≤100 µm, depend- ing on elevation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' This work, executed by MIT Lincoln Laboratory under sponsorship by the Defense Advanced Research Projects Agency (DARPA) and the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Air Force, was part of the upgrade delivering the Haystack Ultrawideband Satellite Imaging Radar (HUSIR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' The HUSIR system is designed around a W-band radar covering a substantial bandwidth of 92–100 GHz, and it also includes an X-band radar operating at 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='5–10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='5 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' The outstanding bandwidth of ∆ν = 8 GHz enables the W-band radar to directly resolve structures of c/(2 · ∆ν) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='9 cm size in range [9], with advanced image processing techniques delivering an effective resolution well below this scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' In 2014, HUSIR’s W-band radar delivered the finest spatial resolution of any imaging radar, while the X-band radar constituted the only system for imaging out to geosynchronous orbits [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' The systems have been upgraded since, and HUSIR continues to be an essential contributing sensor for space situational awareness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Today, NEROC has access to about one-third of the time available on the Haystack Tele- scope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' This time can be used to conduct experiments in astronomy and other fields of funda- mental research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' The primary access windows are weekends, and night hours at 23:00–07:00 local time on Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='–Fri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Access to other periods, as for example needed for time-critical experi- ments in astronomy, is coordinated with MIT Lincoln Laboratory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Such work can currently use K-band (18–26 GHz) and W-band receivers (85–93 GHz) dedicated to astronomical observations that are separate from the HUSIR systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' An existing Q-band system covering 36–50 GHz will be brought online in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' 4 of 11 MIT Haystack Observatory currently studies the expected performance of the telescope at ∼230 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' This is done as part of the ngEHT project (as described elsewhere in this special issue;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' also see https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='ngeht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='org, accessed on 2022 Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' 15), which seeks to deliver a “next generation EHT” by adding new stations and other capabilities to the Event Horizon Telescope (EHT;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' see [10] for a recent description of the system).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Inclusion of the Haystack Telescope into the EHT would enhance the imaging capabilities of the array, as described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' The upgraded dish provides exciting opportunities for astronomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Unfortunately, until re- cently it was not possible to make use of the telescope’s capabilities, given the lack of substantial and systematic funding for astronomical experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' This has changed in the past few years, thanks to a private donation and a grant from the National Science Foundation supporting the ngEHT project (AST-1935980).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' The telescope is currently regularly used to conduct experiments in support of system commissioning and initial experiments into astrophysical research and education.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' This includes three VLBI runs at 86 GHz that have delivered fringe detections on intercontinental baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' The telescope, its current and future instrumentation, and the characteristics of the site are described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' The discussion in Section 3 outlines the case for research, education, and technology development on the Haystack Telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' The connection of the telescope to the EHT and ngEHT projects is described in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' The material is summarized in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Telescope, Instrumentation, and Site 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Telescope and Site Figure 2 summarizes the characteristics of the dish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' The reflector of the Haystack Telescope has a diameter of 120 ft, equivalent to 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='57 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' It is formed by 432 panels that each have an RMS surface accuracy of about 28 µm [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' The main reflector itself is rigged to achieve am RMS surface accuracy of 75 µm at an elevation of 25◦, with larger deformations occurring at higher or lower elevations [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' The moving sections have a mass of 340 t, with 200 t of mass moving in elevation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' The dish is capable of slewing at speeds of 5◦ s−1 in azimuth and 2◦ s−1 in elevation, and it achieves accelerations of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='◦5 s−2 and 2◦ s−2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' By requirement, the pointing accuracy is < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='′′6, with a tracking accuracy < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='′′8 [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' The telescope is housed in a radome of 150 ft diameter that was originally designed for use in extreme arctic environments and is capable of withstanding 130 mph winds (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=', 210 km h−1 or 60 m s−1) [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' The radome is skinned with three-ply ESSCOLAM 10 membranes with a hydrophobic coating, which are characterized by a transmission of about 95% at 90 GHz [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' The receivers are housed in a “box” that is installed about 85 ft above ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' It can be brought down to the floor of the telescope building during dedicated “box-down” periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' The box houses radar equipment as well as the astronomy receivers, and it is very tightly packed with systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' As a consequence, major engineering actives can only be performed during a box-down window, during which the interior of the box can be accessed easily from all sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' The number and duration of box-down periods is minimized in support of high-priority radar observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' The observatory’s land is distributed over the Massachusetts towns of Groton, Tyngsbor- ough, and Westford, with Westford being the administrative home of MIT Haystack Observa- tory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' This thickly forested community is about an hour’s drive away from downtown Boston (MA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' The telescope itself is located at 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='◦62 N vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='◦49 W, at an altitude of 130 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Haystack Observatory experiences extended periods of cold and dry weather during the winter, thus providing the weather conditions needed for observations at millimeter wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Historical measurements of the precipitable water vapor (PWV) column are available from the Suominet3 network for atmospheric research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Archived data give a median PWV column of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='3 mm for the period November 1 to April 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Assuming an outside temperature of 0 ◦C, 5 of 11 modeling of the atmosphere with the AM4 radiative transfer code gives a corresponding optical depth of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='12 at ∼86 GHz under such conditions, equivalent to an atmospheric transmission of 84% at 45◦ elevation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' More realistically, observations by systems like the EHT are triggered in better-than-median atmospheric conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' To give an example, the PWV column is below 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='3 mm for 25% of the winter period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Rich additional documentation about the telescope and the site can be found in Brown and Pensa [1], Whitney et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' [3], Waggener [8], Czerwinski and Usoff [9], Usoff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' [11], MacDonald et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' [12], and Eshbaugh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Current Instrumentation The telescope is equipped with receivers operating in the K (18–26 GHz), Q (36–50 GHz), and W bands (70–115 GHz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' The cryogenic frontends operate at around 20 K in independent dewars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' These are arranged roughly on a vertical line that is offset from the central focal point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' MIT Lincoln Laboratory operates on-axis X-band and W-band radars, so that the three astronomy receivers are offset from boresight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' The sub-reflector on a hexapod is remotely controlled to choose between the three K, Q and W-band receivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Current observing projects make use of the K and W bands, and these receivers are therefore currently kept operational by engineering activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' The K-band frontend is shared between MIT Haystack Observatory and MIT Lincoln Laboratory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' One polarization is available for astronomical observations, while the other polarization is used for holography observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Astronomical observations can be conducted anywhere in the frequency range of 18–26 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' The W-band frontend is currently configured as a single-sideband receiver that senses horizontal and vertical polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Data are taken in a sideband of 8 GHz width that is set by an analog bandpass filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' The system is currently set up to observe at frequencies of 85–93 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Modest upgrades to the hardware would allow to access the full frequency range of 70–115 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' The receiver was recently improved via the installation of a new wideband low noise amplifier (LNA) and components for the sideband rejection scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' These investments were made possible by an NSF MSRI-1 grant to the ngEHT project (AST-1935980).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' The backends are located at the ground level of the telescope building.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' A radiofrequency- over-fiber (RFoF) system is used to transport the signals into this room.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' The RFoF infrastructure is currently being upgraded for transport bandwidths of up to 20 GHz for two polarizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' An up-down converter (UDC) is used to condition the signals for the backends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' The single-dish backend currently processes up to 500 MHz in one polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Further investments in hardware and software would enable processing of larger bandwidths and of a second polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' The backend measures continuum signals, and it currently also produces spectra of up to 500 Hz resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' VLBI data are acquired using a ROACH2 digital backend (R2DBE) connected to a Mark 6 VLBI recorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' A Rakon Oven Controlled Crystal Oscillator (OCXO) is used as a frequency standard for ongoing engineering experiments in VLBI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' The acquisition of the RFoF infrastructure, and the ongoing acquisition of a new OCXO, are supported by an NSF MSRI-1 grant to the ngEHT project (AST-1935980).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Current and Future Instrument Development Current work on the Haystack Telescope focuses on evaluation of the newly upgraded system (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=', after installation of the W-band LNA, RFoF system, and VLBI equipment).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' While characterization of the W-band system is the main activity, the K-band receiver is occasionally used to deliver complementary data on telescope performance under less ideal weather con- ditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' This program consists of single-dish observations of calibrators like planets, as well as participation in observations by VLBI networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' In the area of interferometry the goal is to enable future VLBI observations at ≲90 GHz, and to assess the feasibility of VLBI observations at ∼230 GHz in support of the EHT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' More generally, the observations seek to demonstrate the 6 of 11 capability of the Haystack Telescope to deliver exciting astrophysical research as a single-dish telescope and VLBI station.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Current funding from an NSF MSRI-1 grant (AST-1935980) supports the design of a receiver for VLBI observations with the Haystack Telescope at ∼230 GHz in the context of the ngEHT project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' This undertaking might evolve into the design for a multi-band receiver enabling parallel observations at ∼86 GHz and ∼230 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' This depends on future decisions by the EHT and ngEHT projects concerning the need for multi-band observations in support of “frequency phase transfer” (FPT, [14]), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=', the transfer of VLBI calibration information obtained at one frequency to other bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' The long-term plan for the telescope foresees to support single-dish and VLBI observations at K, Q, and W band, as well as at ∼230 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Ongoing investigations will clarify whether some or all of these receivers need to be able to observe in parallel (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=', to support FPT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' The installation of wideband (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=', ≥8 GHz) receivers and backends for single-dish and VLBI observations, as well as of a maser clock as a frequency standard for VLBI experiments, form part of this long-term plan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' The development of the telescope must be supported via dedicated grants from funding agencies, as the Haystack Telescope receives no general-purpose funding to advance the capabilities of the facility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Astrophysical Research, Education, and Technology Development Section 4 explains how the Haystack Telescope can add new, sensitive, and important baselines to the EHT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' In the same way, the Haystack Telescope can complement the Global Millimeterwave VLBI Array (GMVA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' The increased availability of multi-band receivers on radio observatories, as pioneered by the Korean VLBI Network (KVN) [15], raises the exciting prospect for the Haystack Telescope to join an intercontinental network building on FPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' The outstanding scientific capabilities of large single-dish instruments at millimeter wave- lengths are demonstrated by the high impact of current research on the IRAM 30m-telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Recent work with that telescope includes the study of star formation physics in nearby galax- ies [16], and investigations of molecular cloud structure and evolution in the Milky Way that support the aforementioned extragalactic work [17,18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Many of these studies employ the EMIR receiver system that samples 16 GHz of bandwidth per polarization in the 70–115 GHz range [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Installation of receiver and backend systems matching or exceeding this capability would open up new and exciting capabilities for the US-based community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' The Haystack Telescope offers unique, important, and currently missing educational opportunities in the US.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' The instrument is associated with the NEROC community of 13 research-intensive educational institutions in the Northeast US (see footnote 5 on page 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Several of these are closely involved in the EHT, the Large Millimeter Telescope (LMT), and the Submillimeter Array (SMA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' The Haystack Telescope is within easy driving distance from all these institutions, providing outstanding hands-on experiences for junior researchers within NEROC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' More generally, the telescope can serve as a destination for educational astronomical daytrips that are independent from daytime, and only modestly dependent on the weather, by schools, colleges, and universities in six states of the US (all of MA, RI, CT;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' most of NH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' parts of ME and NY)5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Operations of the Haystack Telescope are exclusively funded by grants with specific objectives and deliverables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' MIT Haystack Observatory receives no general long-term funding that could make the instrument broadly accessible by the community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Future operations are expected to be supported via a mix of funding streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' This will include observations for specific user groups that will pay for having their data taken on the Haystack Telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' The current engineering work undertaken on an NSF MSRI-1 grant supporting the ngEHT project constitutes one example for such observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Similarly, NSF AAG grants could fund data collection for specific astrophysical research projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' It is the ambition of MIT Haystack 7 of 11 Observatory to also make the telescope broadly available to the entire US community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' The feasibility of such a program depends on the grants acquired by the observatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Connection to the Event Horizon Telescope: Current Work and Future Roles Ongoing work on the Haystack Telescope is in part funded by an NSF MSRI-1 grant (AST-1935980).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' This award forms part part of the ngEHT project, and its goal is to evaluate the telescope for inclusion into the EHT at ∼230 GHz via quantitative modeling and VLBI test observations at ∼86 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' A private donation to MIT Haystack Observatory enables parallel activities that enhance the overall capabilities of the instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Figure 3 shows that addition of the Haystack Telescope to future versions of the EHT network would produce new and critical baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' This is specifically demonstrated here for a network that includes the telescopes that are now available for future EHT observations, but that also includes a set of additional antennas enhancing the EHT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' This can be seen in Figure 3 (top right), where dishes enhancing the current EHT constellation are marked by green stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' The resulting significant improvements to the uv-coverage of the array can result in reductions of the inner sidelobes of the synthesized beam by a factor 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='4 (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Akiyama, priv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' This is indicated by imaging simulations assuming the reference array summarised in Figure 3 (top left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' The simulation in particular demonstrates that the addition of the Haystack Telescope to the EHT array would substantially improve the sampling of the uv- domain at baselines of ≲4 × 109 λ (Figure 3 bottom), resulting in the aforementioned reduction of sidelobes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Data from the Haystack Telescope can also improve other practical aspects of VLBI observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' For example, the dish can deliver an important connection between dishes in Europe and the Americas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' The telescope can also add substantial sensitivity to VLBI networks, in particular at low frequencies where the telescope efficiencies are higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' All these factors aid in the overall calibration of VLBI networks, delivering advantages beyond the fundamental improvement in uv-coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Importantly, the specific model shown here demonstates that that the Haystack Telescope would still add substantial value to the EHT array when considering network configurations that include more telescopes than those used today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' The ngEHT project is developing reference arrays for future quantitative array assessments by the collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Such evaluations will in future also include the Haystack Telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' 8 of 11 ngEHT ngEHT + Haystack ngEHT ngEHT + Haystack Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Outline of the impact of observations with the Haystack Telescope on VLBI arrays (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Akiyama, priv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' The upper left panel illustrates an ngEHT reference array used for imaging simulations, as appropriate for a target at a declination of +10◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Specifically, current EHT stations are marked by blue stars, while green stars indicate potential new sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' The Haystack Telescope is marked by a yellow star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' This highlights the important role the Haystack Telescope can play in linking VLBI stations across the Atlantic Ocean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' The upper right panel characterizes how inclusion of the Haystack Telescope into the adopted ngEHT array improves the synthesized beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Inner sidelobes are reduced by a factor 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' The bottom panel illustrates that adding the Haystack Telescope would in particular help to populate the inner area of the uv-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' The Haystack Telescope would add substantial sensitivity to the EHT array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Consider observations at 0 ◦C outside temperature and a better-than-median winter PWV column of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='3 mm (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' In that case the atmospheric transmission is 64% at 230 GHz and 45◦ elevation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Preliminary performance modeling (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Kauffmann, priv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=') further indicates an aperture efficiency of 35% and a radome transmission of 73%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Multiplication of all these factors shows that the telescope’s effective combined efficiency is 16%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' This number is small—but the effective aperture of this telescope is still equivalent to an ideal dish of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='161/2 · 37 m = 15 m diameter above Earth’s atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' This equal to the median dish size of current EHT stations6, and smaller telescope diameters are considered for some future EHT stations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' This underlines the role which the Haystack Telescope can play within the future EHT network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' That said, this calculation purely considers the transmission losses of the system: the impact of ground-pickup and sky brightness on the system temperature are not included, given insufficient modeling at this time, while impact of the atmospheric transmission in the calibration to the T∗ A-scale is taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' For reference, repetition of the analysis for 86 GHz frequency and 45◦ 12 Haystack EHT?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' 10 8 二 6 4 2 0 22021 ngEHT 0 8—9— u(109入)-6 8 10 12 10 8 6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' 2 0 Baseline Length12 EHT 10 8 6 4 22021 ngEHTBase B 8 10 12 12 10 8 6 4 2 0 Baseline Length2 -4 -6 —8 -10-12 u (109入)9 of 11 elevation yields an equivalent diameter of 31 m for an ideal telescope above Earth’s atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' At this frequency the Haystack Telescope could serve as an “anchor station” that can be used to improve the overall calibration of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' In particular, the instrument could serve this purpose at ∼90 GHz in support of FPT to smaller dishes (see Issaoun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=', this volume).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' A key component of ongoing work is to validate the telescope’s abilities via participation in VLBI runs conducted at ∼86 GHz frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' The Haystack Telescope has joined three such experiments since April 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' This has already resulted in the detection of fringes on intercontinental baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Ongoing analysis will quantitatively characterize the value of the Haystack Telescope in VLBI arrays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Summary The reflector of the Haystack Telescope has been upgraded to a dish of 37 m diameter that has a surface accuracy of ≤100 µm, depending on elevation (Section 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' The instrument serves as a radar sensor for space situational awareness, with about one-third of the time available for research by MIT Haystack Observatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Current work funded by an NSF MSRI-1 grant conducts astronomical single-dish and VLBI observations at frequencies of ∼20 GHz and ∼90 GHz to study the inclusion of the telescope into the EHT array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Parallel work enabled via a private donation generally enhances the capabilities of the instrument for research and education.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' The telescope is housed in a radome of 150 ft diameter that is designed to support radar observations at high frequency (Section 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Current data indicate a median precipitable water vapor (PWV) column of about 8 mm during winter months (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=', November 1 to April 30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' These characteristics enable the Haystack Telescope to provide the US-based community with new and important capabilities for research, education, and technology development in radio astronomy (Section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' In particular, the instrument can add new transatlantic baselines to the EHT network that would drastically improve the image quality with a frequency-dependent dish sensitivity equivalent to an ideal telescope of 15 m to 31 m above Earth’s atmosphere (Section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Initial VLBI experiments conducted in April 2022 have resulted in fringe detections on intercontinental baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Author Contributions: Conceptualization, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' writing—original draft preparation, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' writing—review and editing, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=', V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=', C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=', L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=', and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' visualization, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' funding acquisition, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' All authors have read and agreed to the published version of the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Funding: This work was in part enabled by grants from the National Science Foundation, including DUE-1503793 and AST-1935980.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' The development of the Haystack Telescope is further supported by a private donation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Data Availability Statement: Not applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Conflicts of Interest: The authors declare no conflict of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Abbreviations The following abbreviations are used in this manuscript: EHT Event Horizon Telescope LMT Large Millimeter Telescope ngEHT next generation Event Horizon Telescope SMA Submillimeter Array VLBI Very Long Baseline Interferometry 10 of 11 Notes 1 This article includes numerous references to the “Celebrating 50 Years of Haystack” Special Issue of the Lincoln Laboratory Journal, which is available at https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='ll.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='mit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='edu/about/lincoln-laboratory-publications/lincoln-laboratory-journal/lincoln-laboratory- journal-0 (accessed on 2022 Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' 15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' 2 The current NEROC members are Boston College, Boston University, Brandeis University, Dartmouth College, Harvard University, Harvard-Smithsonian Center for Astrophysics, Massachusetts Institute of Technology, Merrimack College, University of Massachusetts at Amherst, University of Massachusetts at Lowell, University of New Hampshire, and Wellesley College.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' NEROC’s mission is to further research, education, and scientific collaboration in the field of radio science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' NEROC is headquartered at MIT Haystack Observatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Also see https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='haystack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='mit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='edu/about/northeast-radio-observatory-corporation-neroc/ (accessed on 2022 Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' 15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' 3 https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='cosmic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='ucar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='edu/what-we-do/suominet-weather-precipitation-data (accessed on 2022 Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' 15) 4 https://lweb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='cfa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='harvard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='edu/~spaine/am/ (accessed on 2022 Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' 15) 5 Permitting a one-way drive time of ≤3 h, following https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='smappen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='com/app/ (accessed on 2022 Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' 15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' 6 The median dish diameter of the EHT array available for future observation cycles is 15 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' This characterizes an array formed from the phased ALMA dishes, with a collection area equivalent to an antenna of 91 m diameter, the phased NOEMA dishes, equivalent to an antenna of 52 m, and the phased dishes of the SMA, equivalent to an antenna of 17 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' The array also includes the LMT of 50 m diameter, the IRAM 30m-telescope, the JCMT of 15 m diameter, the APEX, Kitt Peak, and GLT dishes of 12 m diameter, and the SMT and SPT dishes of 10 m diameter.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Clarke, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Liu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Silver, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Optimizing the HUSIR Antenna Surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Linc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Lab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' 2014, 21, 83–105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' MacDonald, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Anderson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Lee, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Gordon, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' McGrew, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' The HUSIR W-Band Transmitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Linc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Lab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' 2014, 21, 106–114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Eshbaugh, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Morrison, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Hoen, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' ;' metadata={'source': 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observations for mm-VLBI: Astrometry UP to 130 GHz with the kvn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' 2015, 150, 202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='1088/0004-6256/150/6/202.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Korean VLBI Network Receiver Optics for Simultaneous Multifrequency Observation: Evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Publ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Pac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' 2013, 125, 539–547.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='1086/671125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Jiménez-Donaire, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Bigiel, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Leroy, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Usero, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Cormier, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Puschnig, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Gallagher, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Kepley, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Bolatto, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' García-Burillo, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' EMPIRE: The IRAM 30 m Dense Gas Survey of Nearby Galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' 2019, 880, 127, https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='3847/1538- 4357/ab2b95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Kauffmann, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Goldsmith, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Melnick, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Tolls, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Guzman, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Menten, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Molecular Line Emission as a Tool for Galaxy Observations (LEGO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' HCN as a tracer of moderate gas densities in molecular clouds and galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' 2017, 605, L4, https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='1051/0004-6361/201731123.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' 11 of 11 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Barnes, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Kauffmann, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Bigiel, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Brinkmann, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Colombo, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Guzmán, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Kim, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Sz˝ucs, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Wakelam, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Aalto, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' et al.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' 2020, 497, 1972–2001, https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='1093/mnras/staa1814.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Carter, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Lazareff, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Maier, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Chenu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Fontana, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Bortolotti, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Boucher, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Navarrini, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Blanchet, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' Greve, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE0T4oBgHgl3EQf2gJy/content/2301.02713v1.pdf'} +page_content=' et al.' metadata={'source': 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community detection in the light of the Laplacian +Renormalization Group +Pablo Villegas,1 Andrea Gabrielli,1, 2, ∗ Anna Poggialini,1, 3 and Tommaso Gili4 +1‘Enrico Fermi’ Research Center (CREF), Via Panisperna 89A, 00184 - Rome, Italy +2Dipartimento di Ingegneria, Universit`a degli Studi ‘Roma Tre’, Via Vito Volterra 62, 00146 - Rome, Italy +3Dipartimento di Fisica Universit`a “Sapienza”, P.le A. Moro, 2, I-00185 Rome, Italy. +4Networks Unit, IMT Scuola Alti Studi Lucca, Piazza San Francesco 15, 55100- Lucca, Italy. +Heterogeneous and complex networks represent the intertwined interactions between real-world +elements or agents. A fundamental problem of complex network theory involves finding inherent +partitions, clusters, or communities. By taking advantage of the recent Laplacian Renormalization +Group approach, we scrutinize information diffusion pathways throughout networks to shed fur- +ther light on this issue. Based on inter-node communicability, our definition provides a unifying +framework for multiple partitioning measures. Moreover, we account for the elusive behavior of +components bridging different communities within the same network. +Community structures emerge as a ubiquitous feature +of real-world networks. +Detection of relevant network +substructures is of utmost importance to correctly under- +stand their often hidden multiple mesoscopic scales and +particular functionalities [1, 2]. The level of compartmen- +talization of a system, the modularity, has been proposed +as a generic requirement, or an optimal solution, for a +system in a changing environment to be evolvable [3]. +High modularity, therefore, pervades biology on differ- +ent scales from proteins and genes [4] to cells [5, 6], the +human interactome [7] or ecosystems [8]. Modules form +the basis of transcriptional regulatory networks [9], have +been found at every scale in metabolic networks [10], and +have been hypothesized to confer a high efficiency of in- +formation transfer between nodes at low connection cost +in brain networks [11]. +In a seminal paper, Girvan and Newmann [2] proposed +a celebrated algorithm by progressively removing edges +from the original network. The inherent nature of the al- +gorithm led them to introduce a quantitatively ’stopping +criterion’: the modularity, Q, which quickly became the +fundamental ingredient of many clustering methods [12]. +Modularity maximization is one of the most widely used +algorithms to detect communities in a network. However, +an exhaustive modularity maximization implies an NP- +complete problem, whose complexity increases exponen- +tially with the total number of nodes, making it almost +impossible (we refer to [13, 14] for an extensive discussion +on the issue). +More sophisticated procedures rely on the spectral +properties of the graph Laplacian, which encode the net- +work diffusion modes (i.e., allowing to explore the net- +work momentum space [15]). +Indeed, this is the basis +for one of the best-known methods of graph partitioning: +spectral partitioning [16, 17]. A related elegant proposal +combines spectral methods with clustering techniques, +projecting the network nodes into an eigenvector space of +variable (tunable) dimensionality [18]. Note that all the +∗ Corresponding author: andrea.gabrielli@uniroma3.it +algorithms taking advantage of spectral methods, which +operate in Fourier space, need an N-dimensional space +due to a lack of well-defined spatial embedding of the net- +work. Another set of methods is based on the exploration +properties of random walks on top of the network. They +take advantage that, due to the intrinsic network hetero- +geneities, a random walker will remain trapped within +different communities for a relatively long time before +leaving it (see, e.g., [19] for an excellent review on the +field). In particular, the probability of leaving a vertex is +distributed among the outgoing edges according to their +weight [20]. Hence, the concept of Markov stability has +been introduced [20], which quantifies the tendency for a +random walker to stay inside a community for a long time +[19, 21], opening the door to additional quality measures +of a graph partition based on dynamical concepts, there- +fore ranking partitions and establishing their relevance +over time scales. +In either case, detecting mesoscale structures (i.e., +communities) in networks has become a fundamental +problem –and still a matter of debate [22]– in network +science, where many methods have been proposed and +classified according to their performance [23]. +Yet the +problem lacks of an holistic perspective, providing an +unified interpretation, and capable of setting the limi- +tations of the different methods. This is due to the fact +that a proper definition of community depends on the +process on the network for which we want to find the +most suitable partition [24], complicating even further +the issue. Besides, the proper identification of local com- +munities represents a fundamental open problem when +it is tackled through usual community detection meth- +ods [22]. In a nutshell, identifying basic structural and +dynamic communities in a rigorous way raises a general +problem of definition that still needs to be resolved. +Tackling the root of the problem requires following the +evolution of inter-node communication throughout mul- +tiple scales. +In classical statistical physics, the renor- +malization group (RG) –one of the most influential con- +cepts in theoretical physics of the second half of the 20th- +century [25–28]– allows for connecting extremely varied +arXiv:2301.04514v1 [cond-mat.stat-mech] 11 Jan 2023 + +2 +spatiotemporal scales of physical systems embedded in +regular Euclidean spaces. The applications of this frame- +work in its network counterpart, the Laplacian Renor- +malization Group (LRG), is the natural choice to analyze +complex network mesoscales and structures [15]. As LRG +strictly corresponds to the free field theory’s renormal- +ization, it allows for scrutinizing all the possible diffusion +paths at arbitrary time across an arbitrary network. +Here, in the light of the LRG, we develop a new com- +munity detection framework that unravels the complete +hierarchical structures of the network at different reso- +lution levels. +In particular, we reconstruct the hierar- +chical network tree based on diffusion distances among +the nodes by taking advantage of the LRG. Therefore, +we reduce the complexity of quantifying the ’quality’ of +network subdivisions, proposing a natural solution based +on slowing down information diffusion throughout the +network. Finally, we illustrate the emergence of dynami- +cal communities when a set of random walks explore the +networks structure and discuss the bridges with previous +proposals, unifying modularity maximization, Markov +stability and spectral methods in a common framework. +COMMUNITY DETECTION AND LAPLACIAN +RENORMALIZATION GROUP +For an undirected network with N nodes and M edges, +the level of modularity for a given partition (see [13] and +references therein) can be defined as, +Q = +1 +2M +N +� +i,j=1 +(Aij − γPij) δ (gi, gj) , +(1) +where ˆA stands for the network adjacency matrix, γ is a +resolution parameter, and gi is the community to which +node i belongs, being δ(gi, gj) = 1 if gi = gj and 0 other- +wise. Pij is a null-model matrix that critically constrains +the modularity maximization process [29]. Modularity is +strongly related to the concept of Markov stability [19], +where now the standard null model is given by the prob- +ability that a discrete-time random walker visits node i +at t = 0 and node j at t = ∞. One, therefore, obtains +R(t) = +N +� +i,j=1 +�� +p∗ +i e−τ ˆLRW � +ij − p∗ +i p∗ +j +� +δ (gi, gj) , +(2) +where ˆLRW = D−1 ˆL, is the ‘random-walk normalized +Laplacian’, and ˆL = D − A is the Laplacian matrix of +the network, with τ a resolution parameter, which allows +to unravel multiscale structures in the network. +Here, +the null probability p∗ +i p∗ +j corresponds to the the expected +transfer probabilities at stationarity for this Markov pro- +cess [30]. Note that, as previously discussed by the net- +work community [19, 30], Eq.(2) reduces to Eq.(1) under +the very particular assumption of e−τ ˆLRW ≃ I − τ ˆLRW , +a linear approximation which can avoid computational +difficulties to calculate the full matrix exponential. How- +ever, Eq.(2) provides a simple interpretation of the res- +olution parameter (γ = 1/t), in terms of Markov times +[19, 20]. +We point out that the very particular structure of the +network relies, however, on the Laplacian matrix, which +encodes the intrinsic topological properties and manages +renormalization group procedures on top of heteroge- +neous structures [15]. In particular, the Laplacian drives +the temporal evolution of the diffusive paths of a net- +work, s(τ) = e−τ ˆLs(0), giving place to define the density +matrix as [31, 32], +ˆρ(τ) = +e−τ ˆL +Tr(e−τ ˆL) +, +(3) +which opened the door to a thorough analysis of +network structures, +where the entropy [31] (S += +− +1 +log N ˆρ(τ) log ˆρ(τ)) and specific heat [33] (C = − +dS +d log τ ) +revealed characteristic network scales (we refer to [15, 31, +33] for further details). It also makes it possible to scruti- +nize the scale-invariant properties of a network [15], link- +ing the specific heat with the effective spectral dimension +of the network (see also Appendix A). +Figure 1. +Sketch of the network analysis. +The node’s dis- +tances reflects the underlying hidden complex topology, cap- +turing the communicability strength, and giving a natural +merging of nodes at specific times, τ. Finally, the application +of the LRG gives a hierarchical reconstruction of the network. +In particular, the density matrix is defined in terms +of the network propagator, ˆK = e−τ ˆL, which accounts +for the sum of diffusion trajectories along all possible +paths connecting nodes i and j up to a temporal scale +τ [19, 34]. Note that this evolutionary operator essen- +tially contains the same information as Eq. 2 whether +considering ˆL or ˆLRW (for now, we analyze the case of +ˆL, considering later the implications of ˆLRW ). In any +case, ˆρ(τ) gives a robust measure of information com- +municability or diffusion strength between pair of nodes. +This allows us to define in a natural way a Laplacian dis- +tance, Dij [ρ(τ)] = +1 +ρij(τ)−δijρij(τ), as the inverse of the +strength of the information probability paths between +nodes at each particular time τ. Note that such a def- + +3 +10 +410 +2 100 +/ +max +32 +25 +26 +24 +28 +10 +29 +27 +30 +9 +31 +23 +21 +16 +19 +33 +15 +34 +5 +11 +7 +6 +17 +1 +12 +4 +13 +14 +3 +8 +20 +22 +2 +18 +Node index +(a) +10 +2 +100 +0 +1 +1-S +(b) +1 3 5 7 9 +Gap number +0.0 +0.5 +RTR +(c) +0 +1 +log NC/Cmax +Figure 2. LRG communities. (a) Normalized dendrogram +for the weighted Zachary’s karate club with τ = τ ′ = 1/λmax. +Red dashed line reflects the optimal division as stated by the +RTR. Different communities are shaded in different colors. (b) +Entropy parameter (dashed lines, (1 − S)), and specific heat +(solid lines, C), versus the temporal resolution parameter of +the network, τ. Black dashed line indicates τ ′ = 1/λmax. (c) +Retention Time Rate (RTR) versus gap number for τ = τ ′. +Note the high RTR values giving place to the usual division in +two and four communities of the Zachary’s karate club. Inset: +Division into communities of the network. +inition shows an intrinsically ultrametric behavior, sat- +isfying the condition Dij ≤ max (Diz, Dzj), beyond the +standard requests Dij ≥ 0; Dij = 0 ⇔ i = j; Dij = Dji +at each time. +Once we have introduced a way to measure distances +between nodes, a method for grouping nodes into com- +munities is required. We emphasize that, as complex net- +works lack an explicit spatial embedding, ˆD allows us to +reduce the problem to the traditional clustering problems +in Euclidean space [35]. In particular, we choose the av- +erage group clustering algorithm, a compromise between +the sensitivity of complete clustering to outliers and the +tendency of single clustering to inhibit compact clusters +[35, 36]. The output of this algorithm will thus be a hier- +archical tree or dendrogram for each resolution scale, τ. +However, note that D [ρ(τ)] can, in principle, give place +to different dendrograms modifying the specific length of +branches as the diffusion time increases, thus potentially +inducing diverse structural partitions (this situation is +not unique, the same applies, e.g., in Markov stability +[20]). The natural question is how to reconcile the identi- +fication of optimal divisions with preserving the multiple +timescales stemming from the intrinsic heterogeneity of +the network. +A solution for this conundrum comes from the direct +application of the LRG [15]. +As settled by the LRG, +the Laplacian of the network –and its relative eigenvec- +tors |λ⟩– contains the different network diffusive modes +(see Appendix B for further clarifications). In particu- +lar, the LRG scheme takes advantage of the fact that +the Laplacian operator is a sort of telescopic ”scanner” +of the coarse-graining scales, integrating out the differ- +ent network eigenmodes as τ increases. +This suggests +the existence of a resolution window where the different +partitions must be relevant. The smallest possible scale +is given by λmax, which prevents to integrate out any +basal network scales, thus avoiding losing network infor- +mation. This choice of τ ′ = 1/λmax gives us the highest +possible resolution of the network structure from above. +Note that it is possible to consider greater times inte- +grating out network eigenmodes to monitor the stability +of the mesoscopic partitions across diffusion scales, up to +the specific heat peak at long times, where the Fiedler +eigenvalue is integrated out (i.e., defocusing microscopic +details, thus shedding light on the origin of a resolution +limit [37], see below). Finally, for the sake of clarity, we +consider the relative length of the dendrogram D/Dmax. +This results in an optimal normalized dendrogram re- +flecting the intrinsic network mesoscopic structures. Fig. +2(a) shows the application of this method for the partic- +ular case of the weighted version of the Zachary’s karate +club [2, 38], which serves us as a benchmark showing the +usual division in two communities. Fig. 2(b) shows the +entropic phase transition for Zachary’s karate club. +Nevertheless, even if we can scrutinize the whole net- +work hierarchic structure as a function of the distances +between nodes, we need to quantify the ’quality’ of net- +work subdivisions. In other words, the clustering algo- +rithm gives no hint about the ’goodness’ of a specific +partition. We propose the Retention Time Rate (RTR) +as the ratio of the dendrogram gaps and the expected +total diffusion time of the information throughout the +network (i.e., dendrogram length, in logarithmic scale). +Hence, the higher the modularity, the higher the proba- +bility that information slows down, therefore maintaining +the flux trapped in a particular community. The usual +definition of modularity [18, 39], Q =� +i +(eii − ai)2, is +an alternative way of providing a ’quality’ function of +the network subdivision facing the density of links in- +side communities (eii) compared to connections between +communities (ai, weighted with 1/2). However, we stress +that Q only measures the fraction of edges that fall be- +tween communities minus the expected value of the same +quantity in a random graph with the same community di- +vision [18], neglecting other effects (as, e.g., the problem +of loops [40] or the separation of network timescales [41]) +that may have a profound impact on the structural divi- +sion into different communities. Fig. 2(c) shows the best +partitions as a function of the dendrogram gap number +for a particular time τ as indicated by the RTR (see Fig. +3(e) for analysis over multiple timescales). +One important application of the method involves the + +4 +Figure 3. Dynamical communities (LRW ). (a) Normalized dendrogram for Zachary’s karate club with τ = τ ′ = 1/µmax. +Red dashed line reflects the optimal division as stated by the RTR. Different communities are shaded in different colors. (b) +Entropy parameter (dashed lines, (1 − S)), and specific heat (solid lines, C), versus the temporal resolution parameter of the +network, τ. Black dashed line marks τ ′ = 1/µmax. (c) Retention Time Rate (RTR) versus gap number for τ = τ ′. Inset: +Division into communities of the network. (d) Spectrum of eigenvalues for the LRW for a Barabasi-Albert network with m = 10, +a Random Regular graph with κ = 18, an with ⟨κ⟩ = 18 and a Watts-Strogatz network with ⟨κ⟩ = 18 and rewiring probability +p = 0.75. Black dashed line stands for the semicircular law. (e) RTR versus τ/τ ∗ (with τ ∗ corresponding to the maximum in +C). Different colors stand for different time gaps in the dendrograms for L (circles) and LRW (triangles). Distance matrix for: +(f) a hierarchic modular network with core-periphery structure, and (g) a Dorogovstev-Mendes graph. +characterization of heterogeneous hierarchical nested net- +works with multiscale communities. In this light, we an- +alyze both hierarchical-modular networks with a prefer- +ential attachment rule (HM-CP, that produces a core- +periphery structure involving central connector hubs hav- +ing local and global rich clubs [42]) and hierarchical +lattices through the Dorogovtsev-Goltsev-Mendes graph +(DGM), a pseudo-fractal with high clustering properties +[43]. On the one hand, as better illustrated in Appendix +C, the LRG evidences cleanly and concisely the full in- +trinsic modular structure of nested hierarchical networks +(HM-CP), even when the inter-module connections and +communication paths tend to be centralized through the +hubs, as observed in real neural and brain networks [44]. +On the other hand, we highlight that, when applied to +highly hierarchical systems (e.g., DGM networks), usual +community detection methods present severe issues in +unraveling either communities and the complete hierar- +chical structure of the network (see Appendix C for fur- +ther details and illustrative examples). +Therefore, we +emphasize the capability of the LRG to unravel meso- +scopic communities in these types of intricate networks. +In particular, Fig. 3(f) and (g) illustrate the emergent +nested structure already apparent in the distance matrix, +ˆD, for HM-CP and DGM networks. +Finally, we have +successfully tested the Lancichinetti-Fortunato-Radicchi +benchmark [45], a network generator with a priori known +communities, which serves as a stringent criterion, to +compare different community detection methods (see Ap- +pendix C for further details). +THE LAPLACIAN RANDOM-WALK +A different approach can be considered by simply in- +troducing the ’random-walk normalized Laplacian’ of the +network, ˆLRW = D−1 ˆL, into Eq.(3). +This encodes a +traveling dynamics at a unit rate, moving to a particu- +lar neighbor with equal probability for each choice: i.e., +the transition matrix for the random walk dynamics on +top of a graph [46]. It has to be reminded that ˆLRW is +not a stochastic matrix since it does not meet the needed +constraints but, anyway, it represents an evolutionary op- +erator in an L2 space, with all the eigenvalues of ˆLRW +satisfying µi ∈ [0, 2] (and also featuring an identical spec- +trum to the normalized symmetric Laplacian, Lsym [47]). +ˆLRW is also linked with the spectral dimension of a graph +[48], dS, which characterizes the return-time distribution +of the random walker: a walker starting at t = 0 from any +node of the network has probability P0(t) of returning to +the initial node, with P0(t) = +� +dµp(µ)e−µt ∝ t−dS/2 +[49]. This is obtained by the usual Laplace transform, +which automatically links ρ(λ) with the probability den- +sity function of return time distributions in the graph. To +better scrutinize the Laplacian RW spectrum (and conse- +quently, the average trajectories of the walkers [50]), we +propose the following transformation µ′ = 1 +2 + +µ−1 +2(µmax−1), +to ensure the spectrum is upper bounded by one. There- +fore, for high values of dS we expect to recover the +(universal) mean-field theoretical expectation before x(t) +first returns at time T to its initial value x(0); i.e., + +10 +9 +8 +7 +6 +516 +14 +12 +10 +8 +6 +45 +⟨x(s)⟩ = +8 +π +� +s(1 − s), with s = t/T. Indeed, as illus- +trated in Fig. +3(d) for different explicit networks, the +spectrum of ˆLRW converge to this universal shape (the +Wigner semicircle law [51]) over a specific upper criti- +cal dimension (indeed, it must be dS +u = 4, as expected +from the Gaussian model [28, 52, 53]). This is because +the random walk Laplacian spectrum of the network is +closely related to the first-return times and the distribu- +tion of all returns induced by the network [54, 55] (but +not the fluid Laplacian, ˆL, where the eigenvectors form +the Fourier basis of the network where the random walk +dynamics is projected through ˆLRW ). +Figure 3(a) shows the ’Gaussian’ dendrogram for +Zachary’s karate club by considering ˆLRW (it is impor- +tant to stress that Node 3 is placed in a different commu- +nity of those of Fig. 2(c), but we will discuss this aspect +in detail later). Fig. 3(b) shows the entropy parameter +and the specific heat for Zachary’s karate club now using +ˆLRW to characterize the network null ’dynamical’ struc- +ture. Note that the specific heat, C, presents a slightly +different shape to those of Fig. 2(b) as a result of the +dynamical modes of the network, induced by ˆLRW . Fig. +3(c) shows the best partitions for a particular time τ ′ as +indicated by the RTR (see Fig. 3(e) for an analysis of +the two first optimal gaps over multiple timescales). +10 +3 +10 +1 +101 +/ +* +0.0 +0.5 +1.0 +1-S +(a) +L +LRW +10 +2 10 +1 +100 +101 +/ +* +0.0 +0.5 +1.0 +1-S +(b) +0 +1 +2 +log NC +0 +2 +4 +6 +log NC +Figure 4. +Real networks. +Evolution of the spectral en- +tropy using the Laplacian and the Laplacian RW, versus the +normalized time, τ/τ ∗ (where, for the sake of comparison, +τ ∗ corresponds here to the absolute maximum of the specific +heat, for: (a) Mus Musculus PPI network and (b) Drosophila +Melanogaster network. Note how eigenmodes are integrated +out differently in both cases. +We now proceed to apply the methodology of the LRG +to analyze this particular dynamics on top of different +real networks. As we have just discussed, we may ex- +pect that both Laplacians could lead to different cases +of study in the ’fluid case’, ˆρ(τ), and the ’random-walk’ +case, ˆρRW (τ) in certain specific situations. Fig. 4(a) and +(b) shows the comparison of the entropy order parameter +and the specific heat of both cases, whether for the Mus +Musculus PPI network [56] and the color vision circuit in +the medulla of Drosophila Melanogaster [57], showing at +a glance how information (i.e., network modes) is inte- +grated in a different way. In particular, these effects will +strongly depend on the effective (spectral) dimension of +the network, whether at a global or local scale [58]. As +an illustrative example of this effect, we consider the spe- +cific analysis of the clique-star network: a random walker +will be effectively trapped at the star hub leading to two +well-defined dynamical communities. Instead, the fluid +Laplacian will additionally capture the broadcast of in- +formation linked to both hubs, giving place to more ac- +curate information on the topological properties of the +network (see further details in Appendix D). +Finally, we illustrate in Figure 5 the complex hierar- +chical structure, using ˆL, of the E. Coli and M. Musculus +PPI networks [56]. +We identify the functional role of +these communities as indicated in the UniProt Knowl- +edgebase [59], a comprehensive, high-quality, and freely +accessible set of protein sequences annotated with func- +tional information with more than 190 million sequences. +In particular, we use the keywords associated with each +protein to facilitate the characterization of different core +functionalities of the observed modules. In fact, the PPI +network of E.Coli presents a highly nested tree-like struc- +ture with other structural submodules dedicated to dif- +ferent biological functions, as shown in Figure 5(a). For +the sake of clarity, we explore the division into five mod- +ules exemplified in the dendrogram of Figure 5(a). In this +case, we want to stress for the sake of illustration the spe- +cific nature of two main communities: the one dedicated +to RNA-binding (green nodes, 83% of the corresponding +proteins engaged in this process belong to this commu- +nity, constituting the 37.5% of it) and the one dedicated +to cell division and cell cycle (violet nodes, containing +the 62% of the proteins involved in this goal, and con- +forming the 44% of the community). We instead high- +light the interconnected and highly intricate network of +M.Musculus, where, despite this, we can discern three +principal modules, noticing that the smallest one (blue +nodes in Figure 5(b)) is strongly linked to the signal pro- +cesses and immune response of the network. In any case, +the in-depth study of different biological networks is be- +yond the scope of this manuscript. We will explore better +and more refined analyses of these nested structures else- +where. +LOCAL MODULARITY IN HETEROGENEOUS +NETWORKS +Sometimes one might want to know the communities +in only a small region, which does not ensure a global +‘good’ division of the network in terms of an optimization +function [22]. For example, in many real-world networks, +clusters are linked mainly locally among each other, gen- +erating local clusters that are overshadowed by global +network dependency [60]: the coexistence of local mod- +ularity with global nestedness is key to ensure, e.g., evo- +lutionary stability in host-pathogen infection networks +[61], while the human temporal lobe is organized into +spatially compact functional modules at the micro-scale +[62]. Quantifying the local stability across different net- +work scales of these local structures is particularly chal- +lenging when it is only considered an optimization func- + +6 +E.Coli +M.Musculus +#Signal +#Immunity +#RNA-Binding +#Sugar transport +#Transport +#Electron transport +#Signal +#Transport +#Cell division +#Cell cycle +#Nucleotide-binding +#ATP-Binding +Figure 5. Protein-protein interaction networks. +(a) Modular structure of the E.Coli PPI network. The dendrogram +illustrates the hierarchical clustering of the different nodes of the network. Different colors stand for different communities, +along with their corresponding main biological functions. +(b) Modular structure of the M.Musculus PPI network. +The +dendrogram illustrates the hierarchical clustering of the different nodes of the network. Different colors stands for different +modules, as stated by the network dendrogram. Note that, instead of choosing the ’best’ network partition in terms of diffusive +distances, we are now interested here into the complex nested structure and modules of both organisms which gives rise to a +rich structure with multiple overlapping communities. The different functionalities associated to each module are written in the +same color of the nodes it contains. We stress the characterization of a small local module responsible of electron transport as +direct application of the LRT (see black dashed circle). Parameters: We use ˆL with τ = 10, thus integrating out the microscopic +scales of both networks, but without going so far to integrate out the Fiedler eigenvector, therefore ensuring a proper analysis +of the network mesoscopic modules. +tion (e.g., modularity to find the internal structure of the +network). +Figure 6. +Local modularity. +(a) Local modularity in +the normalized dendrogram for Zachary’s karate club with +τ = τ ′ = 1/λmax. Values of the LRTR larger than 0.3 have +been highlighted. (b) Local Retention Time Rate versus τ/τ ∗ +(with τ ∗ corresponding to the maximum in C) for the local +community highlighted in orange color for both the Laplacian +(red line) and the Laplacian RW (green line). +As a natural extension of the method we have de- +scribed before, we propose to measure the Local Reten- +tion Time (LRT) as the length of every particular branch +of groups of three or more network nodes. +We stress +that the branch length, at time τ, represents the com- +munication distance (i.e., sum-over-paths) between two +groups of nodes. +Hence, higher values of the LRT in- +dicates a stable module over a significant amount of in- +trinsic timescales of the network, either from a structural +or a dynamical point of view. Figure 6(a) shows a par- +ticular example for Zachary’s karate club using the fluid +Laplacian, ˆL. +As we illustrate in Fig. +6(b), this par- +ticular module exhibits a large LRT for all the expected +range of significative timescales both for ˆL and ˆLRW . +Another particular application comes from the detailed +study of the dendrogram of the M. Musculus PPI net- +work, which shows a rich structure of heterogeneously +distributed modules. +Finally, the direct application of +the LRT allows detecting a robust, independent module +(see black dashed lines in Fig. 5) detecting a group of +proteins dedicated to electron transport (in particular, +all these proteins are involved in the respiratory chain), +necessary, e.g., for both photosynthesis and aerobic res- +piration. +TEMPORAL DENDROGRAMS AND +METASTABLE NODES +As previously discussed, one particular implication of +the LRG is the ’dynamic’ perspective it gives from the +contribution of some specific nodes to the functional ca- + +7 +pabilities of a heterogeneous structure, i.e., the contri- +bution of diffusive modes at different timescales. We fo- +cus here on nodes that act as ’bridge’ nodes and com- +municate highly connected modules of the network, thus +sharing information (i.e., sending signals) or contributing +to different functional communities at different spatio- +temporal resolution scales of the network structure. In +other words, these nodes can dynamically modify their +diffusive distance to other mesoscopic structures in the +system, which explains why these nodes are extremely +challenging when attempting to classify them through +optimization function techniques that provide a static +picture of the network. +1 +2 +3 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +13 +14 +15 +16 +17 +18 +19 +20 +21 +22 +23 +24 +25 +26 +27 +28 +29 +30 +31 +32 +33 +34 +C1 +C2 +CF1 +CF2 +Community evolution +Figure 7. Sankey diagram. Evolution of the optimal com- +munity division (in terms of RTR) for the Zachary’s Karate +club at three different characteristic times. We emphasize the +ability of ’bridge nodes’ to dinamically change their functional +community at different timescales. +This effect can be illustrated by analyzing Node 3 of +Zachary’s Karate club, whose community differs depend- +ing on the Laplacian we have previously considered (as +discussed above). To shed light on this crucial fact, we +now focus on analyzing this specific node by using ˆL. In +particular, we show in Fig. 7 the best community as- +signment for this particular network at different times, +revealing how this ’bridge’ node dynamically changes the +community it belongs to. This gives it a central role in +managing communication and/or control processes be- +tween independent modules. Even if we represent a sym- +bolic example using this specific network, the relevance +of these nodes in real networks (e.g., the human connec- +tome) for synchronizability and information integration +is expected to be crucial. This metastable way of pop- +ulating modules by specific nodes recalls the so-called +“modular flexibility“ [63]. Modular flexibility represents +how frequently nodes change the modules they belong to +across time. It means that nodes more likely to be con- +nected to multiple modules at different time points are +more flexible. +Modular flexibility has been associated +with network adaptability, giving flexible nodes the role +of pivot of the dynamics processes running on the net- +work. Although the concept of modular flexibility pro- +posed by Khambati et al. is based on the description of +the network’s evolution dynamics across time, it can also +be used to describe the modular properties of a network +according to the dynamic regime of diffusive processes of +information used to investigate it. +DISCUSSION AND CONCLUSION +The lack of a valid source of geometric length-scale +transformations constitutes a fundamental issue in het- +erogeneous systems [64], which has critically constrained, +e.g., renormalization group approaches in heterogeneous +structures [15] and, as a natural consequence, the com- +munity detection problem in complex networks. We pro- +pose a simple but efficient algorithm based on the Lapla- +cian Renormalization Group –taking advantage of the +fact that the Laplacian operator is a sort of telescopic +”scanner” of the mesoscopic scales of a network– which +gives a diffusion-distance-based and free-metric approach +to the community detection problem. In particular, we +also provide a natural and parsimonious interpretation of +the so-called ’resolution limit’ [22, 51], conditioned by the +Taylor expansion of the network propagator, that limits +the maximum number of paths of length n from vertex +i to vertex j to be taken into consideration. This is no +more than the applied zoom once a particular value of τ +has been selected or, in other words, the defocusing: i.e., +the number of diffusive eigenmodes that have been inte- +grated out, with the resultant loss of information about +the microscopic network scales. +We thus highlight important diverse implications of +the Laplacian Renormalization Group. On the one hand, +the LRG sets the formal equivalence between real-space +methods that can be derived from the ˆρ(τ)-density ma- +trix (i.e., Louvain methods for community detection or +Markov stability) and spectral clustering techniques [16– +18], which address the issue in k − space and analyze +diffusion modes of the network. These approaches are +two sides of the same coin. On the other hand, it allows +proposing a unified interpretation of community detec- +tion in terms of structural (ˆL) and dynamical aspects +(ˆLRW , i.e., intrinsic trapping times) providing an overall +vision over different methodologies that have been hith- +erto considered separately. We pinpoint that, given the +connection of the Gaussian model with the random walk +on any graph [52], the use of the ˆLRW constitutes the first +approximation or dynamical reference model in the same +way the Gaussian model provides the starting point for +perturbative RG analyses [28]. As a natural consequence, +we stress the importance of using ˆL for community de- +tection purposes on top of functional matrices because, +otherwise, the underlying dynamics based on ’Markovian’ +random walks is always an implicit assumption and lacks +systematic interpretation. +It is essential to discuss in detail the link between the +Gaussian model and the random walk on any graph (even + +8 +if we refer to [46, 52] for an extended discussion on the +issue). In particular, du = 4 characterizes a critical value +for the intersection properties of two independent random +walks, which is closely connected to the non-triviality +of the φ4 +d field theory [65] (note that over this dimen- +sion, two random walkers are unlikely to intersect, and +the Gaussian results obtained neglecting the intersections +are asymptotically valid [28]). Hence, the spectra of the +Laplacian RW is expected to show some universal behav- +ior [66] dictated by the lowest vertex degree in a network +[54] (i.e., its spectral dimension), which, anyhow, will +control the emergent functional communities and dynam- +ical aspects of the system. We pinpoint the relevance of +this measure intending to discriminate the relevance of +underlying heterogeneity on the system dynamics (e.g., +scale-free networks are expected to be dynamically irrele- +vant for large enough values of m in the Barabasi-Albert +model [54]). +A further crucial issue to consider when we think about +heterogeneous structures is that the effective community +of ’bridge’ nodes can be diverse for different timescales. +Therefore, it would be helpful to understand the proper- +ties these ’bridge’ nodes have in common. This is par- +ticularly relevant for controllability in complex networks: +for example, they are expected to manage switching be- +havior on the large repertoire of attractors, with different +degrees of coherence and stability, present in hierarchic +modular networks [67, 68] or in metastable dynamics in +human brain networks, where hubs have been elucidated +to control the resulting wave patterns [69]. We hypoth- +esize that this issue is, e.g., closely related to the ob- +served disrupted hub organization in the topological dis- +turbances associated with schizophrenia [70], where the +waste of hubs [71] can lead to dysfunctions in controlling +and integrating neuronal signals [72]. Finally, the appli- +cation of the LRG allows us to develop a local detection +method that overcomes resolution limits problems but +gives a global overview of the importance of local com- +munities in the entire network [22]. +Altogether, we propose here a new vision of net- +work modularity based on the Laplacian Renormaliza- +tion Group (LRG) [15], which is the natural extension to +heterogeneous networks of the usual RG approach in sta- +tistical physics and statistical field theory. This permits +to reveal the ’building’ blocks of the network at different +scales without resolution limit constraints. In particular, +we scrutinize the direct links with previous definitions of +network modularity, therefore presenting an overall pic- +ture of different frameworks and revealing their limita- +tions. +Our LRG scheme opens a new route to extend the +study of emergent dynamical communities [24] beyond +the analysis of different ’dynamical’ Laplacian evolution +operators containing the dynamical aspects of diverse +processes running on top of heterogeneous structures [73], +providing much insight into their understanding and fos- +tering future renormalization group analyses. +ACKNOWLEDGMENTS +P.V. acknowledges the Spanish Ministry and Agen- +cia Estatal de investigaci´on (AEI) through Project +I+D+i +Ref. +PID2020-113681GB-I00, +financed +by +MICIN/AEI/10.13039/501100011033 and FEDER ‘A +way to make Europe’. +We also thank G. Cimini, F. +Saracco and D. Garlaschelli for very useful comments. +Appendix A: Statistical physics of information +network diffusion +Let ˆL be the combinatorial Laplacian associated with +the network, namely Lij = [(δij +� +k Aik) − Aij]. +ˆL en- +codes the topological properties of the network. Given a +probability distribution s(τ = 0), its temporal evolution +is given by s(τ) = e−τ ˆLs(τ = 0). In that discrete-states +representation, each element of the propagator e−τ ˆL de- +scribes the sum of diffusion trajectories along all possible +paths connecting nodes i and j at time τ [19, 34, 46]. +Normalizing the propagator, it is possible to define the +ensemble of accessible information diffusion states [31– +33], obtaining +ˆρ(τ) = +e−τ ˆL +Tr +� +e−τ ˆL +�, +(A1) +where one can recognize in ˆρ(τ) the form of a canonical +density operator [74–76]. Note that, in full analogy with +hamiltonian systems in statistical physics, τ and ˆL play +the role of β and H, i.e. the inverse temperature and the +Hamiltonian function, respectively. +It is here important to stress that we assumed to discuss +connected networks to fulfill the ergodic hypothesis. At +that point, one can define the network entropy [31] as +S[ˆρ(τ)] = − Tr[ˆρ(τ) log ˆρ(τ)] += − +1 +log N +N +� +i=1 +µi(τ) log µi(τ), +(A2) +where µi represents the set of eigenvalues of ˆρ. Through +the detailed analysis of the flux of the entropy, it is pos- +sible to track the entropy-driven transition over the net- +work [33]. In particular, this passes from a strict frag- +mentation at τ = 0, where S = 0 and the system lies +in a segregated phase, to a uniformly connected through +diffusion graph, where S = 1 and the system lies in an +integrated and homogeneous phase. The derivative of the +entropy of the logarithm of the diffusion time τ, +C(τ) = − +dS +d log τ +(A3) +is a detector of transition points corresponding to the +intrinsic characteristic diffusion scales of the network [15, +33]. +Indeed, a pronounced peak of C defines τ = τ ∗ + +9 +and reveals the starting point of a strong deceleration +of the information diffusion, separating regions sharing a +rather homogeneous distribution of information from the +rest of the network. Note that, if more well-separated +diffusion timescales exist, then C(τ) can show a multi- +peak structure. +Appendix B: Laplacian Renormalization Group +The renormalization problem is approached here `a la +Wilson (we refer to [15] for a full discussion on the is- +sue), carrying on the comparison with the canonical en- +semble, as shown in Appendix A. The first step consists +in moving to the Fourier space to analyze the network +eigenmodes (as the graph lacks of any spatial embed- +ding). One may anyway keep in mind that ˆL contains +the inverse of the diffusion time scales. As it can be ex- +pected from a discrete version of Gaussian dynamics in +the continuum κ − space, ˆL is diagonalizable, and the +change of basis leads to a decoupling of modes. Since ˆL +is symmetric and real valued, it holds a complete set of +eigenvectors {|λ⟩}, with semi-positive eigenvalues {λ}. +In the bra-ket notation the Laplacian operator can be +decomposed as the projector � +λ λ|λ⟩⟨λ|. The LRG step +consists in integrating out these diffusion eigenmodes +from the Laplacian and appropriately rescaling the net- +work, namely: +1. Reduce the Laplacian operator to the contribution +of the N − n slow eigenvectors with λ < ˜λ, ˜ˆL = +� +λ<˜λ λ|λ⟩⟨λ|; +2. We then rescale the time τ → τ ′′, so that ˜τ in τ +becomes the unitary interval in the rescaled time +variable τ ′′ : τ ′′ = τ/˜τ and, consequently, redefin- +ing the coarsegrained Laplacian as ˆL′′ = ˜τ ˜ˆL. +As for a RG procedure applied to a Gaussian system, one +could expect to recover the same original diagonal form +after step 2. +Concerning the first step, point 1 can be implemented +letting the time run from 0 to a value τ ∗ ∼ 1/λ∗. Look- +ing at the propagator ˆρ(τ) defined in A1, it is possible +to observe that the contribute to the measure of a given +eigenvalue λ start decreasing significantly when τ ∼ 1/λ. +Such soft amputation overcomes the difficulties intro- +duced by the non-euclidean support, moreover, since the +set of eigenvalues is not dense in finite-size networks, a +such soft cut may anyway appear strict, if a sufficient gap +in the eigenvalues set occurs in the shell boundary. Such +request is well satisfied where the specific heat in Eq. A3 +shows a peak. +Appendix C: Unraveling nested structures in +heterogeneous networks +Hierarchic modular networks +We consider a specific case of hierarchical networks +where the connection between modules are not left at +random but with a scale-dependent probability, promot- +ing centralized structures between hubs, and following +the algorithm proposed in [42]. +We create at the be- +ginning 2s blocks of Ns = 16 nodes with mean degree +κ0 = 12 at the deepest level. Once this has taken place, +we give a weight p(i) = i−α/� +j j−α, to the ith node of each +block, i = 1, 2, ..., Ns. At this point, nodes are selected +with probability p(i) and p(j), and connected if they were +not already linked. We use here the same scale-free ex- +ponent α = 2 for all the hierarchical levels except for +the basal one, with α = 1.7 (as suggested to mimick the +empirically supported core-periphery organization with +connector hubs in brain structural networks [77–79]). +10 +6 10 +3 +100 +/ +max +Node index +(a) +10 +1 +101 +/ +* +0 +1 +1-S +(b) +L +LRW +1 3 5 7 911 +Gap number +0.0 +0.5 +RTR +(c) +0 +5 +Clog N +Figure 8. Hierarchic modular network. (a) Normalized +dendrogram for a HM-CP network using τ = 1/λmax. Red +dashed line reflects the division of the network using the third +gap of the RTR. Different communities are shaded in different +colors. (b) Entropy parameter (dashed lines, (1 − S)), and +specific heat (solid lines, C), versus the temporal resolution +parameter of the network, τ. (c) Retention Time Rate (RTR) +versus gap number for τ = τ ′. +Note the high RTR values +indicating the hierarchical structure of the network. Insets: +Adjacency matrix and division into four communities of the +network. +Figure 8(a) illustrates the dendrogram for a specific +network and the nested nature of the different modules, +which present the expected aggregation in the different +hierarchical scales. We also present here the comparison + +10 +of the spectral entropy of the network by the means of ˆL +and ˆLRW , which show no differences in this specific case +due to the high effective dimension of the basal struc- +tures of the network (see red and blue curves in Figure +8(b)). Finally, Figure 8(c) shows the retention time rate +for the different gaps of the dendrogram, together with +the network division into four modules and the adjacency +matrix. +Dorogovtsev-Goltsev-Mendes graph +Dorogovtsev, Goltsev, and Mendes [43] have intro- +duced a hierarchical scale-free network, in a way reminis- +cent of exact fractal lattices. In fact, the DGM network +is a pseudofractal: it contains subnetworks resembling +the whole network, but lacks the affine transformation +of scale which characterizes self-similarity in fractals. As +a result, the DGM network has infinite dimensionality, +containing numerous loops and hence being very far from +tree-like. In particular, the average clustering coefficient +of the network, for the infinite graph, is C = 4 +5, a prop- +erty that is suggestive of a modular organization. +10 +1210 +6 100 +/ +max +Node index +(a) +10 +1 101 +103 +< +> +0 +1 +1-S +(b) +L +LRW +1 +10 20 30 +Gap number +0.0 +0.5 +RTR +(c) +0 +2 +Clog N +Figure 9. Dorogovtsev-Goltsev-Mendes graph. (a) Nor- +malized dendrogram for a DGM network using τ = 1/λmax. +Red dashed line reflects the division of the network using the +second gap of the RTR. Different communities are shaded +in different colors. +(b) Entropy parameter (dashed lines, +(1 − S)), and specific heat (solid lines, C), versus the tem- +poral resolution parameter of the network, τ. (c) Retention +Time Rate (RTR) versus gap number for τ = τ ′. Note ta hat +peaks of the RTR are equally high reflecting the precise hier- +archical structure of the network. Insets: Adjacency matrix +and division into three communities of the network. +Figure 9(a) illustrates the dendrogram for a specific +network and the regular nested nature of the different +modules, which present the expected aggregation in the +different hierarchical scales. +We also present here the +comparison of the spectral entropy of the network by +the means of ˆL and ˆLRW , evidencing .... (see red and +blue curves in Figure 9(b)). Finally, Figure 9(c) shows +the retention time rate for the different gaps of the den- +drogram, together with the network division into three +modules and the adjacency matrix. However, let us re- +mark the serious issues presented by the different usual +community detection algorithms when dealing with these +particular type of hierarchical organization of modularity +(see Fig. 10). We test a set of different algorithms that +have proven to show an excellent performance: the walk- +trap algorithm [80], the Leiden algorithm [81] and In- +fomap [82]. Despite this, they exhibit a great variability +depending on parameters: they do not give an accurate +prediction of the different modules in the DGM network. +We propose the LRG as a way to go beyond the previous +attempts: this consider all the powers of the Laplacian +needed to recover the proper network structure, giving +place to a noticeable quantitative improvement on the +quality of the network subdivision (see Fig. +9(a) and +Fig. 10(a)). +(a) +LRG +(b) +Walktrap +(c) +Leiden +(c) +Infomap +Figure 10. Community detection methods. Comparison +of different community detection methods when applied to +the DGM network: (a) LRG as stated in the previous exam- +ple, (b) Walktrap algorithm (with nsteps = 103), (c) Leiden +algorithm (using γ = 0.004), and (c) Infomap (using 102 trials +to perform the network partition). Note the accuracy of the +LRG to perform network partitions in this specific case. + +11 +Lancichinetti–Fortunato–Radicchi benchmark +There is one further –and final– empirical test that +we can make to properly apply the LRG by using real- +istic benchmarks for community detection that accounts +for the heterogeneity of degree and community size. In +particular, the Lancichinetti–Fortunato–Radicchi bench- +mark, that considers both the degree and the community +size to be distributed as power laws, constitutes a much +harder test for algorithms and makes it easier to dis- +close their limits (we refer to the original work to further +details on the different steps to generate the networks +[45]). +Again, this class of networks represents a chal- +lenging task, even for well-known community detection +algorithms [23]. In particular, we choose the set of pa- +rameters N = 500, τ1 = 2.0, τ2 = 1.5, µ = 0.1 and fix +κmin = 2 and κmax = 50. First, we stress that further +analysis of these benchmark networks based on multiple +peaks on the entropy, together with an in-depth exam- +ination of the parameter space, can also help to under- +stand when these network exhibit or not communities, +as a function, e.g., of the mixing parameter [23]. This +problem will be tackled elsewhere. +10 +2 10 +1 +100 +/ +max +Node index +(a) +100 +102 +, < +> +0 +1 +1-S +(b) +L +LRW +1 +10 20 30 +Gap number +0.0 +0.1 +0.2 +RTR +(c) +0 +2 +Clog N +Figure 11. +Lancichinetti–Fortunato–Radicchi bench- +mark. (a) Normalized dendrogram for a LFR network using +τ = 2. Red dashed line reflects the division of the network +using the optimal gap of the RTR. Different communities are +shaded in different colors. (b) Entropy parameter (dashed +lines, (1 − S)), and specific heat (solid lines, C), versus the +temporal resolution parameter of the network, τ. (c) Reten- +tion Time Rate (RTR) versus gap number for τ = τ ′. Insets: +Network division into communities as set by the dendrogram. +Figure 11 shows that the particular application of +the LRG can precisely predict the predefined commu- +nity of each node. +Note the existence of two peaks +when the spectral entropy is analyzed in this case (see +Figure 11(b)), thus ensuring the presence of two well- +defined network scales. We stress that this benchmark +generates a “flat” community structure without hierar- +chies [23], fully justifying this feature. In our view, the +other significant result is the sharp double-peaked struc- +ture that emerges when we use ˆLRW , resulting from the +conceived trapping-build algorithm that generates the +networks (this is yet another example of different phe- +nomenology between ˆL and ˆLRW , as we detail in the +following example). +Finally, we illustrate the network +communities in Figure 11(c). +Appendix D: The clique-Star network +As we have previously shown in the main text, both the +’fluid’ Laplacian and the Laplacian Random Walk, can +evidence different emergent communities since the last +one encodes the evolution of a Markovian dynamics on +top of the network. This effect can be better understood +on the light on the Clique-Star network illustrated in +Figure 12. +10 +2 +100 +102 +, +0.00 +0.25 +0.50 +0.75 +1.00 +1-S +L +LRW +0 +2 +4 +6 +Clog N +Figure 12. +Clique-Star network. +Entropy parameter +(dashed lines, (1 − S)), and specific heat (solid lines, C), ver- +sus the temporal resolution parameter of the network, τ, for +both Laplacians, ˆL and ˆLRW . Note the explicit differences in +the total number of peaks of the specific heat, because of the +fact that ˆLRW reflects the trapping time into the two main +communities of the network: the star on the one hand, and +the clique on the other. +The Clique-Star network is formed by a dense module +of interconnected nodes connected to a star graph within +one of the clique nodes. This serves as a powerful ex- +ample of the main differences between both Laplacian +matrices. 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U.S.A. 105, 1118 (2008). + diff --git a/WNE3T4oBgHgl3EQfbQqU/content/tmp_files/load_file.txt b/WNE3T4oBgHgl3EQfbQqU/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7226880950ae4dc484a22aef8379805faf2fa05e --- /dev/null +++ b/WNE3T4oBgHgl3EQfbQqU/content/tmp_files/load_file.txt @@ -0,0 +1,1129 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf,len=1128 +page_content='Rethinking network community detection in the light of the Laplacian Renormalization Group Pablo Villegas,1 Andrea Gabrielli,1, 2, ∗ Anna Poggialini,1, 3 and Tommaso Gili4 1‘Enrico Fermi’ Research Center (CREF), Via Panisperna 89A, 00184 - Rome, Italy 2Dipartimento di Ingegneria, Universit`a degli Studi ‘Roma Tre’, Via Vito Volterra 62, 00146 - Rome, Italy 3Dipartimento di Fisica Universit`a “Sapienza”, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content='le A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Moro, 2, I-00185 Rome, Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' 4Networks Unit, IMT Scuola Alti Studi Lucca, Piazza San Francesco 15, 55100- Lucca, Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Heterogeneous and complex networks represent the intertwined interactions between real-world elements or agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' A fundamental problem of complex network theory involves finding inherent partitions, clusters, or communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' By taking advantage of the recent Laplacian Renormalization Group approach, we scrutinize information diffusion pathways throughout networks to shed fur- ther light on this issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Based on inter-node communicability, our definition provides a unifying framework for multiple partitioning measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Moreover, we account for the elusive behavior of components bridging different communities within the same network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Community structures emerge as a ubiquitous feature of real-world networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Detection of relevant network substructures is of utmost importance to correctly under- stand their often hidden multiple mesoscopic scales and particular functionalities [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' The level of compartmen- talization of a system, the modularity, has been proposed as a generic requirement, or an optimal solution, for a system in a changing environment to be evolvable [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' High modularity, therefore, pervades biology on differ- ent scales from proteins and genes [4] to cells [5, 6], the human interactome [7] or ecosystems [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Modules form the basis of transcriptional regulatory networks [9], have been found at every scale in metabolic networks [10], and have been hypothesized to confer a high efficiency of in- formation transfer between nodes at low connection cost in brain networks [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' In a seminal paper, Girvan and Newmann [2] proposed a celebrated algorithm by progressively removing edges from the original network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' The inherent nature of the al- gorithm led them to introduce a quantitatively ’stopping criterion’: the modularity, Q, which quickly became the fundamental ingredient of many clustering methods [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Modularity maximization is one of the most widely used algorithms to detect communities in a network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' However, an exhaustive modularity maximization implies an NP- complete problem, whose complexity increases exponen- tially with the total number of nodes, making it almost impossible (we refer to [13, 14] for an extensive discussion on the issue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' More sophisticated procedures rely on the spectral properties of the graph Laplacian, which encode the net- work diffusion modes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=', allowing to explore the net- work momentum space [15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Indeed, this is the basis for one of the best-known methods of graph partitioning: spectral partitioning [16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' A related elegant proposal combines spectral methods with clustering techniques, projecting the network nodes into an eigenvector space of variable (tunable) dimensionality [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Note that all the ∗ Corresponding author: andrea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content='gabrielli@uniroma3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content='it algorithms taking advantage of spectral methods, which operate in Fourier space, need an N-dimensional space due to a lack of well-defined spatial embedding of the net- work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Another set of methods is based on the exploration properties of random walks on top of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' They take advantage that, due to the intrinsic network hetero- geneities, a random walker will remain trapped within different communities for a relatively long time before leaving it (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=', [19] for an excellent review on the field).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' In particular, the probability of leaving a vertex is distributed among the outgoing edges according to their weight [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Hence, the concept of Markov stability has been introduced [20], which quantifies the tendency for a random walker to stay inside a community for a long time [19, 21], opening the door to additional quality measures of a graph partition based on dynamical concepts, there- fore ranking partitions and establishing their relevance over time scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' In either case, detecting mesoscale structures (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=', communities) in networks has become a fundamental problem –and still a matter of debate [22]– in network science, where many methods have been proposed and classified according to their performance [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Yet the problem lacks of an holistic perspective, providing an unified interpretation, and capable of setting the limi- tations of the different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' This is due to the fact that a proper definition of community depends on the process on the network for which we want to find the most suitable partition [24], complicating even further the issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Besides, the proper identification of local com- munities represents a fundamental open problem when it is tackled through usual community detection meth- ods [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' In a nutshell, identifying basic structural and dynamic communities in a rigorous way raises a general problem of definition that still needs to be resolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Tackling the root of the problem requires following the evolution of inter-node communication throughout mul- tiple scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' In classical statistical physics, the renor- malization group (RG) –one of the most influential con- cepts in theoretical physics of the second half of the 20th- century [25–28]– allows for connecting extremely varied arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content='04514v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content='stat-mech] 11 Jan 2023 2 spatiotemporal scales of physical systems embedded in regular Euclidean spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' The applications of this frame- work in its network counterpart, the Laplacian Renor- malization Group (LRG), is the natural choice to analyze complex network mesoscales and structures [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' As LRG strictly corresponds to the free field theory’s renormal- ization, it allows for scrutinizing all the possible diffusion paths at arbitrary time across an arbitrary network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Here, in the light of the LRG, we develop a new com- munity detection framework that unravels the complete hierarchical structures of the network at different reso- lution levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' In particular, we reconstruct the hierar- chical network tree based on diffusion distances among the nodes by taking advantage of the LRG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Therefore, we reduce the complexity of quantifying the ’quality’ of network subdivisions, proposing a natural solution based on slowing down information diffusion throughout the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Finally, we illustrate the emergence of dynami- cal communities when a set of random walks explore the networks structure and discuss the bridges with previous proposals, unifying modularity maximization, Markov stability and spectral methods in a common framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' COMMUNITY DETECTION AND LAPLACIAN RENORMALIZATION GROUP For an undirected network with N nodes and M edges, the level of modularity for a given partition (see [13] and references therein) can be defined as, Q = 1 2M N � i,j=1 (Aij − γPij) δ (gi, gj) , (1) where ˆA stands for the network adjacency matrix, γ is a resolution parameter, and gi is the community to which node i belongs, being δ(gi, gj) = 1 if gi = gj and 0 other- wise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Pij is a null-model matrix that critically constrains the modularity maximization process [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Modularity is strongly related to the concept of Markov stability [19], where now the standard null model is given by the prob- ability that a discrete-time random walker visits node i at t = 0 and node j at t = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' One, therefore, obtains R(t) = N � i,j=1 �� p∗ i e−τ ˆLRW � ij − p∗ i p∗ j � δ (gi, gj) , (2) where ˆLRW = D−1 ˆL, is the ‘random-walk normalized Laplacian’, and ˆL = D − A is the Laplacian matrix of the network, with τ a resolution parameter, which allows to unravel multiscale structures in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Here, the null probability p∗ i p∗ j corresponds to the the expected transfer probabilities at stationarity for this Markov pro- cess [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Note that, as previously discussed by the net- work community [19, 30], Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' (2) reduces to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' (1) under the very particular assumption of e−τ ˆLRW ≃ I − τ ˆLRW , a linear approximation which can avoid computational difficulties to calculate the full matrix exponential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' How- ever, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' (2) provides a simple interpretation of the res- olution parameter (γ = 1/t), in terms of Markov times [19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' We point out that the very particular structure of the network relies, however, on the Laplacian matrix, which encodes the intrinsic topological properties and manages renormalization group procedures on top of heteroge- neous structures [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' In particular, the Laplacian drives the temporal evolution of the diffusive paths of a net- work, s(τ) = e−τ ˆLs(0), giving place to define the density matrix as [31, 32], ˆρ(τ) = e−τ ˆL Tr(e−τ ˆL) , (3) which opened the door to a thorough analysis of network structures, where the entropy [31] (S = − 1 log N ˆρ(τ) log ˆρ(τ)) and specific heat [33] (C = − dS d log τ ) revealed characteristic network scales (we refer to [15, 31, 33] for further details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' It also makes it possible to scruti- nize the scale-invariant properties of a network [15], link- ing the specific heat with the effective spectral dimension of the network (see also Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Sketch of the network analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' The node’s dis- tances reflects the underlying hidden complex topology, cap- turing the communicability strength, and giving a natural merging of nodes at specific times, τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Finally, the application of the LRG gives a hierarchical reconstruction of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' In particular, the density matrix is defined in terms of the network propagator, ˆK = e−τ ˆL, which accounts for the sum of diffusion trajectories along all possible paths connecting nodes i and j up to a temporal scale τ [19, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Note that this evolutionary operator essen- tially contains the same information as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' 2 whether considering ˆL or ˆLRW (for now, we analyze the case of ˆL, considering later the implications of ˆLRW ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' In any case, ˆρ(τ) gives a robust measure of information com- municability or diffusion strength between pair of nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' This allows us to define in a natural way a Laplacian dis- tance, Dij [ρ(τ)] = 1 ρij(τ)−δijρij(τ), as the inverse of the strength of the information probability paths between nodes at each particular time τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Note that such a def- 3 10 410 2 100 / max 32 25 26 24 28 10 29 27 30 9 31 23 21 16 19 33 15 34 5 11 7 6 17 1 12 4 13 14 3 8 20 22 2 18 Node index (a) 10 2 100 0 1 1-S (b) 1 3 5 7 9 Gap number 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content='5 RTR (c) 0 1 log NC/Cmax Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' LRG communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' (a) Normalized dendrogram for the weighted Zachary’s karate club with τ = τ ′ = 1/λmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Red dashed line reflects the optimal division as stated by the RTR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Different communities are shaded in different colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' (b) Entropy parameter (dashed lines, (1 − S)), and specific heat (solid lines, C), versus the temporal resolution parameter of the network, τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Black dashed line indicates τ ′ = 1/λmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' (c) Retention Time Rate (RTR) versus gap number for τ = τ ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Note the high RTR values giving place to the usual division in two and four communities of the Zachary’s karate club.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Inset: Division into communities of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' inition shows an intrinsically ultrametric behavior, sat- isfying the condition Dij ≤ max (Diz, Dzj), beyond the standard requests Dij ≥ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Dij = 0 ⇔ i = j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Dij = Dji at each time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Once we have introduced a way to measure distances between nodes, a method for grouping nodes into com- munities is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' We emphasize that, as complex net- works lack an explicit spatial embedding, ˆD allows us to reduce the problem to the traditional clustering problems in Euclidean space [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' In particular, we choose the av- erage group clustering algorithm, a compromise between the sensitivity of complete clustering to outliers and the tendency of single clustering to inhibit compact clusters [35, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' The output of this algorithm will thus be a hier- archical tree or dendrogram for each resolution scale, τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' However, note that D [ρ(τ)] can, in principle, give place to different dendrograms modifying the specific length of branches as the diffusion time increases, thus potentially inducing diverse structural partitions (this situation is not unique, the same applies, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=', in Markov stability [20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' The natural question is how to reconcile the identi- fication of optimal divisions with preserving the multiple timescales stemming from the intrinsic heterogeneity of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' A solution for this conundrum comes from the direct application of the LRG [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' As settled by the LRG, the Laplacian of the network –and its relative eigenvec- tors |λ⟩– contains the different network diffusive modes (see Appendix B for further clarifications).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' In particu- lar, the LRG scheme takes advantage of the fact that the Laplacian operator is a sort of telescopic ”scanner” of the coarse-graining scales, integrating out the differ- ent network eigenmodes as τ increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' This suggests the existence of a resolution window where the different partitions must be relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' The smallest possible scale is given by λmax, which prevents to integrate out any basal network scales, thus avoiding losing network infor- mation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' This choice of τ ′ = 1/λmax gives us the highest possible resolution of the network structure from above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Note that it is possible to consider greater times inte- grating out network eigenmodes to monitor the stability of the mesoscopic partitions across diffusion scales, up to the specific heat peak at long times, where the Fiedler eigenvalue is integrated out (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=', defocusing microscopic details, thus shedding light on the origin of a resolution limit [37], see below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Finally, for the sake of clarity, we consider the relative length of the dendrogram D/Dmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' This results in an optimal normalized dendrogram re- flecting the intrinsic network mesoscopic structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' 2(a) shows the application of this method for the partic- ular case of the weighted version of the Zachary’s karate club [2, 38], which serves us as a benchmark showing the usual division in two communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' 2(b) shows the entropic phase transition for Zachary’s karate club.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Nevertheless, even if we can scrutinize the whole net- work hierarchic structure as a function of the distances between nodes, we need to quantify the ’quality’ of net- work subdivisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' In other words, the clustering algo- rithm gives no hint about the ’goodness’ of a specific partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' We propose the Retention Time Rate (RTR) as the ratio of the dendrogram gaps and the expected total diffusion time of the information throughout the network (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=', dendrogram length, in logarithmic scale).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Hence, the higher the modularity, the higher the proba- bility that information slows down, therefore maintaining the flux trapped in a particular community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' The usual definition of modularity [18, 39], Q =� i (eii − ai)2, is an alternative way of providing a ’quality’ function of the network subdivision facing the density of links in- side communities (eii) compared to connections between communities (ai, weighted with 1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' However, we stress that Q only measures the fraction of edges that fall be- tween communities minus the expected value of the same quantity in a random graph with the same community di- vision [18], neglecting other effects (as, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=', the problem of loops [40] or the separation of network timescales [41]) that may have a profound impact on the structural divi- sion into different communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' 2(c) shows the best partitions as a function of the dendrogram gap number for a particular time τ as indicated by the RTR (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' 3(e) for analysis over multiple timescales).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' One important application of the method involves the 4 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Dynamical communities (LRW ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' (a) Normalized dendrogram for Zachary’s karate club with τ = τ ′ = 1/µmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Red dashed line reflects the optimal division as stated by the RTR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Different communities are shaded in different colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' (b) Entropy parameter (dashed lines, (1 − S)), and specific heat (solid lines, C), versus the temporal resolution parameter of the network, τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Black dashed line marks τ ′ = 1/µmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' (c) Retention Time Rate (RTR) versus gap number for τ = τ ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Inset: Division into communities of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' (d) Spectrum of eigenvalues for the LRW for a Barabasi-Albert network with m = 10, a Random Regular graph with κ = 18, an with ⟨κ⟩ = 18 and a Watts-Strogatz network with ⟨κ⟩ = 18 and rewiring probability p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Black dashed line stands for the semicircular law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' (e) RTR versus τ/τ ∗ (with τ ∗ corresponding to the maximum in C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Different colors stand for different time gaps in the dendrograms for L (circles) and LRW (triangles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Distance matrix for: (f) a hierarchic modular network with core-periphery structure, and (g) a Dorogovstev-Mendes graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' characterization of heterogeneous hierarchical nested net- works with multiscale communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' In this light, we an- alyze both hierarchical-modular networks with a prefer- ential attachment rule (HM-CP, that produces a core- periphery structure involving central connector hubs hav- ing local and global rich clubs [42]) and hierarchical lattices through the Dorogovtsev-Goltsev-Mendes graph (DGM), a pseudo-fractal with high clustering properties [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' On the one hand, as better illustrated in Appendix C, the LRG evidences cleanly and concisely the full in- trinsic modular structure of nested hierarchical networks (HM-CP), even when the inter-module connections and communication paths tend to be centralized through the hubs, as observed in real neural and brain networks [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' On the other hand, we highlight that, when applied to highly hierarchical systems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=', DGM networks), usual community detection methods present severe issues in unraveling either communities and the complete hierar- chical structure of the network (see Appendix C for fur- ther details and illustrative examples).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Therefore, we emphasize the capability of the LRG to unravel meso- scopic communities in these types of intricate networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' In particular, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' 3(f) and (g) illustrate the emergent nested structure already apparent in the distance matrix, ˆD, for HM-CP and DGM networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Finally, we have successfully tested the Lancichinetti-Fortunato-Radicchi benchmark [45], a network generator with a priori known communities, which serves as a stringent criterion, to compare different community detection methods (see Ap- pendix C for further details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' THE LAPLACIAN RANDOM-WALK A different approach can be considered by simply in- troducing the ’random-walk normalized Laplacian’ of the network, ˆLRW = D−1 ˆL, into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' This encodes a traveling dynamics at a unit rate, moving to a particu- lar neighbor with equal probability for each choice: i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=', the transition matrix for the random walk dynamics on top of a graph [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' It has to be reminded that ˆLRW is not a stochastic matrix since it does not meet the needed constraints but, anyway, it represents an evolutionary op- erator in an L2 space, with all the eigenvalues of ˆLRW satisfying µi ∈ [0, 2] (and also featuring an identical spec- trum to the normalized symmetric Laplacian, Lsym [47]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' ˆLRW is also linked with the spectral dimension of a graph [48], dS, which characterizes the return-time distribution of the random walker: a walker starting at t = 0 from any node of the network has probability P0(t) of returning to the initial node, with P0(t) = � dµp(µ)e−µt ∝ t−dS/2 [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' This is obtained by the usual Laplace transform, which automatically links ρ(λ) with the probability den- sity function of return time distributions in the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' To better scrutinize the Laplacian RW spectrum (and conse- quently, the average trajectories of the walkers [50]), we propose the following transformation µ′ = 1 2 + µ−1 2(µmax−1), to ensure the spectrum is upper bounded by one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' There- fore, for high values of dS we expect to recover the (universal) mean-field theoretical expectation before x(t) first returns at time T to its initial value x(0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=', 10 9 8 7 6 516 14 12 10 8 6 45 ⟨x(s)⟩ = 8 π � s(1 − s), with s = t/T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Indeed, as illus- trated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' 3(d) for different explicit networks, the spectrum of ˆLRW converge to this universal shape (the Wigner semicircle law [51]) over a specific upper criti- cal dimension (indeed, it must be dS u = 4, as expected from the Gaussian model [28, 52, 53]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' This is because the random walk Laplacian spectrum of the network is closely related to the first-return times and the distribu- tion of all returns induced by the network [54, 55] (but not the fluid Laplacian, ˆL, where the eigenvectors form the Fourier basis of the network where the random walk dynamics is projected through ˆLRW ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Figure 3(a) shows the ’Gaussian’ dendrogram for Zachary’s karate club by considering ˆLRW (it is impor- tant to stress that Node 3 is placed in a different commu- nity of those of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' 2(c), but we will discuss this aspect in detail later).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' 3(b) shows the entropy parameter and the specific heat for Zachary’s karate club now using ˆLRW to characterize the network null ’dynamical’ struc- ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Note that the specific heat, C, presents a slightly different shape to those of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' 2(b) as a result of the dynamical modes of the network, induced by ˆLRW .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' 3(c) shows the best partitions for a particular time τ ′ as indicated by the RTR (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' 3(e) for an analysis of the two first optimal gaps over multiple timescales).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' 10 3 10 1 101 / 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content='0 1-S (a) L LRW 10 2 10 1 100 101 / 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content='0 1-S (b) 0 1 2 log NC 0 2 4 6 log NC Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Real networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Evolution of the spectral en- tropy using the Laplacian and the Laplacian RW, versus the normalized time, τ/τ ∗ (where, for the sake of comparison, τ ∗ corresponds here to the absolute maximum of the specific heat, for: (a) Mus Musculus PPI network and (b) Drosophila Melanogaster network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Note how eigenmodes are integrated out differently in both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' We now proceed to apply the methodology of the LRG to analyze this particular dynamics on top of different real networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' As we have just discussed, we may ex- pect that both Laplacians could lead to different cases of study in the ’fluid case’, ˆρ(τ), and the ’random-walk’ case, ˆρRW (τ) in certain specific situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' 4(a) and (b) shows the comparison of the entropy order parameter and the specific heat of both cases, whether for the Mus Musculus PPI network [56] and the color vision circuit in the medulla of Drosophila Melanogaster [57], showing at a glance how information (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=', network modes) is inte- grated in a different way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' In particular, these effects will strongly depend on the effective (spectral) dimension of the network, whether at a global or local scale [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' As an illustrative example of this effect, we consider the spe- cific analysis of the clique-star network: a random walker will be effectively trapped at the star hub leading to two well-defined dynamical communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Instead, the fluid Laplacian will additionally capture the broadcast of in- formation linked to both hubs, giving place to more ac- curate information on the topological properties of the network (see further details in Appendix D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Finally, we illustrate in Figure 5 the complex hierar- chical structure, using ˆL, of the E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Coli and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Musculus PPI networks [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' We identify the functional role of these communities as indicated in the UniProt Knowl- edgebase [59], a comprehensive, high-quality, and freely accessible set of protein sequences annotated with func- tional information with more than 190 million sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' In particular, we use the keywords associated with each protein to facilitate the characterization of different core functionalities of the observed modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' In fact, the PPI network of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content='Coli presents a highly nested tree-like struc- ture with other structural submodules dedicated to dif- ferent biological functions, as shown in Figure 5(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' For the sake of clarity, we explore the division into five mod- ules exemplified in the dendrogram of Figure 5(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' In this case, we want to stress for the sake of illustration the spe- cific nature of two main communities: the one dedicated to RNA-binding (green nodes, 83% of the corresponding proteins engaged in this process belong to this commu- nity, constituting the 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content='5% of it) and the one dedicated to cell division and cell cycle (violet nodes, containing the 62% of the proteins involved in this goal, and con- forming the 44% of the community).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' We instead high- light the interconnected and highly intricate network of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content='Musculus, where, despite this, we can discern three principal modules, noticing that the smallest one (blue nodes in Figure 5(b)) is strongly linked to the signal pro- cesses and immune response of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' In any case, the in-depth study of different biological networks is be- yond the scope of this manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' We will explore better and more refined analyses of these nested structures else- where.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' LOCAL MODULARITY IN HETEROGENEOUS NETWORKS Sometimes one might want to know the communities in only a small region, which does not ensure a global ‘good’ division of the network in terms of an optimization function [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' For example, in many real-world networks, clusters are linked mainly locally among each other, gen- erating local clusters that are overshadowed by global network dependency [60]: the coexistence of local mod- ularity with global nestedness is key to ensure, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=', evo- lutionary stability in host-pathogen infection networks [61], while the human temporal lobe is organized into spatially compact functional modules at the micro-scale [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Quantifying the local stability across different net- work scales of these local structures is particularly chal- lenging when it is only considered an optimization func- 6 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content='Coli M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content='Musculus #Signal #Immunity #RNA-Binding #Sugar transport #Transport #Electron transport #Signal #Transport #Cell division #Cell cycle #Nucleotide-binding #ATP-Binding Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Protein-protein interaction networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' (a) Modular structure of the E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content='Coli PPI network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' The dendrogram illustrates the hierarchical clustering of the different nodes of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Different colors stand for different communities, along with their corresponding main biological functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' (b) Modular structure of the M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content='Musculus PPI network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' The dendrogram illustrates the hierarchical clustering of the different nodes of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Different colors stands for different modules, as stated by the network dendrogram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Note that, instead of choosing the ’best’ network partition in terms of diffusive distances, we are now interested here into the complex nested structure and modules of both organisms which gives rise to a rich structure with multiple overlapping communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' The different functionalities associated to each module are written in the same color of the nodes it contains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' We stress the characterization of a small local module responsible of electron transport as direct application of the LRT (see black dashed circle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Parameters: We use ˆL with τ = 10, thus integrating out the microscopic scales of both networks, but without going so far to integrate out the Fiedler eigenvector, therefore ensuring a proper analysis of the network mesoscopic modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' tion (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=', modularity to find the internal structure of the network).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Local modularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' (a) Local modularity in the normalized dendrogram for Zachary’s karate club with τ = τ ′ = 1/λmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Values of the LRTR larger than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content='3 have been highlighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' (b) Local Retention Time Rate versus τ/τ ∗ (with τ ∗ corresponding to the maximum in C) for the local community highlighted in orange color for both the Laplacian (red line) and the Laplacian RW (green line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' As a natural extension of the method we have de- scribed before, we propose to measure the Local Reten- tion Time (LRT) as the length of every particular branch of groups of three or more network nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' We stress that the branch length, at time τ, represents the com- munication distance (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=', sum-over-paths) between two groups of nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Hence, higher values of the LRT in- dicates a stable module over a significant amount of in- trinsic timescales of the network, either from a structural or a dynamical point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Figure 6(a) shows a par- ticular example for Zachary’s karate club using the fluid Laplacian, ˆL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' As we illustrate in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' 6(b), this par- ticular module exhibits a large LRT for all the expected range of significative timescales both for ˆL and ˆLRW .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Another particular application comes from the detailed study of the dendrogram of the M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Musculus PPI net- work, which shows a rich structure of heterogeneously distributed modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Finally, the direct application of the LRT allows detecting a robust, independent module (see black dashed lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' 5) detecting a group of proteins dedicated to electron transport (in particular, all these proteins are involved in the respiratory chain), necessary, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=', for both photosynthesis and aerobic res- piration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' TEMPORAL DENDROGRAMS AND METASTABLE NODES As previously discussed, one particular implication of the LRG is the ’dynamic’ perspective it gives from the contribution of some specific nodes to the functional ca- 7 pabilities of a heterogeneous structure, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=', the contri- bution of diffusive modes at different timescales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' We fo- cus here on nodes that act as ’bridge’ nodes and com- municate highly connected modules of the network, thus sharing information (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=', sending signals) or contributing to different functional communities at different spatio- temporal resolution scales of the network structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' In other words, these nodes can dynamically modify their diffusive distance to other mesoscopic structures in the system, which explains why these nodes are extremely challenging when attempting to classify them through optimization function techniques that provide a static picture of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' 1 2 3 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 C1 C2 CF1 CF2 Community evolution Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Sankey diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Evolution of the optimal com- munity division (in terms of RTR) for the Zachary’s Karate club at three different characteristic times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' We emphasize the ability of ’bridge nodes’ to dinamically change their functional community at different timescales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' This effect can be illustrated by analyzing Node 3 of Zachary’s Karate club, whose community differs depend- ing on the Laplacian we have previously considered (as discussed above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' To shed light on this crucial fact, we now focus on analyzing this specific node by using ˆL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' In particular, we show in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' 7 the best community as- signment for this particular network at different times, revealing how this ’bridge’ node dynamically changes the community it belongs to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' This gives it a central role in managing communication and/or control processes be- tween independent modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Even if we represent a sym- bolic example using this specific network, the relevance of these nodes in real networks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=', the human connec- tome) for synchronizability and information integration is expected to be crucial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' This metastable way of pop- ulating modules by specific nodes recalls the so-called “modular flexibility“ [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Modular flexibility represents how frequently nodes change the modules they belong to across time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' It means that nodes more likely to be con- nected to multiple modules at different time points are more flexible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Modular flexibility has been associated with network adaptability, giving flexible nodes the role of pivot of the dynamics processes running on the net- work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Although the concept of modular flexibility pro- posed by Khambati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' is based on the description of the network’s evolution dynamics across time, it can also be used to describe the modular properties of a network according to the dynamic regime of diffusive processes of information used to investigate it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' DISCUSSION AND CONCLUSION The lack of a valid source of geometric length-scale transformations constitutes a fundamental issue in het- erogeneous systems [64], which has critically constrained, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=', renormalization group approaches in heterogeneous structures [15] and, as a natural consequence, the com- munity detection problem in complex networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' We pro- pose a simple but efficient algorithm based on the Lapla- cian Renormalization Group –taking advantage of the fact that the Laplacian operator is a sort of telescopic ”scanner” of the mesoscopic scales of a network– which gives a diffusion-distance-based and free-metric approach to the community detection problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' In particular, we also provide a natural and parsimonious interpretation of the so-called ’resolution limit’ [22, 51], conditioned by the Taylor expansion of the network propagator, that limits the maximum number of paths of length n from vertex i to vertex j to be taken into consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' This is no more than the applied zoom once a particular value of τ has been selected or, in other words, the defocusing: i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=', the number of diffusive eigenmodes that have been inte- grated out, with the resultant loss of information about the microscopic network scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' We thus highlight important diverse implications of the Laplacian Renormalization Group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' On the one hand, the LRG sets the formal equivalence between real-space methods that can be derived from the ˆρ(τ)-density ma- trix (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=', Louvain methods for community detection or Markov stability) and spectral clustering techniques [16– 18], which address the issue in k − space and analyze diffusion modes of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' These approaches are two sides of the same coin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' On the other hand, it allows proposing a unified interpretation of community detec- tion in terms of structural (ˆL) and dynamical aspects (ˆLRW , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=', intrinsic trapping times) providing an overall vision over different methodologies that have been hith- erto considered separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' We pinpoint that, given the connection of the Gaussian model with the random walk on any graph [52], the use of the ˆLRW constitutes the first approximation or dynamical reference model in the same way the Gaussian model provides the starting point for perturbative RG analyses [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' As a natural consequence, we stress the importance of using ˆL for community de- tection purposes on top of functional matrices because, otherwise, the underlying dynamics based on ’Markovian’ random walks is always an implicit assumption and lacks systematic interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' It is essential to discuss in detail the link between the Gaussian model and the random walk on any graph (even 8 if we refer to [46, 52] for an extended discussion on the issue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' In particular, du = 4 characterizes a critical value for the intersection properties of two independent random walks, which is closely connected to the non-triviality of the φ4 d field theory [65] (note that over this dimen- sion, two random walkers are unlikely to intersect, and the Gaussian results obtained neglecting the intersections are asymptotically valid [28]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Hence, the spectra of the Laplacian RW is expected to show some universal behav- ior [66] dictated by the lowest vertex degree in a network [54] (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=', its spectral dimension), which, anyhow, will control the emergent functional communities and dynam- ical aspects of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' We pinpoint the relevance of this measure intending to discriminate the relevance of underlying heterogeneity on the system dynamics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=', scale-free networks are expected to be dynamically irrele- vant for large enough values of m in the Barabasi-Albert model [54]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' A further crucial issue to consider when we think about heterogeneous structures is that the effective community of ’bridge’ nodes can be diverse for different timescales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Therefore, it would be helpful to understand the proper- ties these ’bridge’ nodes have in common.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' This is par- ticularly relevant for controllability in complex networks: for example, they are expected to manage switching be- havior on the large repertoire of attractors, with different degrees of coherence and stability, present in hierarchic modular networks [67, 68] or in metastable dynamics in human brain networks, where hubs have been elucidated to control the resulting wave patterns [69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' We hypoth- esize that this issue is, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=', closely related to the ob- served disrupted hub organization in the topological dis- turbances associated with schizophrenia [70], where the waste of hubs [71] can lead to dysfunctions in controlling and integrating neuronal signals [72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Finally, the appli- cation of the LRG allows us to develop a local detection method that overcomes resolution limits problems but gives a global overview of the importance of local com- munities in the entire network [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Altogether, we propose here a new vision of net- work modularity based on the Laplacian Renormaliza- tion Group (LRG) [15], which is the natural extension to heterogeneous networks of the usual RG approach in sta- tistical physics and statistical field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' This permits to reveal the ’building’ blocks of the network at different scales without resolution limit constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' In particular, we scrutinize the direct links with previous definitions of network modularity, therefore presenting an overall pic- ture of different frameworks and revealing their limita- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Our LRG scheme opens a new route to extend the study of emergent dynamical communities [24] beyond the analysis of different ’dynamical’ Laplacian evolution operators containing the dynamical aspects of diverse processes running on top of heterogeneous structures [73], providing much insight into their understanding and fos- tering future renormalization group analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' ACKNOWLEDGMENTS P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' acknowledges the Spanish Ministry and Agen- cia Estatal de investigaci´on (AEI) through Project I+D+i Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' PID2020-113681GB-I00, financed by MICIN/AEI/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content='13039/501100011033 and FEDER ‘A way to make Europe’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' We also thank G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Cimini, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Saracco and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Garlaschelli for very useful comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Appendix A: Statistical physics of information network diffusion Let ˆL be the combinatorial Laplacian associated with the network, namely Lij = [(δij � k Aik) − Aij].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' ˆL en- codes the topological properties of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Given a probability distribution s(τ = 0), its temporal evolution is given by s(τ) = e−τ ˆLs(τ = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' In that discrete-states representation, each element of the propagator e−τ ˆL de- scribes the sum of diffusion trajectories along all possible paths connecting nodes i and j at time τ [19, 34, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Normalizing the propagator, it is possible to define the ensemble of accessible information diffusion states [31– 33], obtaining ˆρ(τ) = e−τ ˆL Tr � e−τ ˆL �, (A1) where one can recognize in ˆρ(τ) the form of a canonical density operator [74–76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Note that, in full analogy with hamiltonian systems in statistical physics, τ and ˆL play the role of β and H, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' the inverse temperature and the Hamiltonian function, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' It is here important to stress that we assumed to discuss connected networks to fulfill the ergodic hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' At that point, one can define the network entropy [31] as S[ˆρ(τ)] = − Tr[ˆρ(τ) log ˆρ(τ)] = − 1 log N N � i=1 µi(τ) log µi(τ), (A2) where µi represents the set of eigenvalues of ˆρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Through the detailed analysis of the flux of the entropy, it is pos- sible to track the entropy-driven transition over the net- work [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' In particular, this passes from a strict frag- mentation at τ = 0, where S = 0 and the system lies in a segregated phase, to a uniformly connected through diffusion graph, where S = 1 and the system lies in an integrated and homogeneous phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' The derivative of the entropy of the logarithm of the diffusion time τ, C(τ) = − dS d log τ (A3) is a detector of transition points corresponding to the intrinsic characteristic diffusion scales of the network [15, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Indeed, a pronounced peak of C defines τ = τ ∗ 9 and reveals the starting point of a strong deceleration of the information diffusion, separating regions sharing a rather homogeneous distribution of information from the rest of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Note that, if more well-separated diffusion timescales exist, then C(τ) can show a multi- peak structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Appendix B: Laplacian Renormalization Group The renormalization problem is approached here `a la Wilson (we refer to [15] for a full discussion on the is- sue), carrying on the comparison with the canonical en- semble, as shown in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' The first step consists in moving to the Fourier space to analyze the network eigenmodes (as the graph lacks of any spatial embed- ding).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' One may anyway keep in mind that ˆL contains the inverse of the diffusion time scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' As it can be ex- pected from a discrete version of Gaussian dynamics in the continuum κ − space, ˆL is diagonalizable, and the change of basis leads to a decoupling of modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Since ˆL is symmetric and real valued, it holds a complete set of eigenvectors {|λ⟩}, with semi-positive eigenvalues {λ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' In the bra-ket notation the Laplacian operator can be decomposed as the projector � λ λ|λ⟩⟨λ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' The LRG step consists in integrating out these diffusion eigenmodes from the Laplacian and appropriately rescaling the net- work, namely: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Reduce the Laplacian operator to the contribution of the N − n slow eigenvectors with λ < ˜λ, ˜ˆL = � λ<˜λ λ|λ⟩⟨λ|;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' We then rescale the time τ → τ ′′, so that ˜τ in τ becomes the unitary interval in the rescaled time variable τ ′′ : τ ′′ = τ/˜τ and, consequently, redefin- ing the coarsegrained Laplacian as ˆL′′ = ˜τ ˜ˆL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' As for a RG procedure applied to a Gaussian system, one could expect to recover the same original diagonal form after step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Concerning the first step, point 1 can be implemented letting the time run from 0 to a value τ ∗ ∼ 1/λ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Look- ing at the propagator ˆρ(τ) defined in A1, it is possible to observe that the contribute to the measure of a given eigenvalue λ start decreasing significantly when τ ∼ 1/λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Such soft amputation overcomes the difficulties intro- duced by the non-euclidean support, moreover, since the set of eigenvalues is not dense in finite-size networks, a such soft cut may anyway appear strict, if a sufficient gap in the eigenvalues set occurs in the shell boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Such request is well satisfied where the specific heat in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' A3 shows a peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Appendix C: Unraveling nested structures in heterogeneous networks Hierarchic modular networks We consider a specific case of hierarchical networks where the connection between modules are not left at random but with a scale-dependent probability, promot- ing centralized structures between hubs, and following the algorithm proposed in [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' We create at the be- ginning 2s blocks of Ns = 16 nodes with mean degree κ0 = 12 at the deepest level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Once this has taken place, we give a weight p(i) = i−α/� j j−α, to the ith node of each block, i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=', Ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' At this point, nodes are selected with probability p(i) and p(j), and connected if they were not already linked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' We use here the same scale-free ex- ponent α = 2 for all the hierarchical levels except for the basal one, with α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content='7 (as suggested to mimick the empirically supported core-periphery organization with connector hubs in brain structural networks [77–79]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' 10 6 10 3 100 / max Node index (a) 10 1 101 / 0 1 1-S (b) L LRW 1 3 5 7 911 Gap number 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content='5 RTR (c) 0 5 Clog N Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Hierarchic modular network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' (a) Normalized dendrogram for a HM-CP network using τ = 1/λmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Red dashed line reflects the division of the network using the third gap of the RTR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Different communities are shaded in different colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' (b) Entropy parameter (dashed lines, (1 − S)), and specific heat (solid lines, C), versus the temporal resolution parameter of the network, τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' (c) Retention Time Rate (RTR) versus gap number for τ = τ ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Note the high RTR values indicating the hierarchical structure of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Insets: Adjacency matrix and division into four communities of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Figure 8(a) illustrates the dendrogram for a specific network and the nested nature of the different modules, which present the expected aggregation in the different hierarchical scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' We also present here the comparison 10 of the spectral entropy of the network by the means of ˆL and ˆLRW , which show no differences in this specific case due to the high effective dimension of the basal struc- tures of the network (see red and blue curves in Figure 8(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Finally, Figure 8(c) shows the retention time rate for the different gaps of the dendrogram, together with the network division into four modules and the adjacency matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Dorogovtsev-Goltsev-Mendes graph Dorogovtsev, Goltsev, and Mendes [43] have intro- duced a hierarchical scale-free network, in a way reminis- cent of exact fractal lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' In fact, the DGM network is a pseudofractal: it contains subnetworks resembling the whole network, but lacks the affine transformation of scale which characterizes self-similarity in fractals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' As a result, the DGM network has infinite dimensionality, containing numerous loops and hence being very far from tree-like.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' In particular, the average clustering coefficient of the network, for the infinite graph, is C = 4 5, a prop- erty that is suggestive of a modular organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' 10 1210 6 100 / max Node index (a) 10 1 101 103 < > 0 1 1-S (b) L LRW 1 10 20 30 Gap number 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content='5 RTR (c) 0 2 Clog N Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Dorogovtsev-Goltsev-Mendes graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' (a) Nor- malized dendrogram for a DGM network using τ = 1/λmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Red dashed line reflects the division of the network using the second gap of the RTR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Different communities are shaded in different colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' (b) Entropy parameter (dashed lines, (1 − S)), and specific heat (solid lines, C), versus the tem- poral resolution parameter of the network, τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' (c) Retention Time Rate (RTR) versus gap number for τ = τ ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Note ta hat peaks of the RTR are equally high reflecting the precise hier- archical structure of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Insets: Adjacency matrix and division into three communities of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Figure 9(a) illustrates the dendrogram for a specific network and the regular nested nature of the different modules, which present the expected aggregation in the different hierarchical scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' We also present here the comparison of the spectral entropy of the network by the means of ˆL and ˆLRW , evidencing .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content='. (see red and blue curves in Figure 9(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Finally, Figure 9(c) shows the retention time rate for the different gaps of the den- drogram, together with the network division into three modules and the adjacency matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' However, let us re- mark the serious issues presented by the different usual community detection algorithms when dealing with these particular type of hierarchical organization of modularity (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' We test a set of different algorithms that have proven to show an excellent performance: the walk- trap algorithm [80], the Leiden algorithm [81] and In- fomap [82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Despite this, they exhibit a great variability depending on parameters: they do not give an accurate prediction of the different modules in the DGM network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' We propose the LRG as a way to go beyond the previous attempts: this consider all the powers of the Laplacian needed to recover the proper network structure, giving place to a noticeable quantitative improvement on the quality of the network subdivision (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' 9(a) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' 10(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' (a) LRG (b) Walktrap (c) Leiden (c) Infomap Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Community detection methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Comparison of different community detection methods when applied to the DGM network: (a) LRG as stated in the previous exam- ple, (b) Walktrap algorithm (with nsteps = 103), (c) Leiden algorithm (using γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content='004), and (c) Infomap (using 102 trials to perform the network partition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Note the accuracy of the LRG to perform network partitions in this specific case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' 11 Lancichinetti–Fortunato–Radicchi benchmark There is one further –and final– empirical test that we can make to properly apply the LRG by using real- istic benchmarks for community detection that accounts for the heterogeneity of degree and community size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' In particular, the Lancichinetti–Fortunato–Radicchi bench- mark, that considers both the degree and the community size to be distributed as power laws, constitutes a much harder test for algorithms and makes it easier to dis- close their limits (we refer to the original work to further details on the different steps to generate the networks [45]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Again, this class of networks represents a chal- lenging task, even for well-known community detection algorithms [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' In particular, we choose the set of pa- rameters N = 500, τ1 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content='0, τ2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content='5, µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content='1 and fix κmin = 2 and κmax = 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' First, we stress that further analysis of these benchmark networks based on multiple peaks on the entropy, together with an in-depth exam- ination of the parameter space, can also help to under- stand when these network exhibit or not communities, as a function, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=', of the mixing parameter [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' This problem will be tackled elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' 10 2 10 1 100 / max Node index (a) 100 102 , < > 0 1 1-S (b) L LRW 1 10 20 30 Gap number 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content='2 RTR (c) 0 2 Clog N Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Lancichinetti–Fortunato–Radicchi bench- mark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' (a) Normalized dendrogram for a LFR network using τ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Red dashed line reflects the division of the network using the optimal gap of the RTR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Different communities are shaded in different colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' (b) Entropy parameter (dashed lines, (1 − S)), and specific heat (solid lines, C), versus the temporal resolution parameter of the network, τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' (c) Reten- tion Time Rate (RTR) versus gap number for τ = τ ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Insets: Network division into communities as set by the dendrogram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Figure 11 shows that the particular application of the LRG can precisely predict the predefined commu- nity of each node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Note the existence of two peaks when the spectral entropy is analyzed in this case (see Figure 11(b)), thus ensuring the presence of two well- defined network scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' We stress that this benchmark generates a “flat” community structure without hierar- chies [23], fully justifying this feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' In our view, the other significant result is the sharp double-peaked struc- ture that emerges when we use ˆLRW , resulting from the conceived trapping-build algorithm that generates the networks (this is yet another example of different phe- nomenology between ˆL and ˆLRW , as we detail in the following example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Finally, we illustrate the network communities in Figure 11(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Appendix D: The clique-Star network As we have previously shown in the main text, both the ’fluid’ Laplacian and the Laplacian Random Walk, can evidence different emergent communities since the last one encodes the evolution of a Markovian dynamics on top of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' This effect can be better understood on the light on the Clique-Star network illustrated in Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' 10 2 100 102 , 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content='00 1-S L LRW 0 2 4 6 Clog N Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Clique-Star network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Entropy parameter (dashed lines, (1 − S)), and specific heat (solid lines, C), ver- sus the temporal resolution parameter of the network, τ, for both Laplacians, ˆL and ˆLRW .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} +page_content=' Note the explicit differences in the total number of peaks of the specific heat, because of the fact that ˆLRW reflects the trapping time into the two main communities of the network: the star on the one hand, and the clique 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfbQqU/content/2301.04514v1.pdf'} diff --git a/WtFRT4oBgHgl3EQfMzdN/content/tmp_files/2301.13507v1.pdf.txt b/WtFRT4oBgHgl3EQfMzdN/content/tmp_files/2301.13507v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..56e3803aa7fd8ab53988adf12373d4f4463a01f2 --- /dev/null +++ b/WtFRT4oBgHgl3EQfMzdN/content/tmp_files/2301.13507v1.pdf.txt @@ -0,0 +1,458 @@ +arXiv:2301.13507v1 [cs.IR] 31 Jan 2023 +An Analysis of Classification Approaches for Hit +Song Prediction using Engineered Metadata +Features with Lyrics and Audio Features +Mengyisong Zhao1, Morgan Harvey1, David Cameron1, Frank Hopfgartner2, +and Valerie J. Gillet1 +1 The University of Sheffield, Sheffield S10 2TN, UK mzhao18@sheffield.ac.uk, +m.harvey@sheffield.ac.uk, d.s.cameron@sheffield.ac.uk, +v.gillet@sheffield.ac.uk +2 Universit¨at Koblenz, 56072 Koblenz, Germany hopfgartner@uni-koblenz.de +Abstract. Hit song prediction, one of the emerging fields in music in- +formation retrieval (MIR), remains a considerable challenge. Being able +to understand what makes a given song a hit is clearly beneficial to the +whole music industry. Previous approaches to hit song prediction have +focused on using audio features of a record. This study aims to improve +the prediction result of the top 10 hits among Billboard Hot 100 songs +using more alternative metadata, including song audio features provided +by Spotify, song lyrics, and novel metadata-based features (title topic, +popularity continuity and genre class). Five machine learning approaches +are applied, including: k-nearest neighbours, Na¨ıve Bayes, Random For- +est, Logistic Regression and Multilayer Perceptron. Our results show +that Random Forest (RF) and Logistic Regression (LR) with all features +(including novel features, song audio features and lyrics features) outper- +forms other models, achieving 89.1% and 87.2% accuracy, and 0.91 and +0.93 AUC, respectively. Our findings also demonstrate the utility of our +novel music metadata features, which contributed most to the models’ +discriminative performance. +Keywords: Hit song prediction · Music Information Retrieval · Machine +learning · Text processing. +1 +Introduction +Music labels spend more than $4.5 billion every year discovering new talented +artists and producing popular songs [1]. Precipitated by the growing impor- +tance of online digital music platforms and recent advancements in machine +learning and big data technologies, a new research area called hit song science +has attracted increasing attention [2]. A successful hit song prediction approach +could bring considerable benefits to many music lifecycle stakeholders. Early hit +song prediction studies illustrate the complexity of this problem, delivering only +weak classification results [3,4,5,6]. In recent years, more advanced approaches +have been able to accurately predict hits and non-hits using audio features + +2 +M. Zhao et al. +[7,8,9,10,11,12,13]; however, many other potentially useful sources of informa- +tion about the songs are also available. In this study, we employ 12 Spotify au- +dio features (energy, liveness, tempo, speechiness, acousticness, time signature, +key, duration ms, loudness, valence, mode and danceability), these features are +drawn directly from Spotify, together with novel features based on Billboard +music metadata (popularity continuity, genre class and title topic), as well as +the topics extracted from the songs’ lyrics to identify Top 10 hits among Top +100 hits. To our knowledge, this work is the first attempt to improve hit song +prediction by extracting features from the topic of song titles and by using a +song’s prior popularity information. We examine the effectiveness of these novel +features together with song audio and lyrics features for hit song prediction +using a variety of machine learning approaches, including k-nearest neighbours +(kNN), Na¨ıve Bayes (NB), Random Forest (RF), Logistic Regression (LR) and +Multilayer Perceptron (MLP). Our findings demonstrate the utility of the new +features and provide state-of-the-art prediction performance, as well as providing +promising avenues for future work in this area. +2 +Related Work +Hit song prediction (HSP) has been investigated frequently in recent decades. +Much seminal work failed to accurately predict hit songs, with some work even +suggesting that popularity was not predictable [4,5]. An early approach by Dha- +naraj and Logan [3] achieved promising results by using a SVM model to classify +top 1 songs through acoustic and lyrics data. However, they provided only scant +details about their data gathering, feature engineering, model training and pa- +rameter optimization procedure and found textual features to be more predic- +tive than audio analysis. Salganik et al.[4], and Pachet and Roy [5] attempted +to reproduce Dhanaraj and Logan’s work but failed to achieve a similar level of +accuracy. +Various algorithms have been applied to tackle this task, among them: Logis- +tic Regression (LR), Support Vector Machine (SVM) and Neural Networks (NN) +are commonly used [3,6,8,11]. Ni et al. [7] gained promising results in predicting +UK Top 5 hits on the Top 40 single song charts, but again little implementation +detail was provided. Fan and Casey [11] used LR and SVM models to predict +British and Chinese hit songs but found that audio features worked better for +predicting Chinese hits than British ones, and that textual features worked best +overall. +Herremans et al. [6] focussed particularly on dance songs and classified hits +using five machine learning models. Their research affirmed the importance of +audio features; however, they achieved relatively poor accuracy results, perhaps +due to their use of a large number of features without performing any feature +selection. Georgieva et al. [8] compared six machine-learning algorithms when +conducting Billboard hit song prediction; the most successful algorithms were +LR and a NN with a single hidden layer. Their work also demonstrated the utility +of Spotify’s audio features for this task. Nasreldin [12] did similar research but + +An Analysis of Classification Approaches for Hit Song Prediction +3 +identified XGBoost as the top performing classifier; in their study the SVM +model performed the worst. As they only use the raw data without any feature +selection, they only achieved accuracy results similar to those of Herremans et +al. [6] +Recently, Zangerle et al. [15] adopted deep neural networks and treat HSP +as a regression task, and their experimental results show that the wide and +deep neural network-based approach performed best, achieving 72.04% accuracy. +However, the common problem with deep neural networks is that their results +are hard to interpret. Essa et al. [16] tried to solve the HIS task by using both +classification and regression models. They considered audio features alone and, +through adopting seven machine learning models, they achieved results suggest- +ing that both machine learning approaches (classification and regression) can be +used for HSP. +Although previous studies have made a large contribution to this topic, it is +still unclear which features can be used to successfully classify hit songs when +including audio features, music metadata and song lyrics, and in what combina- +tion. Audio features have shown promise, but only raw terms have been used to +construct features to date [5,6,7]. Textual features have rarely been adopted in +hit song prediction tasks and, although Singhi and Brown [17] did attempt to +extract 31 song lyrics features and build SVM model to predict hit songs, the +performance achieved was not inspiring. +3 +Data and Methodology +3.1 +Data collection and preprocessing +To investigate hit song prediction, we obtained Billboard hot 100 songs data from +the open-source platform data.world named “Billboard Hot-100 Songs 2000-2018 +w/Spotify Data+Lyric”3 . The dataset includes all songs in the Billboard hot +100 weekly charts from 2007 to 2017, as well as audio features, metadata and +lyrics of each song provided by Spotify. The raw dataset includes 33 attributes +in total. We firstly remove the irrelevant features (e.g., spotify link, video link, +analysis url). Then, we define “hits” in this context to be songs whose highest +position in the Billboard Hot 100 list was at rank 10 or above to produce a binary +classification of “hit” (1) that at some point reached the Top 10 or “not-hit” (0) +that never reached the Top 10. The features used in this study include those +engineered based on metadata (e.g., weeks, song title, music genre), 12 Spotify +audio features, as well as lyrics of each song. 273 songs had missing audio features +data and/or lyrics, and were subsequently removed as it would not be possible +to extrapolate or estimate such features. This left 3581 unique songs in the final +data set: 507 hits and 3074 non-hits. +3 data.world/typhon/billboard-hot-100-songs-2000-2018-w-spotify-data-lyrics + +4 +M. Zhao et al. +3.2 +Feature Engineering +We engineered several additional features to augment the existing metadata fea- +tures from the original Billboard data and the Spotify audio features. Popularity +continuity was created to represent the sum of each song’s popular duration (i.e., +how many weeks it had already been listed in the hot 100 chart prior to the week +of interest). Songs already present in the chart for more than 50 weeks were as- +signed a 3; those present for between 20 and 50 were assigned to 2; those between +10 and 20 were assigned 1; otherwise, a song was assigned 0. Unlike classical mu- +sic, popular music has relatively rapid iterations [19]. The majority of songs only +remain in the chart for a short period, typically less than 20 weeks. Therefore, +we assign a number based on 3 duration splits where the assigned numbers are +only based on weekly duration data. The song title topic feature was created +based on the song title. We removed symbols, punctuation, short terms (i.e., +fewer than 4 chars) and stopwords from the data, then, inspired by [3], used +a bagofwords representation with Latent Dirichlet Allocation (LDA) to extract +topics from the song titles. In total, ten topics were extracted, and each song +was assigned to the topic number with the highest probability for that song in +θ. The numerical variable named genre class was created to replace the existing +string variable broad genre in which each genre was assigned a numerical value: +1 to 6 representing country, electronic dance music (edm), pop, r&b, rock and +rap music respectively. We treat song lyrics similar to how we treat song titles, +the only difference being the number of topics: 20 topics were extracted from +the lyrics. This is because lyrics are far longer than titles, thus providing suffi- +cient data to extract a larger number of more meaningful topics. Each song was +assigned to the topic number with the highest probability for that song in θ. +3.3 +Training Environment +Min-max normalization method was applied to accelerate the algorithm con- +vergence speed [16]. After preprocessing and feature engineering, a total of 16 +features were used for model building. We treat each song as an individual, tem- +poral factors were not considered in our experiment, the data were split into +training and testing sets using a ratio of 80:20 and, due to the relatively small +size of the overall data set, 5-fold cross validation was applied instead of an in- +dividual validation set. Due to the highly imbalanced classes (i.e., most songs +are not top-10 hits), Synthetic Minority Over-Sampling (SMOTE) was adopted +inspired by Chawla et al. [18], which could effectively increase the accuracy of +minority class (hit song) prediction. In this paper, 5 nearest neighbours have +currently used to over-sampling the minority class (hit songs). resulting in a +final training set of 4918 songs (including hits 2459 and 2459 non-hits). For- +ward feature selection was carried out. All models were trained and tested using +KNIME 4.4.04. +4 https://www.knime.com/ + +An Analysis of Classification Approaches for Hit Song Prediction +5 +3.4 +Model Setup and Optimisation +This study examined five commonly-used machine learning approaches from the +prior literature, and all the model parameter tuning has been using 5-fold cross +validation. The model hyperparameter and their optimum values are shown in +Table 1. +k-Nearest Neighbour (kNN). We tested values of k between 1 to 20 to seek for +more appropriate neighborhood distance when predicting hit songs, and when +we tuned the hyperparameter to k = 1 has achieved most effective accuracy. +Na¨ıve Bayes (NB). We tested the default probability from 0.001 to 1 every +0.01, the best setting was default probability = 0.031. +Random Forest (RF). When using RF to train our model, the different split +criterion algorithm provides varied performance, which includes information +gain, information gain ratio, and Gini index. We tested the number of mod- +els of all algorithms from 50 to 1000 every 50, and the Gini index achieved best +performance at 600 numbers of models. +Logistic Regression (LR). We tested four ways to solve the equation, itera- +tively reweighted least squares with Gauss, iteratively reweighted least squares +with Laplace, stochastic average gradient with Gauss, stochastic average gra- +dient with Laplace. We find out using iteratively reweighted least squares with +Laplace regularization to solve the equation is more effective. The Laplace equals +to 3 has been accepted as best performance. +Neural Network (NN). Multilayer perceptron (MLP) model consisting of an +input layer, a hidden layer, and an output layer has been conducted in this study. +We tested the Maximum number of iterations from 500 to 5000 with 500 stop +sizes, Number of hidden layers and Number of hidden neurons per layer from +1 to 25 was measured every 3. The best parameter tuning result is 4500, 4, 22, +respectively. +Table 1. All model hyperparameter tuning optimization value. +Classifier Hyperparameter +Value +kNN +K value +1 +NB +Default probability +0.031 +RF +Gini index: number of models +600 +LR +Laplace +3 +NN +Maximum number of iterations +4500 +Number of hidden layers +4 +Number of hidden neurons per layer 22 +4 +Findings, Results and Limitations +Our results include an analysis of accuracy, as well as AUC and the number of +features used as a measure of parsimony (see Table 2 and Table 3). We compare +models trained using all features, including our novel engineered ones, audio fea- +tures and lyrics features together, against three “baseline” models, audio features + +6 +M. Zhao et al. +alone, audio features and original metadata features, as well as novel features +and audio features model. It is notable that Random Forest (Accuracy=89.1%, +AUC=0.91) and Logistic Regression (Accuracy=87.2%, AUC=0.93) with all fea- +tures performed best according to both metrics. Logistic Regression with Laplace +regularisation achieves the best AUC score while only using 4 features. Accord- +ing to Han et al. [20], the reason L1 regularisation is more appropriate to this +task could be it capable of reduce the coefficients of some features to zero and +generate a spare solution. Random Forest achieved the best accuracy result, but +required seven features to train the model, which leads to longer training times, +and poorer explainability. MLP shows average performance in this task; this +model requires a maximum number of features according to Table 2, and longest +training time to achieve the best result, perhaps because the volume of the data +available is insufficient to train the network well. Na¨ıve Bayes performs worst on +accuracy, but better on AUC score, which means this model has great ability on +identifying hits but weak on identifying non-hits. +Table 2. Features selected for each model. +Classifier Accepted Feature Combination +kNN +popularity continuity1, song title topic, genre class, energy, liveness, key, lyrics topic2 +NB +popularity continuity, genre class, key, loudness +RF +popularity continuity, genre class, song title topic, key, valence, energy, lyrics topic +LR +popularity continuity, genre class, lyrics topic, danceability +NN +popularity continuity, genre class, key, song title topic, lyrics topic, acousticness, liveness, tempo, danceability +1 Novel features are marked in italics. +2 Lyrics feature is marked in bold. +Table 3. All model training and test results summarisation and comparison. +5-fold CV Accuracy 5-fold CV AUC Model Test Accuracy Model Test AUC +KNN (Audio) +83.98% +0.847 +79.92% +0.530 +KNN (Metadata+audio) +90.85% +0.917 +82.08% +0.748 +KNN (NFE1+audio) +93.79% +0.930 +86.05% +0.7752 +KNN (NFE+audio+lyrics) 94.31% +0.937 +86.38% +0.745 +NB (Audio) +62.71% +0.697 +42.82% +0.609 +NB (Metadata+audio) +82.78% +0.915 +71.13% +0.899 +NB (NFE+audio) +86.26% +0.924 +74.76% +0.922 +NB (NFE+audio+lyrics) +86.23% +0.931 +78.52% +0.900 +RF (Audio) +79.22% +0.876 +71.13% +0.629 +RF (Metadata+audio) +91.62% +0.977 +74.76% +0.869 +RF (NFE+audio) +93.84% +0.980 +87.59% +0.908 +RF (NFE+audio+lyrics) +95.1% +0.989 +89.12% +0.912 +LR (Audio) +61.26% +0.649 +57.88% +0.603 +LR (Metadata+audio) +84.83% +0.917 +83.54% +0.927 +LR (NFE+audio) +86.15% +0.928 +86.47% +0.923 +LR (NFE+audio+lyrics) +87.07% +0.933 +87.17% +0.927 +MLP (Audio) +68.20% +0.756 +63.60% +0.563 +MLP (Metadata+audio) +87.48% +0.923 +76.85% +0.847 +MLP (NFE+audio) +88.0% +0.929 +79.36% +0.734 +MLP (NFE+audio+lyrics) 90.04% +0.931 +84.66% +0.808 +1 NFE stands for abbreviation of novel feature engineering. +2 The best performance has been marked in bold. + +An Analysis of Classification Approaches for Hit Song Prediction +7 +Compared to the baseline method, all the model test accuracy results with +our novel metadata features provided significant performance improvement seen +in Table 3, which proved our novel metadata features have contributed impact to +HSP task. When adding song lyrics topic features, the accuracy score of all mod- +els are slightly increased, the AUC score of kNN and NB are decreased for .030 +and .022 respectively, probably because the lyrics topic increase the complexity +of features, which might be hard for both algorithm to classify the patterns of +hits and non-hits. The novel variables almost frequently in the list of automat- +ically selected features as shown in Table 2, demonstrating their discriminative +power. The utility of popularity continuity indicates that the longer a song in a +particular genre can maintain a position in the charts, the more likely it is to +become a hit song. Certain topically-coherent sets of terms, such as love, girls, +life, and hearts are more likely to appear in the hits than non-hits, as captured +in the song title topic and lyrics topic feature. Based on the ablation studies, +some of the Spotify audio features such as key, liveness, energy, and danceability +are also important when classifying hit songs but less consistently so than our +novel features and song lyrics feature. The contributed features are varied be- +tween each model. Compared to song title topic, song lyrics feature shows more +contribution when using these two features together to identify hit songs. +The result of this study supports the findings of [6,8,13,14] that music meta- +data, audio features and lyrics can be used to classify hit songs through machine +learning approaches. Adding all features together has achieved the best perfor- +mance of all models. Moreover, we have been able to outperform the baseline +results of [8,9,10], as their work achieved an accuracy score around 60% to 87% +compared to our work, which gave accuracy scores around 79% to 89%. +As future work, we intend to further enrich our models by developing more +features based on, for example, music reviews and social tags. More complex +and granular genre classifications such as different types of music from various +cultures, like Latin music or dance songs from India could be used to extend our +model. Furthermore, a larger dataset covering a longer period will be examined. +As the hit songs identified in our study can be defined as extremely popular +songs (top 10 among100), the model generalisation ability may need more tests, +particularly adding songs never achieved in billboard top 100. The substance of +a hit song may change over time, and we will consider more complex models that +include temporal aspects to model changes in genres and topical popularity over +time. Other audio-based features could also be considered, such as Mel-frequency +Cepstral Coefficients (MFCC) and compared with the Spotify audio features. +References +1. IFPI Global Music Report Homepage, http://www.ifpicr.cz/ifpi-global-music-report-2016/. +last accessed 21 Nov 2022 +2. Greenberg, D.M., and Rentfrow, P.J.: Music and big data: a new frontier. Current +opinion in behavioral sciences 18, 50-56 (2017) +3. Dhanaraj, R, and Logan, B.: Automatic Prediction of Hit Songs. In ISMIR, 488-491 +(2005) + +8 +M. Zhao et al. +4. Salganik, M.J., Dodds, P.S. and Watts, J.D.: Experimental study of inequality and +unpredictability in an artificial cultural market. science 311, 5762, 854-856 (2006) +5. Pachet, F., and Roy, P.: Hit Song Science Is Not Yet a Science. In ISMIR, 355-360 +(2008) +6. 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Song Popularity Predictor Homepage, https://towardsdatascience.com/song-popularity-predictor-1ef69735e380. +last accessed 17th October 2021 +13. Kawawa-Beaudan, J. and Garza, G.: Predicting Billboard Top 100 Songs (2015) +14. Borg, N. and Hokkanen, G.: What makes for a hit pop song? What makes for a +pop song. Unpublished thesis, Stanford University, California, USA (2011) +15. Zangerle, E., V¨otter, M., Huber, R. and Yang, Y. H.: Hit Song Prediction: Lever- +aging Low-and High-Level Audio Features. In ISMIR 319-326 (2019) +16. Essa, Y., Usman, A., Garg, T. and Singh, M. K.: Predicting the Song Popularity +Using Machine Learning Algorithm (2022) +17. Singhi, A. and Brown, D. G.: Can song lyrics predict hits. In Proceedings of the +11th International Symposium on Computer Music Multidisciplinary Research 457- +471 (2015) +18. Chawla, N. V., Bowyer, K. W., Hall, L. O. and Kegelmeyer, W. P.: SMOTE: +synthetic minority over-sampling technique. 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Han, J., Kamber, M. and Pei, J.: Data Mining: Concepts and Techniques (3rd ed.). +Elsevier Inc (2012) + diff --git a/WtFRT4oBgHgl3EQfMzdN/content/tmp_files/load_file.txt b/WtFRT4oBgHgl3EQfMzdN/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9dec577347778e25ec33048a4aaa3761fdaf0b5d --- /dev/null +++ b/WtFRT4oBgHgl3EQfMzdN/content/tmp_files/load_file.txt @@ -0,0 +1,409 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf,len=408 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content='13507v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content='IR] 31 Jan 2023 An Analysis of Classification Approaches for Hit Song Prediction using Engineered Metadata Features with Lyrics and Audio Features Mengyisong Zhao1, Morgan Harvey1, David Cameron1, Frank Hopfgartner2, and Valerie J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' Gillet1 1 The University of Sheffield, Sheffield S10 2TN, UK mzhao18@sheffield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content='uk, m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content='harvey@sheffield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content='uk, d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content='cameron@sheffield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content='uk, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content='gillet@sheffield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content='uk 2 Universit¨at Koblenz, 56072 Koblenz, Germany hopfgartner@uni-koblenz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content='de Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' Hit song prediction, one of the emerging fields in music in- formation retrieval (MIR), remains a considerable challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' Being able to understand what makes a given song a hit is clearly beneficial to the whole music industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' Previous approaches to hit song prediction have focused on using audio features of a record.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' This study aims to improve the prediction result of the top 10 hits among Billboard Hot 100 songs using more alternative metadata, including song audio features provided by Spotify, song lyrics, and novel metadata-based features (title topic, popularity continuity and genre class).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' Five machine learning approaches are applied, including: k-nearest neighbours, Na¨ıve Bayes, Random For- est, Logistic Regression and Multilayer Perceptron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' Our results show that Random Forest (RF) and Logistic Regression (LR) with all features (including novel features, song audio features and lyrics features) outper- forms other models, achieving 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content='1% and 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content='2% accuracy, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content='91 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content='93 AUC, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' Our findings also demonstrate the utility of our novel music metadata features, which contributed most to the models’ discriminative performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' Keywords: Hit song prediction · Music Information Retrieval · Machine learning · Text processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' 1 Introduction Music labels spend more than $4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content='5 billion every year discovering new talented artists and producing popular songs [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' Precipitated by the growing impor- tance of online digital music platforms and recent advancements in machine learning and big data technologies, a new research area called hit song science has attracted increasing attention [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' A successful hit song prediction approach could bring considerable benefits to many music lifecycle stakeholders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' Early hit song prediction studies illustrate the complexity of this problem, delivering only weak classification results [3,4,5,6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' In recent years, more advanced approaches have been able to accurately predict hits and non-hits using audio features 2 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' [7,8,9,10,11,12,13];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' however, many other potentially useful sources of informa- tion about the songs are also available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' In this study, we employ 12 Spotify au- dio features (energy, liveness, tempo, speechiness, acousticness, time signature, key, duration ms, loudness, valence, mode and danceability), these features are drawn directly from Spotify, together with novel features based on Billboard music metadata (popularity continuity, genre class and title topic), as well as the topics extracted from the songs’ lyrics to identify Top 10 hits among Top 100 hits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' To our knowledge, this work is the first attempt to improve hit song prediction by extracting features from the topic of song titles and by using a song’s prior popularity information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' We examine the effectiveness of these novel features together with song audio and lyrics features for hit song prediction using a variety of machine learning approaches, including k-nearest neighbours (kNN), Na¨ıve Bayes (NB), Random Forest (RF), Logistic Regression (LR) and Multilayer Perceptron (MLP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' Our findings demonstrate the utility of the new features and provide state-of-the-art prediction performance, as well as providing promising avenues for future work in this area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' 2 Related Work Hit song prediction (HSP) has been investigated frequently in recent decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' Much seminal work failed to accurately predict hit songs, with some work even suggesting that popularity was not predictable [4,5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' An early approach by Dha- naraj and Logan [3] achieved promising results by using a SVM model to classify top 1 songs through acoustic and lyrics data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' However, they provided only scant details about their data gathering, feature engineering, model training and pa- rameter optimization procedure and found textual features to be more predic- tive than audio analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' Salganik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' [4], and Pachet and Roy [5] attempted to reproduce Dhanaraj and Logan’s work but failed to achieve a similar level of accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' Various algorithms have been applied to tackle this task, among them: Logis- tic Regression (LR), Support Vector Machine (SVM) and Neural Networks (NN) are commonly used [3,6,8,11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' Ni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' [7] gained promising results in predicting UK Top 5 hits on the Top 40 single song charts, but again little implementation detail was provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' Fan and Casey [11] used LR and SVM models to predict British and Chinese hit songs but found that audio features worked better for predicting Chinese hits than British ones, and that textual features worked best overall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' Herremans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' [6] focussed particularly on dance songs and classified hits using five machine learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' Their research affirmed the importance of audio features;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' however, they achieved relatively poor accuracy results, perhaps due to their use of a large number of features without performing any feature selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' Georgieva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' [8] compared six machine-learning algorithms when conducting Billboard hit song prediction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' the most successful algorithms were LR and a NN with a single hidden layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' Their work also demonstrated the utility of Spotify’s audio features for this task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' Nasreldin [12] did similar research but An Analysis of Classification Approaches for Hit Song Prediction 3 identified XGBoost as the top performing classifier;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' in their study the SVM model performed the worst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' As they only use the raw data without any feature selection, they only achieved accuracy results similar to those of Herremans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' [6] Recently, Zangerle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' [15] adopted deep neural networks and treat HSP as a regression task, and their experimental results show that the wide and deep neural network-based approach performed best, achieving 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content='04% accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' However, the common problem with deep neural networks is that their results are hard to interpret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' Essa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' [16] tried to solve the HIS task by using both classification and regression models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' They considered audio features alone and, through adopting seven machine learning models, they achieved results suggest- ing that both machine learning approaches (classification and regression) can be used for HSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' Although previous studies have made a large contribution to this topic, it is still unclear which features can be used to successfully classify hit songs when including audio features, music metadata and song lyrics, and in what combina- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' Audio features have shown promise, but only raw terms have been used to construct features to date [5,6,7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' Textual features have rarely been adopted in hit song prediction tasks and, although Singhi and Brown [17] did attempt to extract 31 song lyrics features and build SVM model to predict hit songs, the performance achieved was not inspiring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' 3 Data and Methodology 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content='1 Data collection and preprocessing To investigate hit song prediction, we obtained Billboard hot 100 songs data from the open-source platform data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content='world named “Billboard Hot-100 Songs 2000-2018 w/Spotify Data+Lyric”3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' The dataset includes all songs in the Billboard hot 100 weekly charts from 2007 to 2017, as well as audio features, metadata and lyrics of each song provided by Spotify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' The raw dataset includes 33 attributes in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' We firstly remove the irrelevant features (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=', spotify link, video link, analysis url).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' Then, we define “hits” in this context to be songs whose highest position in the Billboard Hot 100 list was at rank 10 or above to produce a binary classification of “hit” (1) that at some point reached the Top 10 or “not-hit” (0) that never reached the Top 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' The features used in this study include those engineered based on metadata (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=', weeks, song title, music genre), 12 Spotify audio features, as well as lyrics of each song.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' 273 songs had missing audio features data and/or lyrics, and were subsequently removed as it would not be possible to extrapolate or estimate such features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' This left 3581 unique songs in the final data set: 507 hits and 3074 non-hits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' 3 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content='world/typhon/billboard-hot-100-songs-2000-2018-w-spotify-data-lyrics 4 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content='2 Feature Engineering We engineered several additional features to augment the existing metadata fea- tures from the original Billboard data and the Spotify audio features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' Popularity continuity was created to represent the sum of each song’s popular duration (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=', how many weeks it had already been listed in the hot 100 chart prior to the week of interest).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' Songs already present in the chart for more than 50 weeks were as- signed a 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' those present for between 20 and 50 were assigned to 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' those between 10 and 20 were assigned 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' otherwise, a song was assigned 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' Unlike classical mu- sic, popular music has relatively rapid iterations [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' The majority of songs only remain in the chart for a short period, typically less than 20 weeks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' Therefore, we assign a number based on 3 duration splits where the assigned numbers are only based on weekly duration data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' The song title topic feature was created based on the song title.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' We removed symbols, punctuation, short terms (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=', fewer than 4 chars) and stopwords from the data, then, inspired by [3], used a bagofwords representation with Latent Dirichlet Allocation (LDA) to extract topics from the song titles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' In total, ten topics were extracted, and each song was assigned to the topic number with the highest probability for that song in θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' The numerical variable named genre class was created to replace the existing string variable broad genre in which each genre was assigned a numerical value: 1 to 6 representing country, electronic dance music (edm), pop, r&b, rock and rap music respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' We treat song lyrics similar to how we treat song titles, the only difference being the number of topics: 20 topics were extracted from the lyrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' This is because lyrics are far longer than titles, thus providing suffi- cient data to extract a larger number of more meaningful topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' Each song was assigned to the topic number with the highest probability for that song in θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content='3 Training Environment Min-max normalization method was applied to accelerate the algorithm con- vergence speed [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' After preprocessing and feature engineering, a total of 16 features were used for model building.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' We treat each song as an individual, tem- poral factors were not considered in our experiment, the data were split into training and testing sets using a ratio of 80:20 and, due to the relatively small size of the overall data set, 5-fold cross validation was applied instead of an in- dividual validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' Due to the highly imbalanced classes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=', most songs are not top-10 hits), Synthetic Minority Over-Sampling (SMOTE) was adopted inspired by Chawla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' [18], which could effectively increase the accuracy of minority class (hit song) prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' In this paper, 5 nearest neighbours have currently used to over-sampling the minority class (hit songs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' resulting in a final training set of 4918 songs (including hits 2459 and 2459 non-hits).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' For- ward feature selection was carried out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' All models were trained and tested using KNIME 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' 4 https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content='knime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content='com/ An Analysis of Classification Approaches for Hit Song Prediction 5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content='4 Model Setup and Optimisation This study examined five commonly-used machine learning approaches from the prior literature, and all the model parameter tuning has been using 5-fold cross validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' The model hyperparameter and their optimum values are shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' k-Nearest Neighbour (kNN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' We tested values of k between 1 to 20 to seek for more appropriate neighborhood distance when predicting hit songs, and when we tuned the hyperparameter to k = 1 has achieved most effective accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' Na¨ıve Bayes (NB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' We tested the default probability from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content='001 to 1 every 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content='01, the best setting was default probability = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content='031.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' Random Forest (RF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' When using RF to train our model, the different split criterion algorithm provides varied performance, which includes information gain, information gain ratio, and Gini index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' We tested the number of mod- els of all algorithms from 50 to 1000 every 50, and the Gini index achieved best performance at 600 numbers of models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' Logistic Regression (LR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' We tested four ways to solve the equation, itera- tively reweighted least squares with Gauss, iteratively reweighted least squares with Laplace, stochastic average gradient with Gauss, stochastic average gra- dient with Laplace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' We find out using iteratively reweighted least squares with Laplace regularization to solve the equation is more effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' The Laplace equals to 3 has been accepted as best performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' Neural Network (NN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' Multilayer perceptron (MLP) model consisting of an input layer, a hidden layer, and an output layer has been conducted in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' We tested the Maximum number of iterations from 500 to 5000 with 500 stop sizes, Number of hidden layers and Number of hidden neurons per layer from 1 to 25 was measured every 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' The best parameter tuning result is 4500, 4, 22, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' All model hyperparameter tuning optimization value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' Classifier Hyperparameter Value kNN K value 1 NB Default probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content='031 RF Gini index: number of models 600 LR Laplace 3 NN Maximum number of iterations 4500 Number of hidden layers 4 Number of hidden neurons per layer 22 4 Findings, Results and Limitations Our results include an analysis of accuracy, as well as AUC and the number of features used as a measure of parsimony (see Table 2 and Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' We compare models trained using all features, including our novel engineered ones, audio fea- tures and lyrics features together, against three “baseline” models, audio features 6 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' alone, audio features and original metadata features, as well as novel features and audio features model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' It is notable that Random Forest (Accuracy=89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content='1%, AUC=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content='91) and Logistic Regression (Accuracy=87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content='2%, AUC=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content='93) with all fea- tures performed best according to both metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' Logistic Regression with Laplace regularisation achieves the best AUC score while only using 4 features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' Accord- ing to Han et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' [20], the reason L1 regularisation is more appropriate to this task could be it capable of reduce the coefficients of some features to zero and generate a spare solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' Random Forest achieved the best accuracy result, but required seven features to train the model, which leads to longer training times, and poorer explainability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' MLP shows average performance in this task;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' this model requires a maximum number of features according to Table 2, and longest training time to achieve the best result, perhaps because the volume of the data available is insufficient to train the network well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' Na¨ıve Bayes performs worst on accuracy, but better on AUC score, which means this model has great ability on identifying hits but weak on identifying non-hits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' Features selected for each model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' Classifier Accepted Feature Combination kNN popularity continuity1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' song title topic,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' genre class,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' energy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' liveness,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' key,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' lyrics topic2 NB popularity continuity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' genre class,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' key,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' loudness RF popularity continuity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' genre class,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' song title topic,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' key,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' valence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' energy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' lyrics topic LR popularity continuity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' genre class,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' lyrics topic,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' danceability NN popularity continuity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' genre class,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' key,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' song title topic,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' lyrics topic,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' acousticness,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' liveness,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' tempo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' danceability 1 Novel features are marked in italics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' 2 Lyrics feature is marked in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' All model training and test results summarisation and comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' 5-fold CV Accuracy 5-fold CV AUC Model Test Accuracy Model Test AUC KNN (Audio) 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content='98% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content='847 79.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content='923 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content='85% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content='847 MLP (NFE+audio) 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content='929 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content='36% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content='734 MLP (NFE+audio+lyrics) 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content='04% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content='931 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content='66% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content='808 1 NFE stands for abbreviation of novel feature engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' 2 The best performance has been marked in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' An Analysis of Classification Approaches for Hit Song Prediction 7 Compared to the baseline method, all the model test accuracy results with our novel metadata features provided significant performance improvement seen in Table 3, which proved our novel metadata features have contributed impact to HSP task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' When adding song lyrics topic features, the accuracy score of all mod- els are slightly increased, the AUC score of kNN and NB are decreased for .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content='030 and .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content='022 respectively, probably because the lyrics topic increase the complexity of features, which might be hard for both algorithm to classify the patterns of hits and non-hits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' The novel variables almost frequently in the list of automat- ically selected features as shown in Table 2, demonstrating their discriminative power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' The utility of popularity continuity indicates that the longer a song in a particular genre can maintain a position in the charts, the more likely it is to become a hit song.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' Certain topically-coherent sets of terms, such as love, girls, life, and hearts are more likely to appear in the hits than non-hits, as captured in the song title topic and lyrics topic feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' Based on the ablation studies, some of the Spotify audio features such as key, liveness, energy, and danceability are also important when classifying hit songs but less consistently so than our novel features and song lyrics feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' The contributed features are varied be- tween each model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' Compared to song title topic, song lyrics feature shows more contribution when using these two features together to identify hit songs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' The result of this study supports the findings of [6,8,13,14] that music meta- data, audio features and lyrics can be used to classify hit songs through machine learning approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' Adding all features together has achieved the best perfor- mance of all models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' Moreover, we have been able to outperform the baseline results of [8,9,10], as their work achieved an accuracy score around 60% to 87% compared to our work, which gave accuracy scores around 79% to 89%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' As future work, we intend to further enrich our models by developing more features based on, for example, music reviews and social tags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' More complex and granular genre classifications such as different types of music from various cultures, like Latin music or dance songs from India could be used to extend our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' Furthermore, a larger dataset covering a longer period will be examined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' As the hit songs identified in our study can be defined as extremely popular songs (top 10 among100), the model generalisation ability may need more tests, particularly adding songs never achieved in billboard top 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' The substance of a hit song may change over time, and we will consider more complex models that include temporal aspects to model changes in genres and topical popularity over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' Other audio-based features could also be considered, such as Mel-frequency Cepstral Coefficients (MFCC) and compared with the Spotify audio features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' IFPI Global Music Report Homepage, http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content='ifpicr.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=': Data Mining: Concepts and Techniques (3rd ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} +page_content=' Elsevier Inc (2012)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFRT4oBgHgl3EQfMzdN/content/2301.13507v1.pdf'} diff --git a/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf b/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..b31dbebf9b9c0235379aa2292ce8a3eeb0b36d11 --- /dev/null +++ b/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:998973073ef42a430b24316e83f398cb9de509e4ddd7b70b90f644c715556ce0 +size 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b/XNAzT4oBgHgl3EQfmf09/content/tmp_files/2301.01563v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a9be739bdd7007b699200ef1e6d3f1d9db377d5d --- /dev/null +++ b/XNAzT4oBgHgl3EQfmf09/content/tmp_files/2301.01563v1.pdf.txt @@ -0,0 +1,1849 @@ +A Posterior Error Estimator for Mixed Interior Penalty Discontinuous +Galerkin Finite Element Method for the H(curl)-Elliptic Problems +Ming Tanga, Xiaoqing Xinga,∗, Liuqiang Zhonga +aSchool of Mathematical Sciences, South China Normal University, Guangzhou 510631, China +Abstract +In this paper, we design the first residual type a posteriori error estimator for mixed interior penalty +discontinuous Galerkin method for the H(curl)-elliptic problems. Then we prove that our residual +based a posteriori error indicator is both reliable and efficient. At last, we present some numeri- +cal experiments to validate the performance of the indicator within an adaptive mesh refinement +procedure. +Keywords: +H(curl)-elliptic problems, mixed interior penalty discontinuous Galerkin method, a +posterior error estimator, reliability, efficiency. +1. Introduction +In this work, we consider the H(curl)-elliptic problems as follows: find the electric or magnetic +field u satisfy +curl(αcurl u) + βu = f, +in Ω, +(1.1) +u · t = 0, +on ∂Ω, +(1.2) +where Ω ⊂ R2 be a simply connected bounded Lipschitz polygon with boundary ∂Ω and is partitioned +into non-overlapping subdomains Ωi, 1 ≤ i ≤ m, f is a given vector field depending on a given +external source field, t is the unit tangent on ∂Ω oriented counter-clockwisely, α ≥ α0 > 0 and +β ≥ β0 > 0 are piecewise constants in Ωi, α0 and β0 are constans. +We recall that, curl v = +∂v2/∂x−∂v1/∂y for a vector field v = (v1, v2), while curl φ = (∂φ/∂y, −∂φ/∂x) for a scalar function +φ. Our numerical scheme and the a posteriori error analysis are based on a mixed formulation of +(1.1)-(1.2), which is obtained by introducing an auxiliary variable p = curl u +curl (αp) + βu = f, +in Ω, +(1.3) +p − curl u = 0, +in Ω, +(1.4) +∗Corresponding author +Email addresses: mingtang@m.scnu.edu.cn (Ming Tang), xingxq@scnu.edu.cn (Xiaoqing Xing), +zhong@scnu.edu.cn (Liuqiang Zhong) +Preprint submitted to Elsevier +January 5, 2023 +arXiv:2301.01563v1 [math.NA] 4 Jan 2023 + +u · t = 0, +on ∂Ω. +(1.5) +Discontinuous Galerkin (DG) finite element method is one of popular methods for numerical +solution of partial differential equations. Compared with the traditional conforming finite element +method, the DG finite element method has advantages as follows: to allow incompatible with sus- +pension point grid, to deal with complex boundary and interface problems easily, and to implement +partial encryption and each unit of polynomial independent selection easily. One of the key fea- +tures of the DG method is that the discontinuous approximation at element interfaces naturally +allows jump discontinuities in the solution if element boundaries are placed along them [20]. DG +method has been developed to solve many equations, such as elliptic problems [3], parabolic equa- +tions [23], advection-diffusion-reaction problems [17]. The DG methods include locally DG(LDG) +method [7], interior penalty DG(IPDG) method [1]. The discontinuous finite element method for +H(curl)−elliptic problems is still in its infancy. Chung and Kim [10] proposed an improved Feti-DP +algorithm and convergence analysis for the mixed interleaved discontinuous finite element method +for the two-dimensional H(curl)−elliptic problems. +On the other hand, in practical engineering applications and scientific calculations, there are +many factors that may cause strong singularities in the propagation of electromagnetic fields. For +example, the material coefficient of the medium in the electromagnetic wave propagation area is +discontinuous, or the source term of the generated electromagnetic field is not smooth [12, 13]. +Although these singularities can be overcomed by uniformly densifying the grid when performing +numerical solutions, consistent densification can lead to a sharp increase in computational cost. +Hence, adaptive finite element emerges as the times require. In the past few decades, adaptive finite +element method have been proven to be a useful and effective tool in scientific computing. The +standard adaptative process is as follows SOLVE → ESTIMATE → MARK → REFINE. The +adaptive finite element method is based on a posteriori error estimation. It automatically refines +and optimizes mesh generation according to the local posteriori error indicator on the element. It +is a numerical calculation method with high reliability and efficiency. +Most of the work on the convergence of the adaptive method for the H(curl)-elliptic equations +focuses on the edge finite element. For example, using the so-called interior node property and +oscillation marker as technical assumptions, the convergence of the lowest order edge elements of the +N´ed´elec’s first family of adaptive for two-dimensional and three-dimensional eddy current equations +are proved in [4,14], respectively. Chen, Xu and Zou [8] proved that an adaptive method for three +dimensional static Maxwell equations without additional marking of oscillation terms and gives +corresponding proof of convergence with the lowest order edge elements of N´ed´elec’s first family. +Zhong, Shu, Chen and Xu [29] proved that the three-dimensional H(curl)-elliptic problem with +variable coefficients is convergent by using high order and the two family of N´ed´elec edge elements. +There are also some studies on the posteriori error estimator of the adaptive DG finite element +method for H(curl)-elliptic problem [16, 26]. Houston, Perugia and Schotzau [16] gave the residual- +2 + +type posteriori error estimator and proved the reliability and efficiency of the error estimator. Xing +and Zhong [26] gave a simplified posteriori error indicator and proved corresponding upper bound. +Recently, Zhong, Chen and Xing [28] proved the convergence of the adaptive interior penalty DG +methods. +Meanwhile, there are many successful works of solving the Maxwell’s equations by the mixed +finite element method, e.g. [18,19,21,15]. However, For adaptive mixed finite element method solving +Maxwell’s equations, there are only few research results for a posterior error estimator. For example, +Carstensen, Hoppe, Sharma and Warburton [6] studied a posteriori error estimation of the hybridized +finite element method and proved the reliability of the estimator up to a consistency error. Chung, +Yuen and Zhong [11] studied a posteriori error estimation of the staggered discontinuous Galerkin +method for time-harmonic Maxwell’s equations and proved that residual based a posteriori error +indicator is both reliable and efficient. As far as we know, there are not any published literatures +on the posteriori error estimation of the adaptive mixed finite element method for H(curl)−elliptic +problems (1.1)-(1.2). The main idea of the manuscript comes from [11]. However, one of main tool, +a Cl´ement-type quasi-interpolation operator given by [24], can not be used for 2D finite element +space. Here, we use the Helmholtz decomposition and operators in articles [25, 4] for estimation. +Here is some notation used throughout the paper. The following shorthand notation will be used +to avoid the repeated constants, following [27], x ≲ y and x ≈ y means x ≤ C1y and C2x ≤ y ≤ C3x, +where C1, C2 and C3 are generic positive constants. +The rest of the article is organized as follows. In Section 2, we introduce some basic notations, +present the variational form of the model problem (1.1)-(1.2), and design a residual type a posteriori +error estimator. In Section 3 and Section 4, we show that this indicator is reliable and effective, +respectively. In Section 5, we report some numerical results in support of theoretical results. +2. Mixed IPDG method and a posteriori error indicator +In this section, we give the continuous variational problem, the discrete variational problem of +mixed IPDG method, and the definition of the a posteriori error indicator. +2.1. Continuous variational problem +For any domain D ⊂ R2, we use standard definitions for the Sobolev spaces Hs(D) and Hs(D) +of scalar and vector-valued square integrable functions with inner products (·, ·)s,D and associated +norms ∥·∥s,D for s ≥ 0, respectively. We refer to L2(D) and L2(D) as the Hilbert spaces of scalar and +vector-valued square integrable functions with inner products (·, ·)0,D and associated norms ∥ · ∥0,D, +respectively. For simplicity, we drop the subscript when G = D. Then, the spaces are defined by +H(curl, Ω) := +� +v : v ∈ L2(Ω), curl v ∈ L2(Ω) +� +, +H0(curl, Ω) := {v : v ∈ H(curl, Ω), v · t = 0 on ∂Ω} . +3 + +The space H(curl, Ω) is equipped with norm ∥v∥2 +curl,Ω := ∥v∥2 +0,Ω+∥curl v∥2 +0,Ω for any v ∈ H(curl, Ω). +We simplify the symbols H0(curl, Ω) and L2(Ω) to U and Q, respectively. In this manuscript, we +assume that f ∈ H(div, Ω) = {v : v ∈ L2(Ω), ∇ · v ∈ L2(Ω)}. The variational form for (1.3)-(1.5) +is to find (u, p) ∈ U × Q such that +a(p, q) − b(u, q) = ℓ1(q), +∀q ∈ Q, +(2.6) +d(v, p) + c(u, v) = ℓ2(v), ∀v ∈ U, +(2.7) +where the four bilinear forms given by +a(p, q) := (p, q), +(2.8) +b(u, q) := (curl u, q), +(2.9) +c(u, v) := (βu, v), +(2.10) +d(v, p) := (curl v, αp), +(2.11) +and two linear functionals ℓ1(·) ∈ Q∗, ℓ2(·) ∈ U ∗, where Q∗ and U ∗ are the dual spaces of Q and +U, respectively, as follows +ℓ1(q) := 0, +(2.12) +ℓ2(v) := (f, v). +(2.13) +In order to prove the well-posedness of continuous variational problem (2.6)-(2.7) and the relia- +bility of a posteriori error indicator(see Lemma 3.1). We also define the operator A : (U × Q) �→ +(U × Q)∗ by +(A(u, p))(v, q) := a(p, q) − b(u, q) + d(v, p) + c(u, v), +for all u, v ∈ U, p, q ∈ Q. +Thus, the operator form of the equations (2.6)-(2.7) is obtained +(A(u, p))(v, q) = ℓ(v, q), +(2.14) +where ℓ(v, q) = ℓ2(v) + ℓ1(q). +The following lemma provides the existence and uniqueness of solutions to the variational problem +(2.6)-(2.7). +Lemma 2.1 ([11], Lemma 2.1). Let Ω be a bounded Lipschitz polygon with connected boundary ∂Ω. +Then A is a continuous and bijective linear operator. Moreover, for any (ℓ1, ℓ2) ∈ Q∗ × U ∗ given +by (2.16) and (2.17), respectively, then the system (2.6)-(2.7) has a unique solution (u, p) ∈ U × Q +such that +∥(u, p)∥U×Q := +� +∥u∥2 +U + ∥p∥2 +Q +�1/2 ≲ ∥ℓ1∥Q∗ + ∥ℓ2∥U ∗, +(2.15) +where ∥ · ∥Q∗ and ∥ · ∥U ∗ are dual norms in Q∗ and U ∗, respectively. +4 + +2.2. Discrete variational problem +Before presenting the discrete variational problem, we introduce some preliminaries. Given a +shape-regular triangulation Th for Ω. For τ ∈ Th, we write hτ = |τ|1/2 to denote the local mesh size +of the element τ, where |τ| is the Lebesgue measure of τ. Let h = maxτ∈Th hτ. +Let Eh be the set of all the edges, E0 +h = Eh\∂Ω be the set of all the interior edges, and E∂ +h = Eh∩∂Ω +be the set of all the boundary edges, then Eh = E0 +h +� E∂ +h. +For T ′ +h ⊆ Th and E′ +h ⊆ Eh,the discrete L2 inner product and norm are given by +(v, w)T ′ +h = +� +τ∈T ′ +h +(v, w)τ = +� +τ∈T ′ +h +� +τ +v · wdx, +∥v∥2 +T ′ +h = (v, v)T ′ +h, +⟨v, w⟩E′ +h = +� +e∈E′ +h +⟨v, w⟩e = +� +e∈E′ +h +� +e +v · wds, +∥v∥2 +E′ +h = ⟨v, v⟩E′ +h. +For any e ∈ E0 +h with e = ∂τ1 ∩ ∂τ2, we define the average, tangential jump and normal jump for +a vector function w by +{{w}}e = (w|τ1 + w|τ2)/2, +[[w]]e = w|τ1 · t1 + w|τ2 · t2, +[w]e = w|τ1 · n1 + w|τ2 · n2, +where w|τi denotes the value of w on τi, ti and ni are the unit tangential vectors and the outward +unit normal vectors on e for τi (i = 1, 2), respectively. +Similarly, we define the average and the tangential jump on e for a scalar function φ as +{{φ}}e = (φ|τ1 + φ|τ2)/2, [[φ]]e = φ|τ1 t1 + φ|τ2 t2, +where φ|τi denotes the value of φ on τi, i = 1, 2. +For any e ∈ E∂ +h, there is a element τ ∈ Th such that e ∈ ∂τ ∩∂Ω, we define the average, tangential +jump and normal jump for a vector function w are defined as +{{w}}e = w|τ, [[w]]e = w|τ · t, [w]e = w|τ · n +where w|τ denotes the value of w on τ and n denotes the outward unit normal vectors on e for τ. +For a scalar function φ, its average and tangential jump on e are defined as +{{φ}}e = φ|τ, [[φ]]e = φ|τt, +where φ|τ denote the value of φ on τ. +The DG methods are based on the approximation of the vector field u and p by elementwise +polynomials, thus giving rise to the finite dimensional function spaces +U h := +� +vh ∈ L2(Ω) | vh|τ ∈ R1(τ), vh|e = 0, ∀τ ∈ Th +� +, +5 + +Qh := +� +qh ∈ L2(Ω) | qh|τ ∈ P0(τ), ∀τ ∈ Th +� +, +where R1(τ) = +� +∃α ∈ R2, ∃β ∈ R, ∀x = (x1, x2) ∈ τ : q(x) = α + β (−x2, x1) +� +, and P0(τ) denotes +the constant in τ. +Now, we present the mixed interior penalty discontinuous Galerkin(MIPDG) finite element +method for the system (1.3)-(1.4): find (uh, ph) ∈ U h × Qh such that +ah(ph, qh) − bh(uh, qh) += +ℓ1,h(qh) + d1,h(uh, qh), +∀qh ∈ Qh, +(2.16) +dh(vh, ph) + ch(uh, vh) += +ℓ2,h(vh) + d2,h(uh, vh), +∀vh ∈ U h, +(2.17) +where +ah(ph, qh) := (ph, qh)Th, +(2.18) +bh(uh, qh) := (curlh uh, qh)Th, +(2.19) +ch(uh, vh) := (βuh, vh)Th, +(2.20) +dh(vh, ph) := (curlh vh, αph)Th, +(2.21) +ℓ1,h(qh) := 0, +(2.22) +ℓ2,h(vh) := (f, vh)Th, +(2.23) +d1,h (uh, qh) := − < {{qh}}, [[uh]] >Eh, +(2.24) +d2,h (uh, vh) :=< {{αcurlh uh}} − κh−1 +e [[uh]], [[vh]] >Eh, +(2.25) +with κ > 0 is a penalty parameter and should be taken large enough. +Remark 2.1. +• Comparing with the continuous variational problem (2.6)-(2.7) and the discrete +variational problem (2.16)-(2.17), the definitions of the bilinear terms, which without including +curlh, are the same. In order to be consistent with other symbols, we add the subscript h to +the bilinear terms in the discrete variational form. +• The calculation of curlh in the bilinear terms of the discrete variational problem is piecewise +derivation. +• Compared with the continuous variational form, the discrete variational form adds two terms +d1,h and d2,h. +In order to give the well-posedness of the discrete variational problems, we need to introduce the +suitable IPDG form of the H(curl)−elliptic problems: find uh ∈ U h, such that +aIP (uh, vh) = (f, vh)Th , +(2.26) +where +aIP (uh, vh) += +(βuh, vh)Th + (αcurl uh, curl vh)Th − < {{curl vh}}, [[αuh]] >Eh +− < {{αcurl uh}}, [[vh]] >Eh +κ < h−1 +e [[uh]], [[vh]] >Eh . +(2.27) +6 + +Remark 2.2. Let qh = αcurl vh in (2.16)-(2.17), and subtract (2.16) from (2.17) then lead to +(2.26). +To provide the existence and uniqueness of solutions to the variational problem (2.26), we need +to introduce the following norm. +|||vh|||2 +h = ∥curl vh∥2 +Th + ∥vh∥2 +Th + κ∥h +− 1 +2 +e +[[vh]]∥2 +Eh, +∀vh ∈ (H1 (Th))2, κ > 0. +Similar to [2], by using the Cauchy-Schwarz inequality, trace inequality and inverse inequality, it +is easy to verify that ah(·, ·) is bounded by ∥| · |∥h, i.e., +ah (wh, vh) ⩽ C∥|wh|∥h∥|vh|∥h, +∀wh, vh ∈ Vh. +(2.28) +Furthermore, for the coercivity of the bilinear forms ah(·.·) on Vh, we have +ah (vh, vh) ⩾ C∥|vh|∥2 +h, +∀vh ∈ Vh. +(2.29) +Combining (2.28) and (2.29), we obtain the well-posedness of the discrete variational problem (2.26). +Furthermore, Lemma 2.2 shows that the variational problem (2.16)-(2.17) and the variational +problem (2.26) have equivalent form. The proof use similar arguments in [5] and is skipped here. +Lemma 2.2. If (uh, ph) ∈ (U h, Qh) is the solution of equation (2.16)-(2.17), then uh ∈ U h is +the solution of the variational problem (2.26). On the contrary, if uh ∈ U h is the solution of the +variational problem (2.26), then there is a corresponding ph ∈ Qh makes (uh, ph) ∈ (U h, Qh) is the +solution of (2.16)-(2.17). +2.3. A posteriori error indicator +For any τ ∈ Th, e ∈ Eh and (vh, qh) ∈ Qh ×U h, we introduce the following element-wise residuals +and edge-wise jump residuals as +R1 (vh, qh) |τ := qh|τ − curlhvh|τ, +R2 (vh, qh) |τ := f|τ − (curlh αqh + βvh) |τ, +R3 (vh) |τ := ∇ · (f − βvh) |τ, +J1 (qh) |e := [[αqh]]e, +J2 (vh) |e := [f − βvh]e, +J3 (vh) |e := [[vh]]e. +The local error estimator on τ ∈ Th is defined as +η2 (vh, qh; τ) :=∥R1 (vh, qh) ∥2 +0,τ + h2 +τ +� +∥R2 (vh, qh) ∥2 +0,τ + ∥R3 (vh) ∥2 +0,τ +� ++ +� +e∈∂τ +he +� +∥J1 (qh) ∥2 +0,e + ∥J2 (vh) ∥2 +0,e +� ++ κ +� +e∈∂τ +h−1 +e +∥J3(vh)∥2 +0,e , +7 + +where hτ denotes the diameter of the element τ. The mesh Th is shape-regular which implies that +hτ ≈ he. +Then the global error estimator on Th is defined as +η2 (vh, qh; Th) = +� +τ∈Th +η2 (vh, qh; τ) . +(2.30) +3. Reliability analysis +For any (v, q) ∈ U × Q and (vh, qh) ∈ U h × Qh, we define the following error +∥(v, q) − (vh, qh)∥2 +DG +:= +∥q − qh∥2 +0,Ω + ∥v − vh∥2 +0,Ω + ∥curlh (v − vh)∥2 +0,Ω ++κ +� +e∈Eh +h−1 +e ∥[[vh]]∥2 +0,e. +(3.31) +Remark 3.1. Here we use [[vh]]e instead of [[v − vh]]e, since [[v]]e = 0 for v ∈ U. +Next, we focus on proving the reliability of the error indicator defined in (2.30). The key of our +argument is to use the space decomposition technique: decompose the DG finite element solution uh +into two parts: one is conforming part uconf +h +∈ U conf +h +:= U h ∩ U and the other is its L2 orthogonal +part u⊥ +h ∈ U ⊥ +h . Therefore, we need to take care of continuous error ∥(u, p) − (uconf +h +, ph)∥DG instead +of ∥(u, p) − (uh, ph)∥DG. The main analysis tools for continuous error are Helmholtz decomposition +and the two interpolations. We prove the reliability of the error indicator. +The following lemmas provide some estimates related to the continuous error. +Lemma 3.1. Let (u, p) ∈ U × Q be solution of system (2.6)-(2.7), then for any (vconf +h +, ph) ∈ +U conf +h +× Qh, we have +∥(u − vconf +h +, p − ph)∥U×Q ≲ ∥˜ℓ1∥Q∗ + ∥˜ℓ2∥U ∗, +(3.32) +where +˜ℓ1(q) = −a (ph, q) + b(vconf +h +, q), ∀q ∈ Q, +(3.33) +˜ℓ2(v) = ℓ2(v) − d (v, ph) − c(vconf +h +, v), ∀v ∈ U. +(3.34) +Proof. For the operator A given by (2.14), it is easy to obtain the linearity, namely, for any q1, q2, q ∈ +Q and v1, v2, v ∈ U, we have +(A (q1 + q2, v1 + v2)) (q, v) = (A (q1, v1)) (q, v) + (A (q2, v2)) (q, v). +Hence, we have +(A(p − ph, u − vconf +h +))(q, v) +8 + += +(A(p, u))(q, v) − (A(ph, vconf +h +))(q, v) += +ℓ2(v) − (a(ph, q) − b(vconf +h +, q) + d(v, ph) + c(vconf +h +, v)) +:= +˜ℓ1(q) + ˜ℓ2(v). +At last, noting that (u − vconf +h +, p − ph) ∈ U × Q and using the definition of the operator norm, this +completes the proof. +In the following lemmas, our purpose is to obtian upper bounds for ∥˜ℓ1∥Q∗ and ∥˜ℓ2∥U ∗ in Lemmas +3.2 and 3.3, respectively. +Lemma 3.2. Let (uh, ph) ∈ U h × Qh be solution of (2.16)-(2.17). For any vconf +h +∈ U conf +h +, we have +∥˜ℓ1∥Q∗ ≲ +� � +τ∈Th +∥R1 (uh, ph) ∥2 +0,τ +�1/2 ++ +� � +τ∈Th +∥curlh(vconf +h +− uh)∥2 +0,τ +�1/2 +. +Proof. For any q ∈ Q, by using (3.33), (2.8) and (2.9), we have +˜ℓ1(q) += +−a (ph, q) + b(vconf +h +, q) += +− (ph, q) + (curl vconf +h +, q) += +(curlh uh − ph, q)Th + (curlh(vconf +h +− uh), q)Th. +Applying H¨older inequality and Cauchy-Schwarz inequality leads to +|˜ℓ1(q)| ≤ +� +τ∈Th +∥curlh uh − ph∥0,τ ∥q∥0,τ + +� +τ∈Th +∥curlh(vconf +h +− uh)∥0,τ∥q∥0,τ +≤ 2 +� +� +� � +τ∈Th +∥R1 (uh, ph) ∥2 +0,τ +�1/2 ++ +� � +τ∈Th +∥curlh(uh − vconf +h +)∥2 +0,τ +�1/2� +� ∥q∥0,Ω. +The proof is completed. +In order to estimate the term ∥˜ℓ2∥U ∗ in Lemma 3.3, we shall use the following two interpolation +operators with the corresponding approximations. +(1) Scott-Zhang quasi-interpolation Sh : H1 +0(Ω) → {v ∈ C(Ω)| v|τ ∈ P1(τ), v|∂Ω = 0, ∀τ ∈ Th}, +where P1(τ) represents a linear polynomial space. The definition and approximation properties +of Scott-Zhang quasi-interpolation interpolation were first proposed in [25]. For ψ ∈ H1 +0(Ω), +there hold +∥∇Shψ∥0,τ ≲ ∥∇ψ∥0,ωτ , +∀τ ∈ Th, +(3.35) +∥ψ − Shψ∥0,τ ≲ hτ∥∇ψ∥0,ωτ , +∀τ ∈ Th, +(3.36) +∥ψ − Shψ∥0,e ≲ h +1 +2e ∥∇ψ∥0,ωe, +∀e ∈ Eh, +(3.37) +where ωτ := +� +τ ′∩τ̸=∅ +τ ′ and ωe := +� +τ∩e̸=∅ +τ. +9 + +(2) Vector-Valued operator P h : H1(Ω) ∩ U → U conf +h +. The definition and approximation proper- +ties of Vector-Valued operator were proposed in [4]. For q ∈ H1(Ω) ∩ U, there hold +∥P hq∥0,τ ≲ ∥q∥1,˜ωτ , +∀τ ∈ Th, +(3.38) +∥q − P hq∥0,τ ≲ hτ∥q∥1,˜ωτ , +∀τ ∈ Th, +(3.39) +∥q − P hq∥0,e ≲ h +1 +2e ∥q∥1,˜ωe, +∀e ∈ Eh, +(3.40) +where ˜ωe := � {τ ∈ Th(Ω) | e ∈ Eh(T)} and ˜ωτ := � {ωe | e ∈ Eh(T)}. +Lemma 3.3. Let (uh, ph) ∈ U h × Qh be solution of system (2.16)-(2.17). For any vconf +h +∈ U conf +h +, +then there exists a constant C1 > 0 depending only on ∥β∥0,∞, we have +∥˜ℓ2∥U ∗ +≤ +C1 +� � +τ∈Th +h2 +τ +� +∥R2 (uh, ph) ∥2 +0,τ + ∥R3 (uh) ∥2 +0,τ +� ++ +� +e∈Eh +he +� +∥J1 (ph) ∥2 +0,e + ∥J2 (uh) ∥2 +0,e +� ++ +� +τ∈Th +∥uh − vconf +h +∥2 +0,τ +�1/2 +. +Proof. For any v ∈ U, we use Helmholtz decomposition v as follows (see the Theorem 2.1 of [4]) +v := v0 + v⊥, +(3.41) +and +∥v0∥2 +0,Ω + ∥v⊥∥2 +0,Ω = ∥v∥2 +0,Ω, +(3.42) +where v0 ∈ H0(curl0, Ω) := {v : v ∈ H0(curl, Ω), curl v = 0} and v⊥ ∈ H⊥(curl, Ω) := {v : v ∈ +H0(curl, Ω), (v, v0) = 0, v0 ∈ H0(curl0, Ω)}. We make use of the representation H0(curl0, Ω) = +grad H1 +0(Ω), hence, we have v0 = ∇ψ for some ψ ∈ H1 +0(Ω). Applying the definition of ˜ℓ2 (3.34), +we have +˜ℓ2(v) += +˜ℓ2(v0 + v⊥) = ˜ℓ2(∇ψ + v⊥) += +˜ℓ2(∇Shψ + P hv⊥) + ˜ℓ2(∇ψ + v⊥ − ∇Shψ − P hv⊥). +(3.43) +We first analyze the first item on the right hand side of (3.43). According to the definition of ˜ℓ2 +(3.34), we have +˜ℓ2 +� +∇Shψ + P hv⊥� += +ℓ2 +� +∇Shψ + P hv⊥� +− d +� +∇Shψ + P hv⊥, ph +� +− c(vconf +h +, ∇Shψ + P hv⊥) += +ℓ2 +� +∇Shψ + P hv⊥� +− d +� +∇Shψ + P hv⊥, ph +� +− c +� +uh, ∇Shψ + P hv⊥� ++c(uh − vconf +h +, ∇Shψ + P hv⊥). +(3.44) +By using Remark 2.1, for any e ∈ Eh, noting that [[∇Shψ + P hv⊥]]e = 0 since ∇Shψ + P hv⊥ ⊂ U, +and choosing vh = P hv⊥ + ∇Shψ in (2.17), we have +ℓ2 +� +P hv⊥ + ∇Shψ +� +− d +� +P hv⊥ + ∇Shψ, ph +� +− c +� +uh, P hv⊥ + ∇Shψ +� += 0. +(3.45) +10 + +Combining (3.44) and (3.45), we have +˜ℓ2 +� +∇Shψ + P hv⊥� += +c(uh − vconf +h +, ∇Shψ + P hv⊥). +Applying Cauchy-Schwarz inequality, (3.35), (3.38) and H⊥ +0 (curl, Ω) is continuously imbedded in +H1(Ω) ∩ H0(curl, Ω)(see [4]), we deduce +˜ℓ2 +� +∇Shψ + P hv⊥� +≲ +C1∥uh − vconf +h +∥0,Ω(∥∇Shψ∥0,Ω + ∥P hv⊥∥0,Ω) +≲ +C1∥uh − vconf +h +∥0,Ω(∥∇ψ∥0,Ω + ∥v⊥∥0,Ω) +≲ +C1∥uh − vconf +h +∥0,Ω(∥∇ψ∥0,Ω + ∥curl v⊥∥0,Ω) +≲ +C1∥uh − vconf +h +∥0,Ω∥v∥curl,Ω, +(3.46) +where C1 > 0 depends only on ∥β∥0,∞. +Next, we analyze the second item on the right hand side of (3.43), using the definition of ˜ℓ2 +(3.34), the fact (curlh(∇ψ − ∇Shψ), αph)Th = 0 and Green formula, we have +˜ℓ2 +� +∇ψ + v⊥ − ∇Shψ − P hv⊥� += +˜ℓ2 (∇ψ − ∇Shψ) + ˜ℓ2 +� +v⊥ − P hv⊥� += +(f, ∇ψ − ∇Shψ)Th − (curlh(∇ψ − ∇Shψ), αph)Th − (βvconf +h +, ∇ψ − ∇Shψ)Th ++(f, v⊥ − P hv⊥)Th − (curlh(v⊥ − P hv⊥), αph)Th − (βvconf +h +, v⊥ − P hv⊥)Th += +(f, ∇ψ − ∇Shψ)Th − (βuh, ∇ψ − ∇Shψ)Th + (β(uh − vconf +h +), ∇ψ − ∇Shψ)Th ++(f, v⊥ − P hv⊥)Th − (curlh(v⊥ − P hv⊥), αph)Th − (βuh, v⊥ − P hv⊥)Th ++(β(uh − vconf +h +), v⊥ − P hv⊥)Th += +− (∇ · (f − βuh) , ψ − Shψ)Th + ⟨[f − βuh]e, ψ − Shψ⟩Eh ++(β(uh − vconf +h +), ∇ψ − ∇Shψ)Th + +� +f − curlh αph − βuh, v⊥ − P hv⊥� +Th +− +� +[[αph]]e, v⊥ − P hv⊥� +Eh + (β(uh − vconf +h +), v⊥ − P hv⊥)Th. +(3.47) +Applying Cauchy-Schwarz inequality, (3.36)-(3.40) and H⊥ +0 (curl, Ω) is continuously imbedded in +H1(Ω) ∩ H0(curl, Ω), we deduce +|˜ℓ2(v)| ≲ C1 +� � +τ∈Th +h2 +τ +� +∥R2 (uh, ph) ∥2 +0,τ + ∥R3(uh) ∥2 +0,τ +� ++ +� +e∈Eh +he +� +∥J1 (ph) ∥2 +0,e + ∥J2 (uh) ∥2 +0,e +� ++ +� +τ∈T +∥uh − vconf +h +∥2 +0,τ +�1/2 +∥v∥curl,Ω, +which complete the proof. +In the end of this section, we give the upper bound of the error estimator. +11 + +Theorem 3.1. Let (u, p) ∈ U × Q and (uh, ph) ∈ U h × Qh be the solutions of (2.6)-(2.7) and +(2.16)-(2.17), respectively. Then there exists a constant C1 > 0 depending only on ∥β∥∞,Ω and the +grid shape regularity, we have +∥(u, p) − (uh, ph)∥2 +DG ≲ C1η2(uh, ph; Th). +Proof. For any vconf +h +∈ U conf +h +, using the definition of ∥ · ∥DG in (3.31), triangle inequality, we get +∥(u, p) − (uh, ph) ∥2 +DG += ∥p − ph∥2 +0,Ω + ∥u − uh∥2 +0,Ω + ∥curl u − curlh uh∥2 +0,Ω + κ +� +e∈Eh +h−1 +e ∥[[uh]]∥2 +0,e += |p − ph∥2 +0,Ω + ∥u − uh∥2 +curl,Ω + κ +� +e∈Eh +h−1 +e ∥[[uh]]∥2 +0,e +≤ ∥p − ph∥2 +0,Ω + ∥u − vconf +h +∥2 +curl,Ω + ∥vconf +h +− uh∥2 +curl,Ω + κ +� +e∈Eh +h−1 +e ∥[[uh]]∥2 +0,e += ∥(u − vconf +h +, p − ph)∥U×Q + ∥vconf +h +− uh∥2 +curl,Ω + κ +� +e∈Eh +h−1 +e ∥[[uh]]∥2 +0,e. +(3.48) +For uh ∈ U h, there exists uconf +h +∈ U conf +h +, u⊥ +h ∈ U ⊥ +h satisfing (see Proposition 4.10 of [22]) +uh = uconf +h ++ u⊥ +h , +and +h2 +τ∥curlh(uh − uconf +h +)∥2 +0,Ω + ∥uh − uconf +h +∥2 +0,Ω ≲ h2 +τ +� +e∈Eh +h−1 +e +∥[[uh]]e∥2 +0,e . +(3.49) +At last, choosing vconf +h += uconf +h +in (3.48) and using Lemmas 3.1, 3.2, 3.3 and (3.49), we have +∥(u, p) − (uh, ph) ∥2 +DG +≤ +∥(u − uconf +h +, p − ph)∥U×Q + ∥uconf +h +− uh∥2 +curl,Ω + κ +� +e∈Eh +h−1 +e ∥[[uh]]∥2 +0,e +≲ +∥˜ℓ1∥2 +Q∗ + ∥˜ℓ2∥2 +U ∗ + ∥uconf +h +− uh∥2 +curl,Ω + κ +� +e∈Eh +h−1 +e ∥[[uh]]∥2 +0,e +≲ +C1η2(vh, ph; Th), +which completes the proof. +4. Efficiency analysis +In this section, we will prove that the error estimator is efficient. In the Section 3, we have +already proved that the error estimator is reliable. Hence, the error estimator is a good indicator of +energy error, that is +η2 (uh, ph; Th) ≈ ∥ (p − ph, u − uh) ∥2 +DG. +(4.50) +We fitst introduce the following theorem, which is the efficiency of the error estimator. +12 + +Theorem 4.1. Let (u, p) ∈ U × Q and (uh, ph) ∈ U h × Qh be the solutions of (2.6)-(2.7) and +(2.16)-(2.17), respectively. The space of all piecewise polynomial of arbitrary order on Th is denoted +by P(Th). Then for any f h ∈ Sconf +h +:= P(Th) ∩ H(div; Ω), we have +η2 (uh, ph; Th) ≲ C2 +� +∥ (p − ph, u − uh) ∥2 +DG + ∥f − f h∥2 +0,Ω + +� +τ∈Th +h2 +τ∥∇ · (f − f h) ∥2 +0,τ +� +, +where C2 is a constant depending on ∥α∥0,∞ and ∥β∥0,∞. +Remark 4.1. We assume f ∈ H(div; Ω) to show the proof of the Theorem 4.1. While, we should +note that the conditional f h ∈ Sconf +h +is only required in the Lemma 4.7 ( See (4.78)). +For any τ ∈ Th, e ∈ Eh, let bτ and be be standard the interior bubble functions and edge bubble +functions, respectively. We introduce the following two lemmas which are used in later proofs. +Lemma 4.1 ([16], Lemma 5.1). Let χ be a scalar polynomial function on τ, then +∥bτχ∥0,τ ≲ ∥χ∥0,τ, +(4.51) +∥χ∥0,τ ≲ ∥b +1 +2τ χ∥0,τ, +(4.52) +∥∇ (bτχ) ∥0,τ ≲ h−1 +τ ∥χ∥0,τ. +(4.53) +Moreover, let e be an edge shared by two triangles τ1 and τ2, let φ be a scalar polynomial function +on e, the +∥φ∥0,e ≲ ∥b +1 +2e φ∥0,e. +(4.54) +In addition, there exists an extension of beφ as ˜φb ∈ H1 +0 +� +(τ 1 ∪ τ 2)◦� +such that ˜φb|e = beφ and +∥˜φb∥0,τ ≲ h +1 +2e ∥φ∥0,e, +∀τ ∈ ωe, +(4.55) +∥∇˜φb∥0,τ ≲ h +− 1 +2 +e +∥φ∥0,e, +∀τ ∈ ωe. +(4.56) +Lemma 4.2 ([16], Lemma 5.2). Let w be a vector-valued polynomial function on τ, then +∥bτw∥0,τ ≲ ∥w∥0,τ, +(4.57) +∥w∥0,τ ≲ ∥b +1 +2τ w∥0,τ, +(4.58) +∥curl (bτw) ∥0,τ ≲ h−1 +τ ∥w∥0,τ. +(4.59) +Moreover, let e be an edge shared by two triangles on τ1 and τ2, and let w be a vector-valued +polynomial function on e, then +∥w∥0,e ≲ ∥b +1 +2e w∥0,e. +(4.60) +In addition, there exists an extension of bew as ˜wb ∈ +� +H1 +0 +� +(τ 1 ∪ τ 2)◦��2, such that ˜wb|e = bew, +and +∥ ˜wbc∥0,τ ≲ h +1 +2e ∥w∥0,e, +∀τ ∈ ωe, +(4.61) +∥curl ˜wb∥0,τ ≲ h +− 1 +2 +e +∥w∥0,e, +∀τ ∈ ωe. +(4.62) +13 + +Next, we estimate each term of the error indicator η(uh, ph; Th) given by (2.30), separately. We +first estimate the first term R1 of the error estimator. +Lemma 4.3. Let (u, p) ∈ U × Q and (uh, ph) ∈ U h × Qh be the solutions of (2.6)-(2.7) and +(2.16)-(2.17), respectively. Then we obtain +� +τ∈Th +∥R1 (uh, ph) ∥2 +0,τ ⩽ ∥ (p − ph, u − uh) ∥2 +DG. +(4.63) +Proof. We can observe that R1 (uh, ph) = ph − curlhuh ∈ Qh ⊂ Q, and taking q = R1 (uh, ph) in +(2.6), we have +(p − curl u, R1 (uh, ph))Th = 0. +(4.64) +Thus, by (4.64) and Cauchy-Schwarz inequality, we obtain +� +τ∈Th +∥R1 (uh, ph) ∥2 +0,τ += +(ph − curlh uh, R1 (uh, ph))Th += +(ph − p − curlh (uh − u) , R1 (uh, ph))Th +⩽ +(∥ph − p∥0,Ω + ∥curlh (uh − u) ∥0,Ω) ∥R1 (uh, ph) ∥0,Ω, +which implies (4.63). +We estimate the second term R2 of the error estimator η(uh, ph; Th). +Lemma 4.4. Let (u, p) ∈ U × Q and (uh, ph) ∈ U h × Qh be the solutions of (2.6)-(2.7) and +(2.16)-(2.17), respectively. Then we have +� +τ∈Th +h2 +τ∥R2 (uh, ph) ∥2 +0,τ ≲ C2 +� +∥(u, p) − (uh, ph)∥2 +DG + +� +τ∈Th +h2 +τ ∥f − f h∥2 +0,τ +� +, +where C2 is a constant depending on ∥α∥0,∞ and ∥β∥0,∞. +Proof. Let f h ∈ Sconf +h +and let w := f h − curlh αph − βuh. Hence w is also a polynomial on τ. +Let wb = bτw, we can observe that wb ∈ H1 +0(τ) ⊂ H1 +0(Ω) ⊂ U. Setting v = wb in (2.7) and using +Green’s formula, we get +(f − curl αp − βu, wb)τ = 0. +(4.65) +Then by using (4.58), the definitons of w and wb, (4.65) and Cauchy-Schwarz inequality, we have +∥w∥2 +0,τ +≲ +∥b1/2 +τ +w∥2 +0,τ = (w, bτw)τ += +((f h − curlh αph − βuh) , wb)τ += +((f h − f) − curlh α (ph − p) − β (uh − u) , wb)τ +≲ +� +∥f h − f∥0,τ + ∥β∥0,∞ ∥uh − u∥0,τ +� +∥wb∥0,τ +− (curlh α (ph − p) , wb)τ . +(4.66) +14 + +According to the last term of right hand side (4.66), using Green’s formula with the fact wb = 0 +on ∂τ, Cauchy-Schwarz inequality and (4.59), we have +(curlh α (ph − p) , wb)τ = (α(ph − p), curlh wb)τ +⩽ +∥α∥0,∞ ∥ph − p∥0,τ ∥curlh wb∥0,τ ≲ h−1 +τ ∥α∥0,∞ ∥ph − p∥0,τ ∥w∥0,τ. +(4.67) +Using (4.66) and (4.67), we get +∥w∥0,τ ≲ ∥f h − f∥0,τ + ∥β∥0,∞ ∥uh − u∥0,τ + h−1 +τ ∥α∥0,∞ ∥ph − p∥0,τ . +(4.68) +By the definition of R2 (uh, ph) and (4.68), we obtain +∥R2 (uh, ph) ∥0,τ +⩽ +∥w∥0,τ + ∥f h − f∥0,τ +≲ +C2 +� +∥f h − f∥0,τ + ∥uh − u∥0,τ + h−1 +τ +∥ph − p∥0,τ +� +, +which implies +� +τ∈Th +h2 +τ∥R2 (uh, ph) ∥2 +0,τ +≲ C2 +� � +τ∈Th +� +h2 +τ ∥f h − f∥2 +0,τ + h2 +τ ∥uh − u∥2 +0,τ + ∥ph − p∥2 +0,τ +� � +≲ C2 +� +∥(u, p) − (uh, ph)∥2 +DG + +� +τ∈Th +h2 +τ ∥f h − f∥2 +0,τ +� +. +In the following, we turn to estimate the third term R3 of the error estimator. +Lemma 4.5. Let (u, p) ∈ U × Q and (uh, ph) ∈ U h × Qh be the solutions of (2.6)-(2.7) and +(2.16)-(2.17), respectively. Then we have +� +τ∈Th +h2 +τ∥R3 (uh) ∥2 +0,τ ≲ C1 +� +∥u − uh∥2 +0,Ω + ∥f − f h∥2 +0,Ω + +� +τ∈Th +h2 +τ∥∇ · (f − f h) ∥2 +0,τ +� +, +(4.69) +where C1 is a constant depending on ∥β∥0,∞. +Proof. Let f h ∈ Sconf +h +, χ := ∇ · (f h − βuh) and χb = bτχ, where χ is a polynomial on each τ ∈ T +and χb ∈ H1 +0(τ) ⊂ H1 +0(Ω). Setting v = ∇χb ∈ U in (2.7) and with the fact χb|∂τ = 0, we get +(f − βu, ∇χb)τ = 0. +(4.70) +Then, using (4.52), the Green’s formula and (4.70), we obtain +∥χ∥2 +0,τ +≲ +∥b1/2 +τ +χ∥2 +0,τ = (χ, bτχ) = (∇ · (f h − βuh) , χb)τ += +−((f h − βuh) , ∇χb)τ = ((f − f h) − β (u − uh) , ∇χb)τ . +15 + +According to the inverse inequalities, there holds +∥χ∥2 +0,τ +⩽ +� +∥f h − f∥0,τ + ∥β∥0,∞ ∥uh − u∥0,τ +� +∥∇χb∥0,τ +≲ +C1h−1 +τ +� +∥f h − f∥0,τ + ∥uh − u∥0,τ +� +∥χ∥0,τ, +which imples +hτ∥χ∥ ≲ C1 +� +∥f h − f∥0,τ + ∥uh − u∥0,τ +� +. +(4.71) +Then combing the definition of R3(uh), triangle inequality and (4.71), we get +hτ∥R3 (uh) ∥0,τ +⩽ +hτ∥χ∥0,τ + hτ∥∇ · (f − f h) ∥0,τ +≲ +C1 (∥f h − f∥0,τ + ∥uh − u∥0,τ) + hτ∥∇ · (f − f h) ∥0,τ. +Summing over all elements τ ∈ T , the result (4.69) follows directly. +Next, we turn to bound the fourth term J1 of the error estimator. +Lemma 4.6. Let (u, p) ∈ U × Q and (uh, ph) ∈ U h × Qh be the solutions of (2.6)-(2.7) and +(2.16)-(2.17), respectively. Then we have +� +e∈Eh +he∥J1 (ph) ∥2 +0,e +≲ C2 +� � +τ∈Th +∥R1 (uh, ph) ∥2 +0,τ + +� +τ∈Th +h2 +e∥R2 (uh, ph) ∥2 +0,τ + ∥(u, p) − (uh, ph)∥2 +DG +� +, +where C2 is a constant depending on ∥α∥0,∞ and ∥β∥0,∞. +Proof. Without loss of generality, let e ∈ Eh be the common edge of τ1 ∈ Th and τ2 ∈ Th, we give +the following definition +wh = [[αph]]e, +˜wb = bewh ∈ H1 +0(τ1 ∪ τ2). +(4.72) +Applying (4.72), [[αcurl u]]e = 0, Green’s fromula, (2.6) and (2.7), we get +∥b +1 +2e wh∥2 +0,e = ⟨wh, bewh⟩e += +⟨[[αph]]e, ˜wb⟩e += +⟨[[α(ph − curl u)]]e, ˜wb⟩e += +� +τ∈ωe +[(curl (α(ph − curl u)) , ˜wb)τ − (α(ph − curl u), curl ˜wb)τ] += +� +τ∈ωe +[(curl αph − curl αcurl u + f − f + βu − βu + βuh − βuh, ˜wb)τ +− (α(ph − curl u), curl ˜wb)τ] += +� +τ∈ωe +[(curl αph + βuh − f, ˜wb)τ + (βu − βuh, ˜wb)τ +16 + +− (α(ph − curl u), curl ˜wb)τ] . +(4.73) +Using (4.60), (4.73), Cauchy-Schwarz inequality, (4.61) and (4.62), there holds +∥wh∥2 +0,e ≲ ∥b +1 +2e wh∥2 +0,e +≲ +� +τ∈ωe +[(curl αph + βuh − f, ˜wb)τ + (βu − βuh, ˜wb)τ +− (α(ph − curl u), curl ˜wb)τ] +≲ +� +τ∈ωe +� +∥curl αph + βuh − f∥0,τ · ∥ ˜wb∥0,τ + ∥β∥0,∞ ∥u − uh∥0,τ · ∥ ˜wb∥0,τ ++∥α∥0,∞ ∥ph − curl u∥0,τ · ∥curl ˜wb∥0,τ +� +≲ +C2 +� +τ∈ωe +� +∥curl αph + βuh − f∥0,τ · h +1 +2e ∥wh∥0,e ++ ∥u − uh∥0,τ · h +1 +2e ∥wh∥0,e + ∥ph − curl u∥0,τ · h +− 1 +2 +e +∥wh∥0,e +� +≲ +C2 +� +τ∈ωe +� +h +1 +2e ∥curl αph + βuh − f∥0,τ + h +1 +2e ∥u − uh∥0,τ ++h +− 1 +2 +e +∥ph − curlh uh∥0,τ + h +− 1 +2 +e +∥curl u − curlh uh∥0,τ +� +· ∥wh∥0,e , +which imples +∥wh∥0,e +≲ +C2 +� +τ∈ωe +� +h +1 +2e ∥f − curl αph − βuh∥0,τ + h +1 +2e ∥u − uh∥0,τ ++h +− 1 +2 +e +∥ph − curlh uh∥0,τ + h +− 1 +2 +e +∥curl u − curlh uh∥0,τ +� +. +(4.74) +Applying (4.72) and (4.74) results in +he∥J1(ph)∥2 +0,e = he∥[[αph]]τ∥2 +0,e = he ∥wh∥2 +0,e +≲ C2 +� � +τ∈ωe +� +∥R1(uh, ph)∥0,τ + h2 +e∥R2(uh, ph)∥0,τ +� ++ ∥ (p − ph, u − uh) ∥2 +DG +� +. +At last, we turn to bound the fifth term J2 of the error estimator. +Lemma 4.7. Let (u, p) ∈ U × Q and (uh, ph) ∈ U h × Qh be the solutions of (2.6)-(2.7) and +(2.16)-(2.17), respectively. Then we have +� +e∈Eh +he∥J2 (uh) ∥2 +0,e ≲ C2 +� +∥u − uh∥2 +0,Ω + ∥f − f h∥2 +0,Ω + +� +τ∈Th +h2 +τ∥∇ · (f − f h) ∥2 +0,Ω +� +, +where C1 is a constant depending on ∥β∥0,∞. +17 + +Proof. Without loss of generality, let e ∈ Eh be the common edge of τ1 and τ2. +Setting φ = +[(f h − βuh)]e and ˜φb = beφ, then according to (4.54), there holds +∥φ∥2 +0,e +≲ +∥b1/2 +e +φ∥2 +0,e = (φ, beφ) = ⟨[(f h − βuh)]e, ˜φb⟩e += +� +i=1,2 +� +(∇ · (f h − βuh) , ˜φb)τi + (f h − βuh, ∇˜φb)τi +� +, +(4.75) +where we have used the fact that ˜φb ∈ H1 +0 (τ1 ∪ τ2). Applying the definition of R3 (uh) and (4.55) +to the first term (∇ · (f h − βuh) , ˜φb)τi of (4.75) gives +(∇ · (f h − βuh) , ˜φb)τi += +(R3 (uh) , ˜φb)τi + (∇ · (f h − f) , ˜φb)τi +⩽ +(∥R3 (uh) ∥0,τi + ∥∇ · (f − f h) ∥0,τi) · ∥˜φb∥0,τi +≲ +h +1 +2e (∥R3 (uh) ∥0,τi + ∥∇ · (f − f h) ∥0,τi) · ∥φ∥0,e. +(4.76) +Using the Cauchy-Schwarz inequality and (4.56) to the second term of (4.75) gives +� +i=1,2 +� +f h − βuh, ∇˜φb +� +τi = +� +i=1,2 +� +f h − f − β (uh − u) , ∇˜φb +� +τi +≲ +C1 +� +i=1,2 +� +∥f − f h∥0,τi + ∥u − uh∥0,τi +� +· ∥∇˜φb∥0,τi +≲ +C1 +� +i=1,2 +h−1/2 +e +� +∥f − f h∥0,τi + ∥u − uh∥0,τi +� +· ∥φ∥0,e, +(4.77) +where we have used +� +f − βu, ∇˜φb +� +τ1∪τ2 = 0 when we set v = ∇˜φb ∈ U in (2.7). +According to the assumptions f ∈ H(div, Ω) and f h ∈ Sconf +h +, there holds +[f − f h]e = 0. +(4.78) +Due to the definition of J2 (uh) and (4.75)-(4.78), we arrive at +∥J2 (uh) ∥0,e += +∥φ∥0,e + ∥[(f − f h)]e∥0,e +≲ +C1 +� +i=1,2 +� +h1/2 +e +(∥R3 (uh) ∥0,τi + ∥∇ · (f − f h) ∥0,τi) ++ h−1/2 +e +� +∥f − f h∥0,τi + ∥u − uh∥0,τi +� � +. +Hence, combining Lemma 4.5, we have +� +e +he∥J2 (uh) ∥2 +0,e +≲ +C1 +� +τ +� +h2 +τi +� +∥R3 (uh) ∥2 +0,τi + ∥∇ · (f − f h) ∥2 +0,τi +� ++ ∥f − f h∥2 +0,τi + ∥u − uh∥2 +0,τi +� +≲ +C1 +� +∥u − uh∥2 +0,Ω + ∥f − f h∥2 +0,Ω + h2 +τ∥∇ · (f − f h) ∥2 +0,Ω +� +. +18 + +Next, we present the proof of Theorem 4.1. +Proof of Theorem 4.1. +The following result follows immediately by a direct application of Lemmas +4.3-4.7, +κ +� +e∈Th +� +e∈∂τ +h−1 +e +∥J3(uh)∥2 +0,e ≲ ∥(u, p) − (uh, ph)∥DG. +5. Numerical experiment +In this section, we report some experiments to show the performance of the error indicator and +the adaptive algorithm AMIPDG. We carry out these numerical experiments by using the MATLAB +software package iFEM [9]. In Experiments 5.1 and 5.2, we take p = curl u. +In Example 5.1, we discuss the influence of the penalty parameter κ on the error both in L2 and +∥ · ∥DG norms, and observe the dependency of the condition number of stiffness matrix on κ. At +last, we verify the reliability and efficiency of the constructed error indicator (2.30). +Example 5.1. Let Ω := [−1, 1] × [−1, 1], we construct the following analytical solution of the model +(1.1)-(1.2): +u = +� +cos(πx) sin(πy) +− cos(πy) sin(πx) +� +with coefficients +α = 1, +β = 1, +which corresponds to a right hand source term +f(x, y) = +� +� +2π2 − 1 +� +cos(πx) sin(πy) +− +� +2π2 − 1 +� +cos(πy) sin(πx) +� +. +It is easy to see that the solution u satisfies the boundary condition u · t = 0 on ∂Ω. +In this example, we get an uniform meshes by partitioning the x− and y− into equally distributed +M(M ≥ 2) subintervals, and then dividing one square into two triangles. Let h = 1/M be mesh +sizes for different triangular meshes. Firstly, we fixed mesh with h = 1/32 and report the error +estimates in both L2 and ∥ · ∥DG norm for different penalty parameters κ = 1, 50, 100, 150 and 200 +in Table 1. We note that ∥u − uh∥0 increases slightly as the penalty parameter κ increases. On the +contrary, ∥ (p − ph, u − uh) ∥DG decreases slightly as κ increases. +Next, we also use fixed mesh with h = 1/32, and observe the influence of different κ on the +condition numbers of stiffness matrices in Table 2. It is easy to see that the condition numbers of +stiffness matrices increase with the increase of penalty parameters κ. +As a way to balance, in the following numerical tests, we always choose κ = 50. +19 + +Table 1: The errors in both L2 and ∥ · ∥DG norms with h = 1/32. +κ +∥u − uh∥0 +∥ (p − ph, u − uh) ∥DG +1 +2.00399e-02 +1.51333e-01 +50 +2.00391e-02 +1.46739e-01 +100 +2.00397e-02 +1.46733e-01 +150 +2.00400e-02 +1.46732e-01 +200 +2.00401e-02 +1.46732e-01 +Table 2: Condition number of stiffness matrices with different κ. +κ +1 +50 +100 +150 +200 +Cond +3.40461e+05 +1.11946e+07 +2.42288e+07 +3.73104e+07 +5.04046e+07 +Now, we verify the reliability and efficiency of the error estimate by comparing η (uh, ph; Th) +with ∥(p − ph, u − uh)∥DG. The numerical results are given in Table 3 which also provide the values +for the effectivity index σ = ∥ (p − ph, u − uh) ∥DG/η (uh, ph; Th). We observe that the convergence +rate of ∥ (p − ph, u − uh) ∥DG is first order and the effectivity index σ ≈ 0.286. This shows that the +error indicator is effective and reliable, which is (4.50). +Table 3: Rate of convergence of the ∥ (p − ph, u − uh) ∥DG and η (uh, ph; Th) on uniform triangular meshes. +h +∥ (p − ph, u − uh) ∥DG +η (uh, ph; Th) +index +Error +order +Error +order +σ +1/16 +2.93259E-01 +N/A +1.02621E-00 +N/A +0.286 +1/32 +1.46739E-01 +0.9989 +5.13690E-01 +0.9984 +0.286 +1/64 +7.33830E-02 +0.9997 +2.56920E-01 +0.9996 +0.286 +1/128 +3.66932E-02 +0.9999 +1.28470E-02 +0.9999 +0.286 +1/256 +1.83468E-02 +1.0000 +6.42365E-02 +1.0000 +0.286 +Noting that we only consider uniform meshes and the constant coefficients in Example 5.1. Next +we test adaptive meshes and the jump coefficients. Our adaptive cycle can be implemented by the +following algorithm: +Example 5.2. Let Ω := [−1, 1] × [−1, 1], we construct the following analytical solution of the model +(1.1)-(1.2) +u = +� +� +y(x2−1)(y2−1) +x2+y2+0.02 +−x(x2−1)(y2−1) +x2+y2+0.02 +� +� , +with the jump coefficients +α = 1.0, β = 1.0, +on Ω1, +α = 1.0, β = 100, +on Ω\Ω1, +20 + +Algorithm 1 An adaptive mixed interior penalty discontinuous method (AMIPDG) cycle +Input initial triangulation T0; data f; tolerance tol; marking parameter θ ∈ (0, 1). +Output a triangulation TJ; MIPDG solution (uJ, pJ). +η = 1; k = 0; +while η ≥ tol +SOLVE solve discrete varational problem (2.16)-(2.17) on Tk to get the solution (uk, pk); +ESTIMATE compute the posterior error estimator η = η(uk, pk, Tk) by using (2.30); +MARK seek a minimum cardinality Mk ⊂ Tk such that +η2 (uk, pk, Mk) ≥ θη2 (uk, pk, Tk) ; +REFINE Bisect elements in Mk and the neighboring elements to form a conforming Tk+1; +k = k + 1; +end +uJ = uk; pJ = pk; TJ = Tk; +where Ω1 = (−0.5, 0.5)2(see the left of Figure 2). Note that the solution u satisfies the condition +u · t = 0 on ∂Ω. +Here, we can observe that u has a relatively large change at (0, 0). Figure 1 shows the contours +of the exact solution. +Figure 1: Left: the first component of the analytical solution. Right: the second component of the analytical solution. +We get an initial mesh T0 by partitioning the x- and y-axes into equally distributed eight subin- +tervals and then dividing one square into two triangles, see the middle of Figure 2. The right of +Figure 2 shows an adaptively refined mesh with marking parameter θ = 0.5 after k = 8, and we can +see that the grid is locally refined near both the origin and at ∂Ω1. +21 + +N +5 +1 +0.5 +0 +0.5 +-0.5 +0 +-0.5 +： +-1 +x5 +N +1 +0.5 +0 +0.5 +0 +-0.5 +-0.5 +y +-1Figure 2: Left: regional diagram. Middle: the initial mesh with 512 DoFs. Right: the adaptive mesh(θ = 0.5) with +20416 DoFs after 8 refinements. +Figure 3 shows the curves of ln N − ln ∥(p − pk, u − uk)∥DG for different marking parame- +ters θ = 0.3, 0.5 and 0.7, where N is the number of degrees of freedom. +The curves indicate +the convergence and the quasi-optimality of the adaptive algorithm AMIPDG of the energy error +∥ (p − ph, u − uh) ∥DG, i.e. +∥(p − pk, u − uk)∥DG ≲ N −1/2. +From these curves, it seems that the convergence rate are robust for θ changing from 0.3 to 0.7. +Figure 3: Quasi optimality of the AMIPDG of the error ∥ (p − ph, u − uh) ∥DG with different marking parameters θ. +The examples above are considered convex domain and the jump coefficients. Finally we consider +variable coefficients, ‘L-shaped’ domain and unknown exact solution. +22 + +1 +212 +0.5 +0 +-0.5 +-0.5 +0 +0.5 +1Rate of convergence is CN-0.5 +adaptive refiniment =0.3 +adaptive refiniment =0.5 +adaptive refiniment =0.7 +10 +C-0.5 +JOJE +100 +104 +Number of unknownsExample 5.3. Let domain Ω = (−1, 1)2/([0, 1) × [0, 1)) and let the variable coefficients +α = +1 +1 + x2 + y2 , +β = +� +1 + x2 +xy +xy +1 + y2 +� +. +We set the homogeneous Dirichlet boundary condition u·t = 0 on ∂Ω, the source f = ( +1 +x2+y2+0.01, +1 +x2+y2+0.01). +We get an initial mesh T0 by partitioning the x- and y-axes into equally distributed eight subin- +tervals and then dividing one square into two triangles, see the left of Figure 4. The right of Figure +4 shows an adaptively refined mesh with marking parameter- θ = 0.5 after k = 8. The grid is locally +refined near the origin. +Figure 4: Left: the initial mesh with 384 DoFs. +Right: the adaptive mesh(θ = 0.5) with 16032 DoFs after 8 +refinements. +The Figure 5 shows the curves of ln N − ln η (uk, pk; Tk) for parameters θ = 0.3, 0.5, 0.7. The +curves indicate the convergence and the quasi-optimality of the adaptive algorithm AMIPDG of +η (uk, pk; Tk). +Acknowledgments +The authors are supported by the National Natural Science Foundation of China (Grant No. +12071160). The second author is also supported by the National Natural Science Foundation of +China (Grant No. 11901212). +References +[1] D. Arnold, An interior penalty finite element method with discontinuous elements, Siam J. +Numer. Anal. 19(1982) 742-760. +[2] A. Bonito, R. H. Nochetto, +Quasi-optimal convergence rate of an adaptive discontinuous +Galerkin method, Siam J. Numer. 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Xing, Convergence of adaptive interior penalty discontinuous +Galerkin methos for H(curl)-elliptic problems, +Journal of South China Normal University +(Natural Science Edition), 48(2016) 92-98. +[29] L. Q. Zhong, S. Shu, L. Chen, J. C. Xu, Convergence of adaptive edge finite element methods +for H(curl)-elliptic problems, Numer. Linear Algebra Appl. 17(2010) 415-432. +26 + diff --git a/XNAzT4oBgHgl3EQfmf09/content/tmp_files/load_file.txt b/XNAzT4oBgHgl3EQfmf09/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c63c93637aea41d01dba01813da2c89803a54c71 --- /dev/null +++ b/XNAzT4oBgHgl3EQfmf09/content/tmp_files/load_file.txt @@ -0,0 +1,955 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf,len=954 +page_content='A Posterior Error Estimator for Mixed Interior Penalty Discontinuous Galerkin Finite Element Method for the H(curl)-Elliptic Problems Ming Tanga, Xiaoqing Xinga,∗, Liuqiang Zhonga aSchool of Mathematical Sciences, South China Normal University, Guangzhou 510631, China Abstract In this paper, we design the first residual type a posteriori error estimator for mixed interior penalty discontinuous Galerkin method for the H(curl)-elliptic problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Then we prove that our residual based a posteriori error indicator is both reliable and efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' At last, we present some numeri- cal experiments to validate the performance of the indicator within an adaptive mesh refinement procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Keywords: H(curl)-elliptic problems, mixed interior penalty discontinuous Galerkin method, a posterior error estimator, reliability, efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Introduction In this work, we consider the H(curl)-elliptic problems as follows: find the electric or magnetic field u satisfy curl(αcurl u) + βu = f, in Ω, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='1) u · t = 0, on ∂Ω, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='2) where Ω ⊂ R2 be a simply connected bounded Lipschitz polygon with boundary ∂Ω and is partitioned into non-overlapping subdomains Ωi, 1 ≤ i ≤ m, f is a given vector field depending on a given external source field, t is the unit tangent on ∂Ω oriented counter-clockwisely, α ≥ α0 > 0 and β ≥ β0 > 0 are piecewise constants in Ωi, α0 and β0 are constans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' We recall that, curl v = ∂v2/∂x−∂v1/∂y for a vector field v = (v1, v2), while curl φ = (∂φ/∂y, −∂φ/∂x) for a scalar function φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Our numerical scheme and the a posteriori error analysis are based on a mixed formulation of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='1)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='2), which is obtained by introducing an auxiliary variable p = curl u curl (αp) + βu = f, in Ω, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='3) p − curl u = 0, in Ω, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='4) ∗Corresponding author Email addresses: mingtang@m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='scnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='cn (Ming Tang), xingxq@scnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='cn (Xiaoqing Xing), zhong@scnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='cn (Liuqiang Zhong) Preprint submitted to Elsevier January 5, 2023 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='01563v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='NA] 4 Jan 2023 u · t = 0, on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='5) Discontinuous Galerkin (DG) finite element method is one of popular methods for numerical solution of partial differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Compared with the traditional conforming finite element method, the DG finite element method has advantages as follows: to allow incompatible with sus- pension point grid, to deal with complex boundary and interface problems easily, and to implement partial encryption and each unit of polynomial independent selection easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' One of the key fea- tures of the DG method is that the discontinuous approximation at element interfaces naturally allows jump discontinuities in the solution if element boundaries are placed along them [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' DG method has been developed to solve many equations, such as elliptic problems [3], parabolic equa- tions [23], advection-diffusion-reaction problems [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' The DG methods include locally DG(LDG) method [7], interior penalty DG(IPDG) method [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' The discontinuous finite element method for H(curl)−elliptic problems is still in its infancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Chung and Kim [10] proposed an improved Feti-DP algorithm and convergence analysis for the mixed interleaved discontinuous finite element method for the two-dimensional H(curl)−elliptic problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' On the other hand, in practical engineering applications and scientific calculations, there are many factors that may cause strong singularities in the propagation of electromagnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' For example, the material coefficient of the medium in the electromagnetic wave propagation area is discontinuous, or the source term of the generated electromagnetic field is not smooth [12, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Although these singularities can be overcomed by uniformly densifying the grid when performing numerical solutions, consistent densification can lead to a sharp increase in computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Hence, adaptive finite element emerges as the times require.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' In the past few decades, adaptive finite element method have been proven to be a useful and effective tool in scientific computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' The standard adaptative process is as follows SOLVE → ESTIMATE → MARK → REFINE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' The adaptive finite element method is based on a posteriori error estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' It automatically refines and optimizes mesh generation according to the local posteriori error indicator on the element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' It is a numerical calculation method with high reliability and efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Most of the work on the convergence of the adaptive method for the H(curl)-elliptic equations focuses on the edge finite element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' For example, using the so-called interior node property and oscillation marker as technical assumptions, the convergence of the lowest order edge elements of the N´ed´elec’s first family of adaptive for two-dimensional and three-dimensional eddy current equations are proved in [4,14], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Chen, Xu and Zou [8] proved that an adaptive method for three dimensional static Maxwell equations without additional marking of oscillation terms and gives corresponding proof of convergence with the lowest order edge elements of N´ed´elec’s first family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Zhong, Shu, Chen and Xu [29] proved that the three-dimensional H(curl)-elliptic problem with variable coefficients is convergent by using high order and the two family of N´ed´elec edge elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' There are also some studies on the posteriori error estimator of the adaptive DG finite element method for H(curl)-elliptic problem [16, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Houston, Perugia and Schotzau [16] gave the residual- 2 type posteriori error estimator and proved the reliability and efficiency of the error estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Xing and Zhong [26] gave a simplified posteriori error indicator and proved corresponding upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Recently, Zhong, Chen and Xing [28] proved the convergence of the adaptive interior penalty DG methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Meanwhile, there are many successful works of solving the Maxwell’s equations by the mixed finite element method, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' [18,19,21,15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' However, For adaptive mixed finite element method solving Maxwell’s equations, there are only few research results for a posterior error estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' For example, Carstensen, Hoppe, Sharma and Warburton [6] studied a posteriori error estimation of the hybridized finite element method and proved the reliability of the estimator up to a consistency error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Chung, Yuen and Zhong [11] studied a posteriori error estimation of the staggered discontinuous Galerkin method for time-harmonic Maxwell’s equations and proved that residual based a posteriori error indicator is both reliable and efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' As far as we know, there are not any published literatures on the posteriori error estimation of the adaptive mixed finite element method for H(curl)−elliptic problems (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='1)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' The main idea of the manuscript comes from [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' However, one of main tool, a Cl´ement-type quasi-interpolation operator given by [24], can not be used for 2D finite element space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Here, we use the Helmholtz decomposition and operators in articles [25, 4] for estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Here is some notation used throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' The following shorthand notation will be used to avoid the repeated constants, following [27], x ≲ y and x ≈ y means x ≤ C1y and C2x ≤ y ≤ C3x, where C1, C2 and C3 are generic positive constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' The rest of the article is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' In Section 2, we introduce some basic notations, present the variational form of the model problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='1)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='2), and design a residual type a posteriori error estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' In Section 3 and Section 4, we show that this indicator is reliable and effective, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' In Section 5, we report some numerical results in support of theoretical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Mixed IPDG method and a posteriori error indicator In this section, we give the continuous variational problem, the discrete variational problem of mixed IPDG method, and the definition of the a posteriori error indicator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Continuous variational problem For any domain D ⊂ R2, we use standard definitions for the Sobolev spaces Hs(D) and Hs(D) of scalar and vector-valued square integrable functions with inner products (·, ·)s,D and associated norms ∥·∥s,D for s ≥ 0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' We refer to L2(D) and L2(D) as the Hilbert spaces of scalar and vector-valued square integrable functions with inner products (·, ·)0,D and associated norms ∥ · ∥0,D, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' For simplicity, we drop the subscript when G = D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Then, the spaces are defined by H(curl, Ω) := � v : v ∈ L2(Ω), curl v ∈ L2(Ω) � , H0(curl, Ω) := {v : v ∈ H(curl, Ω), v · t = 0 on ∂Ω} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' 3 The space H(curl, Ω) is equipped with norm ∥v∥2 curl,Ω := ∥v∥2 0,Ω+∥curl v∥2 0,Ω for any v ∈ H(curl, Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' We simplify the symbols H0(curl, Ω) and L2(Ω) to U and Q, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' In this manuscript, we assume that f ∈ H(div, Ω) = {v : v ∈ L2(Ω), ∇ · v ∈ L2(Ω)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' The variational form for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='3)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='5) is to find (u, p) ∈ U × Q such that a(p, q) − b(u, q) = ℓ1(q), ∀q ∈ Q, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='6) d(v, p) + c(u, v) = ℓ2(v), ∀v ∈ U, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='7) where the four bilinear forms given by a(p, q) := (p, q), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='8) b(u, q) := (curl u, q), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='9) c(u, v) := (βu, v), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='10) d(v, p) := (curl v, αp), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='11) and two linear functionals ℓ1(·) ∈ Q∗, ℓ2(·) ∈ U ∗, where Q∗ and U ∗ are the dual spaces of Q and U, respectively, as follows ℓ1(q) := 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='12) ℓ2(v) := (f, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='13) In order to prove the well-posedness of continuous variational problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='6)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='7) and the relia- bility of a posteriori error indicator(see Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' We also define the operator A : (U × Q) �→ (U × Q)∗ by (A(u, p))(v, q) := a(p, q) − b(u, q) + d(v, p) + c(u, v), for all u, v ∈ U, p, q ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Thus, the operator form of the equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='6)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='7) is obtained (A(u, p))(v, q) = ℓ(v, q), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='14) where ℓ(v, q) = ℓ2(v) + ℓ1(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' The following lemma provides the existence and uniqueness of solutions to the variational problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='6)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='1 ([11], Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Let Ω be a bounded Lipschitz polygon with connected boundary ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Then A is a continuous and bijective linear operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Moreover, for any (ℓ1, ℓ2) ∈ Q∗ × U ∗ given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='16) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='17), respectively, then the system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='6)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='7) has a unique solution (u, p) ∈ U × Q such that ∥(u, p)∥U×Q := � ∥u∥2 U + ∥p∥2 Q �1/2 ≲ ∥ℓ1∥Q∗ + ∥ℓ2∥U ∗, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='15) where ∥ · ∥Q∗ and ∥ · ∥U ∗ are dual norms in Q∗ and U ∗, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Discrete variational problem Before presenting the discrete variational problem, we introduce some preliminaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Given a shape-regular triangulation Th for Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' For τ ∈ Th, we write hτ = |τ|1/2 to denote the local mesh size of the element τ, where |τ| is the Lebesgue measure of τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Let h = maxτ∈Th hτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Let Eh be the set of all the edges, E0 h = Eh\\∂Ω be the set of all the interior edges, and E∂ h = Eh∩∂Ω be the set of all the boundary edges, then Eh = E0 h � E∂ h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' For T ′ h ⊆ Th and E′ h ⊆ Eh,the discrete L2 inner product and norm are given by (v, w)T ′ h = � τ∈T ′ h (v, w)τ = � τ∈T ′ h � τ v · wdx, ∥v∥2 T ′ h = (v, v)T ′ h, ⟨v, w⟩E′ h = � e∈E′ h ⟨v, w⟩e = � e∈E′ h � e v · wds, ∥v∥2 E′ h = ⟨v, v⟩E′ h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' For any e ∈ E0 h with e = ∂τ1 ∩ ∂τ2, we define the average, tangential jump and normal jump for a vector function w by {{w}}e = (w|τ1 + w|τ2)/2, [[w]]e = w|τ1 · t1 + w|τ2 · t2, [w]e = w|τ1 · n1 + w|τ2 · n2, where w|τi denotes the value of w on τi, ti and ni are the unit tangential vectors and the outward unit normal vectors on e for τi (i = 1, 2), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Similarly, we define the average and the tangential jump on e for a scalar function φ as {{φ}}e = (φ|τ1 + φ|τ2)/2, [[φ]]e = φ|τ1 t1 + φ|τ2 t2, where φ|τi denotes the value of φ on τi, i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' For any e ∈ E∂ h, there is a element τ ∈ Th such that e ∈ ∂τ ∩∂Ω, we define the average, tangential jump and normal jump for a vector function w are defined as {{w}}e = w|τ, [[w]]e = w|τ · t, [w]e = w|τ · n where w|τ denotes the value of w on τ and n denotes the outward unit normal vectors on e for τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' For a scalar function φ, its average and tangential jump on e are defined as {{φ}}e = φ|τ, [[φ]]e = φ|τt, where φ|τ denote the value of φ on τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' The DG methods are based on the approximation of the vector field u and p by elementwise polynomials, thus giving rise to the finite dimensional function spaces U h := � vh ∈ L2(Ω) | vh|τ ∈ R1(τ), vh|e = 0, ∀τ ∈ Th � , 5 Qh := � qh ∈ L2(Ω) | qh|τ ∈ P0(τ), ∀τ ∈ Th � , where R1(τ) = � ∃α ∈ R2, ∃β ∈ R, ∀x = (x1, x2) ∈ τ : q(x) = α + β (−x2, x1) � , and P0(τ) denotes the constant in τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Now, we present the mixed interior penalty discontinuous Galerkin(MIPDG) finite element method for the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='3)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='4): find (uh, ph) ∈ U h × Qh such that ah(ph, qh) − bh(uh, qh) = ℓ1,h(qh) + d1,h(uh, qh), ∀qh ∈ Qh, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='16) dh(vh, ph) + ch(uh, vh) = ℓ2,h(vh) + d2,h(uh, vh), ∀vh ∈ U h, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='17) where ah(ph, qh) := (ph, qh)Th, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='18) bh(uh, qh) := (curlh uh, qh)Th, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='19) ch(uh, vh) := (βuh, vh)Th, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='20) dh(vh, ph) := (curlh vh, αph)Th, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='21) ℓ1,h(qh) := 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='22) ℓ2,h(vh) := (f, vh)Th, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='23) d1,h (uh, qh) := − < {{qh}}, [[uh]] >Eh, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='24) d2,h (uh, vh) :=< {{αcurlh uh}} − κh−1 e [[uh]], [[vh]] >Eh, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='25) with κ > 0 is a penalty parameter and should be taken large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Comparing with the continuous variational problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='6)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='7) and the discrete variational problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='16)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='17), the definitions of the bilinear terms, which without including curlh, are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' In order to be consistent with other symbols, we add the subscript h to the bilinear terms in the discrete variational form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' The calculation of curlh in the bilinear terms of the discrete variational problem is piecewise derivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Compared with the continuous variational form, the discrete variational form adds two terms d1,h and d2,h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' In order to give the well-posedness of the discrete variational problems, we need to introduce the suitable IPDG form of the H(curl)−elliptic problems: find uh ∈ U h, such that aIP (uh, vh) = (f, vh)Th , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='26) where aIP (uh, vh) = (βuh, vh)Th + (αcurl uh, curl vh)Th − < {{curl vh}}, [[αuh]] >Eh − < {{αcurl uh}}, [[vh]] >Eh +κ < h−1 e [[uh]], [[vh]] >Eh .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='27) 6 Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Let qh = αcurl vh in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='16)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='17), and subtract (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='16) from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='17) then lead to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' To provide the existence and uniqueness of solutions to the variational problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='26), we need to introduce the following norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' |||vh|||2 h = ∥curl vh∥2 Th + ∥vh∥2 Th + κ∥h − 1 2 e [[vh]]∥2 Eh, ∀vh ∈ (H1 (Th))2, κ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Similar to [2], by using the Cauchy-Schwarz inequality, trace inequality and inverse inequality, it is easy to verify that ah(·, ·) is bounded by ∥| · |∥h, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=', ah (wh, vh) ⩽ C∥|wh|∥h∥|vh|∥h, ∀wh, vh ∈ Vh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='28) Furthermore, for the coercivity of the bilinear forms ah(·.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='·) on Vh, we have ah (vh, vh) ⩾ C∥|vh|∥2 h, ∀vh ∈ Vh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='29) Combining (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='28) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='29), we obtain the well-posedness of the discrete variational problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Furthermore, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='2 shows that the variational problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='16)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='17) and the variational problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='26) have equivalent form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' The proof use similar arguments in [5] and is skipped here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' If (uh, ph) ∈ (U h, Qh) is the solution of equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='16)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='17), then uh ∈ U h is the solution of the variational problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' On the contrary, if uh ∈ U h is the solution of the variational problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='26), then there is a corresponding ph ∈ Qh makes (uh, ph) ∈ (U h, Qh) is the solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='16)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' A posteriori error indicator For any τ ∈ Th, e ∈ Eh and (vh, qh) ∈ Qh ×U h, we introduce the following element-wise residuals and edge-wise jump residuals as R1 (vh, qh) |τ := qh|τ − curlhvh|τ, R2 (vh, qh) |τ := f|τ − (curlh αqh + βvh) |τ, R3 (vh) |τ := ∇ · (f − βvh) |τ, J1 (qh) |e := [[αqh]]e, J2 (vh) |e := [f − βvh]e, J3 (vh) |e := [[vh]]e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' The local error estimator on τ ∈ Th is defined as η2 (vh, qh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' τ) :=∥R1 (vh, qh) ∥2 0,τ + h2 τ � ∥R2 (vh, qh) ∥2 0,τ + ∥R3 (vh) ∥2 0,τ � + � e∈∂τ he � ∥J1 (qh) ∥2 0,e + ∥J2 (vh) ∥2 0,e � + κ � e∈∂τ h−1 e ∥J3(vh)∥2 0,e , 7 where hτ denotes the diameter of the element τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' The mesh Th is shape-regular which implies that hτ ≈ he.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Then the global error estimator on Th is defined as η2 (vh, qh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Th) = � τ∈Th η2 (vh, qh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' τ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='30) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Reliability analysis For any (v, q) ∈ U × Q and (vh, qh) ∈ U h × Qh, we define the following error ∥(v, q) − (vh, qh)∥2 DG := ∥q − qh∥2 0,Ω + ∥v − vh∥2 0,Ω + ∥curlh (v − vh)∥2 0,Ω +κ � e∈Eh h−1 e ∥[[vh]]∥2 0,e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='31) Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Here we use [[vh]]e instead of [[v − vh]]e, since [[v]]e = 0 for v ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Next, we focus on proving the reliability of the error indicator defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' The key of our argument is to use the space decomposition technique: decompose the DG finite element solution uh into two parts: one is conforming part uconf h ∈ U conf h := U h ∩ U and the other is its L2 orthogonal part u⊥ h ∈ U ⊥ h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Therefore, we need to take care of continuous error ∥(u, p) − (uconf h , ph)∥DG instead of ∥(u, p) − (uh, ph)∥DG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' The main analysis tools for continuous error are Helmholtz decomposition and the two interpolations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' We prove the reliability of the error indicator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' The following lemmas provide some estimates related to the continuous error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Let (u, p) ∈ U × Q be solution of system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='6)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='7), then for any (vconf h , ph) ∈ U conf h × Qh, we have ∥(u − vconf h , p − ph)∥U×Q ≲ ∥˜ℓ1∥Q∗ + ∥˜ℓ2∥U ∗, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='32) where ˜ℓ1(q) = −a (ph, q) + b(vconf h , q), ∀q ∈ Q, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='33) ˜ℓ2(v) = ℓ2(v) − d (v, ph) − c(vconf h , v), ∀v ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='34) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' For the operator A given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='14), it is easy to obtain the linearity, namely, for any q1, q2, q ∈ Q and v1, v2, v ∈ U, we have (A (q1 + q2, v1 + v2)) (q, v) = (A (q1, v1)) (q, v) + (A (q2, v2)) (q, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Hence, we have (A(p − ph, u − vconf h ))(q, v) 8 = (A(p, u))(q, v) − (A(ph, vconf h ))(q, v) = ℓ2(v) − (a(ph, q) − b(vconf h , q) + d(v, ph) + c(vconf h , v)) := ˜ℓ1(q) + ˜ℓ2(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' At last, noting that (u − vconf h , p − ph) ∈ U × Q and using the definition of the operator norm, this completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' In the following lemmas, our purpose is to obtian upper bounds for ∥˜ℓ1∥Q∗ and ∥˜ℓ2∥U ∗ in Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Let (uh, ph) ∈ U h × Qh be solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='16)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' For any vconf h ∈ U conf h , we have ∥˜ℓ1∥Q∗ ≲ � � τ∈Th ∥R1 (uh, ph) ∥2 0,τ �1/2 + � � τ∈Th ∥curlh(vconf h − uh)∥2 0,τ �1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' For any q ∈ Q, by using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='33), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='8) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='9), we have ˜ℓ1(q) = −a (ph, q) + b(vconf h , q) = − (ph, q) + (curl vconf h , q) = (curlh uh − ph, q)Th + (curlh(vconf h − uh), q)Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Applying H¨older inequality and Cauchy-Schwarz inequality leads to |˜ℓ1(q)| ≤ � τ∈Th ∥curlh uh − ph∥0,τ ∥q∥0,τ + � τ∈Th ∥curlh(vconf h − uh)∥0,τ∥q∥0,τ ≤ 2 � � � � τ∈Th ∥R1 (uh, ph) ∥2 0,τ �1/2 + � � τ∈Th ∥curlh(uh − vconf h )∥2 0,τ �1/2� � ∥q∥0,Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' The proof is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' In order to estimate the term ∥˜ℓ2∥U ∗ in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='3, we shall use the following two interpolation operators with the corresponding approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' (1) Scott-Zhang quasi-interpolation Sh : H1 0(Ω) → {v ∈ C(Ω)| v|τ ∈ P1(τ), v|∂Ω = 0, ∀τ ∈ Th}, where P1(τ) represents a linear polynomial space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' The definition and approximation properties of Scott-Zhang quasi-interpolation interpolation were first proposed in [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' For ψ ∈ H1 0(Ω), there hold ∥∇Shψ∥0,τ ≲ ∥∇ψ∥0,ωτ , ∀τ ∈ Th, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='35) ∥ψ − Shψ∥0,τ ≲ hτ∥∇ψ∥0,ωτ , ∀τ ∈ Th, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='36) ∥ψ − Shψ∥0,e ≲ h 1 2e ∥∇ψ∥0,ωe, ∀e ∈ Eh, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='37) where ωτ := � τ ′∩τ̸=∅ τ ′ and ωe := � τ∩e̸=∅ τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' 9 (2) Vector-Valued operator P h : H1(Ω) ∩ U → U conf h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' The definition and approximation proper- ties of Vector-Valued operator were proposed in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' For q ∈ H1(Ω) ∩ U, there hold ∥P hq∥0,τ ≲ ∥q∥1,˜ωτ , ∀τ ∈ Th, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='38) ∥q − P hq∥0,τ ≲ hτ∥q∥1,˜ωτ , ∀τ ∈ Th, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='39) ∥q − P hq∥0,e ≲ h 1 2e ∥q∥1,˜ωe, ∀e ∈ Eh, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='40) where ˜ωe := � {τ ∈ Th(Ω) | e ∈ Eh(T)} and ˜ωτ := � {ωe | e ∈ Eh(T)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Let (uh, ph) ∈ U h × Qh be solution of system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='16)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' For any vconf h ∈ U conf h , then there exists a constant C1 > 0 depending only on ∥β∥0,∞, we have ∥˜ℓ2∥U ∗ ≤ C1 � � τ∈Th h2 τ � ∥R2 (uh, ph) ∥2 0,τ + ∥R3 (uh) ∥2 0,τ � + � e∈Eh he � ∥J1 (ph) ∥2 0,e + ∥J2 (uh) ∥2 0,e � + � τ∈Th ∥uh − vconf h ∥2 0,τ �1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' For any v ∈ U, we use Helmholtz decomposition v as follows (see the Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='1 of [4]) v := v0 + v⊥, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='41) and ∥v0∥2 0,Ω + ∥v⊥∥2 0,Ω = ∥v∥2 0,Ω, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='42) where v0 ∈ H0(curl0, Ω) := {v : v ∈ H0(curl, Ω), curl v = 0} and v⊥ ∈ H⊥(curl, Ω) := {v : v ∈ H0(curl, Ω), (v, v0) = 0, v0 ∈ H0(curl0, Ω)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' We make use of the representation H0(curl0, Ω) = grad H1 0(Ω), hence, we have v0 = ∇ψ for some ψ ∈ H1 0(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Applying the definition of ˜ℓ2 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='34), we have ˜ℓ2(v) = ˜ℓ2(v0 + v⊥) = ˜ℓ2(∇ψ + v⊥) = ˜ℓ2(∇Shψ + P hv⊥) + ˜ℓ2(∇ψ + v⊥ − ∇Shψ − P hv⊥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='43) We first analyze the first item on the right hand side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' According to the definition of ˜ℓ2 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='34), we have ˜ℓ2 � ∇Shψ + P hv⊥� = ℓ2 � ∇Shψ + P hv⊥� − d � ∇Shψ + P hv⊥, ph � − c(vconf h , ∇Shψ + P hv⊥) = ℓ2 � ∇Shψ + P hv⊥� − d � ∇Shψ + P hv⊥, ph � − c � uh, ∇Shψ + P hv⊥� +c(uh − vconf h , ∇Shψ + P hv⊥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='44) By using Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='1, for any e ∈ Eh, noting that [[∇Shψ + P hv⊥]]e = 0 since ∇Shψ + P hv⊥ ⊂ U, and choosing vh = P hv⊥ + ∇Shψ in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='17), we have ℓ2 � P hv⊥ + ∇Shψ � − d � P hv⊥ + ∇Shψ, ph � − c � uh, P hv⊥ + ∇Shψ � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='45) 10 Combining (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='44) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='45), we have ˜ℓ2 � ∇Shψ + P hv⊥� = c(uh − vconf h , ∇Shψ + P hv⊥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Applying Cauchy-Schwarz inequality, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='35), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='38) and H⊥ 0 (curl, Ω) is continuously imbedded in H1(Ω) ∩ H0(curl, Ω)(see [4]), we deduce ˜ℓ2 � ∇Shψ + P hv⊥� ≲ C1∥uh − vconf h ∥0,Ω(∥∇Shψ∥0,Ω + ∥P hv⊥∥0,Ω) ≲ C1∥uh − vconf h ∥0,Ω(∥∇ψ∥0,Ω + ∥v⊥∥0,Ω) ≲ C1∥uh − vconf h ∥0,Ω(∥∇ψ∥0,Ω + ∥curl v⊥∥0,Ω) ≲ C1∥uh − vconf h ∥0,Ω∥v∥curl,Ω, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='46) where C1 > 0 depends only on ∥β∥0,∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Next, we analyze the second item on the right hand side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='43), using the definition of ˜ℓ2 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='34),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' the fact (curlh(∇ψ − ∇Shψ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' αph)Th = 0 and Green formula,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' we have ˜ℓ2 � ∇ψ + v⊥ − ∇Shψ − P hv⊥� = ˜ℓ2 (∇ψ − ∇Shψ) + ˜ℓ2 � v⊥ − P hv⊥� = (f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' ∇ψ − ∇Shψ)Th − (curlh(∇ψ − ∇Shψ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' αph)Th − (βvconf h ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' ∇ψ − ∇Shψ)Th +(f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' v⊥ − P hv⊥)Th − (curlh(v⊥ − P hv⊥),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' αph)Th − (βvconf h ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' v⊥ − P hv⊥)Th = (f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' ∇ψ − ∇Shψ)Th − (βuh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' ∇ψ − ∇Shψ)Th + (β(uh − vconf h ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' ∇ψ − ∇Shψ)Th +(f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' v⊥ − P hv⊥)Th − (curlh(v⊥ − P hv⊥),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' αph)Th − (βuh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' v⊥ − P hv⊥)Th +(β(uh − vconf h ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' v⊥ − P hv⊥)Th = − (∇ · (f − βuh) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' ψ − Shψ)Th + ⟨[f − βuh]e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' ψ − Shψ⟩Eh +(β(uh − vconf h ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' ∇ψ − ∇Shψ)Th + � f − curlh αph − βuh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' v⊥ − P hv⊥� Th − � [[αph]]e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' v⊥ − P hv⊥� Eh + (β(uh − vconf h ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' v⊥ − P hv⊥)Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='47) Applying Cauchy-Schwarz inequality, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='36)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='40) and H⊥ 0 (curl, Ω) is continuously imbedded in H1(Ω) ∩ H0(curl, Ω), we deduce |˜ℓ2(v)| ≲ C1 � � τ∈Th h2 τ � ∥R2 (uh, ph) ∥2 0,τ + ∥R3(uh) ∥2 0,τ � + � e∈Eh he � ∥J1 (ph) ∥2 0,e + ∥J2 (uh) ∥2 0,e � + � τ∈T ∥uh − vconf h ∥2 0,τ �1/2 ∥v∥curl,Ω, which complete the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' In the end of this section, we give the upper bound of the error estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' 11 Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Let (u, p) ∈ U × Q and (uh, ph) ∈ U h × Qh be the solutions of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='6)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='7) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='16)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='17), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Then there exists a constant C1 > 0 depending only on ∥β∥∞,Ω and the grid shape regularity, we have ∥(u, p) − (uh, ph)∥2 DG ≲ C1η2(uh, ph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Th).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' For any vconf h ∈ U conf h , using the definition of ∥ · ∥DG in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='31), triangle inequality, we get ∥(u, p) − (uh, ph) ∥2 DG = ∥p − ph∥2 0,Ω + ∥u − uh∥2 0,Ω + ∥curl u − curlh uh∥2 0,Ω + κ � e∈Eh h−1 e ∥[[uh]]∥2 0,e = |p − ph∥2 0,Ω + ∥u − uh∥2 curl,Ω + κ � e∈Eh h−1 e ∥[[uh]]∥2 0,e ≤ ∥p − ph∥2 0,Ω + ∥u − vconf h ∥2 curl,Ω + ∥vconf h − uh∥2 curl,Ω + κ � e∈Eh h−1 e ∥[[uh]]∥2 0,e = ∥(u − vconf h , p − ph)∥U×Q + ∥vconf h − uh∥2 curl,Ω + κ � e∈Eh h−1 e ∥[[uh]]∥2 0,e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='48) For uh ∈ U h, there exists uconf h ∈ U conf h , u⊥ h ∈ U ⊥ h satisfing (see Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='10 of [22]) uh = uconf h + u⊥ h , and h2 τ∥curlh(uh − uconf h )∥2 0,Ω + ∥uh − uconf h ∥2 0,Ω ≲ h2 τ � e∈Eh h−1 e ∥[[uh]]e∥2 0,e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='49) At last, choosing vconf h = uconf h in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='48) and using Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='1, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='3 and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='49), we have ∥(u, p) − (uh, ph) ∥2 DG ≤ ∥(u − uconf h , p − ph)∥U×Q + ∥uconf h − uh∥2 curl,Ω + κ � e∈Eh h−1 e ∥[[uh]]∥2 0,e ≲ ∥˜ℓ1∥2 Q∗ + ∥˜ℓ2∥2 U ∗ + ∥uconf h − uh∥2 curl,Ω + κ � e∈Eh h−1 e ∥[[uh]]∥2 0,e ≲ C1η2(vh, ph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Th), which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Efficiency analysis In this section, we will prove that the error estimator is efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' In the Section 3, we have already proved that the error estimator is reliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Hence, the error estimator is a good indicator of energy error, that is η2 (uh, ph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Th) ≈ ∥ (p − ph, u − uh) ∥2 DG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='50) We fitst introduce the following theorem, which is the efficiency of the error estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' 12 Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Let (u, p) ∈ U × Q and (uh, ph) ∈ U h × Qh be the solutions of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='6)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='7) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='16)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='17), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' The space of all piecewise polynomial of arbitrary order on Th is denoted by P(Th).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Then for any f h ∈ Sconf h := P(Th) ∩ H(div;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Ω), we have η2 (uh, ph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Th) ≲ C2 � ∥ (p − ph, u − uh) ∥2 DG + ∥f − f h∥2 0,Ω + � τ∈Th h2 τ∥∇ · (f − f h) ∥2 0,τ � , where C2 is a constant depending on ∥α∥0,∞ and ∥β∥0,∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' We assume f ∈ H(div;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Ω) to show the proof of the Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' While, we should note that the conditional f h ∈ Sconf h is only required in the Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='7 ( See (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='78)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' For any τ ∈ Th, e ∈ Eh, let bτ and be be standard the interior bubble functions and edge bubble functions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' We introduce the following two lemmas which are used in later proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='1 ([16], Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Let χ be a scalar polynomial function on τ, then ∥bτχ∥0,τ ≲ ∥χ∥0,τ, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='51) ∥χ∥0,τ ≲ ∥b 1 2τ χ∥0,τ, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='52) ∥∇ (bτχ) ∥0,τ ≲ h−1 τ ∥χ∥0,τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='53) Moreover, let e be an edge shared by two triangles τ1 and τ2, let φ be a scalar polynomial function on e, the ∥φ∥0,e ≲ ∥b 1 2e φ∥0,e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='54) In addition, there exists an extension of beφ as ˜φb ∈ H1 0 � (τ 1 ∪ τ 2)◦� such that ˜φb|e = beφ and ∥˜φb∥0,τ ≲ h 1 2e ∥φ∥0,e, ∀τ ∈ ωe, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='55) ∥∇˜φb∥0,τ ≲ h − 1 2 e ∥φ∥0,e, ∀τ ∈ ωe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='56) Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='2 ([16], Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Let w be a vector-valued polynomial function on τ, then ∥bτw∥0,τ ≲ ∥w∥0,τ, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='57) ∥w∥0,τ ≲ ∥b 1 2τ w∥0,τ, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='58) ∥curl (bτw) ∥0,τ ≲ h−1 τ ∥w∥0,τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='59) Moreover, let e be an edge shared by two triangles on τ1 and τ2, and let w be a vector-valued polynomial function on e, then ∥w∥0,e ≲ ∥b 1 2e w∥0,e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='60) In addition, there exists an extension of bew as ˜wb ∈ � H1 0 � (τ 1 ∪ τ 2)◦��2, such that ˜wb|e = bew, and ∥ ˜wbc∥0,τ ≲ h 1 2e ∥w∥0,e, ∀τ ∈ ωe, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='61) ∥curl ˜wb∥0,τ ≲ h − 1 2 e ∥w∥0,e, ∀τ ∈ ωe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='62) 13 Next, we estimate each term of the error indicator η(uh, ph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Th) given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='30), separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' We first estimate the first term R1 of the error estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Let (u, p) ∈ U × Q and (uh, ph) ∈ U h × Qh be the solutions of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='6)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='7) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='16)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='17), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Then we obtain � τ∈Th ∥R1 (uh, ph) ∥2 0,τ ⩽ ∥ (p − ph, u − uh) ∥2 DG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='63) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' We can observe that R1 (uh, ph) = ph − curlhuh ∈ Qh ⊂ Q, and taking q = R1 (uh, ph) in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='6), we have (p − curl u, R1 (uh, ph))Th = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='64) Thus, by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='64) and Cauchy-Schwarz inequality, we obtain � τ∈Th ∥R1 (uh, ph) ∥2 0,τ = (ph − curlh uh, R1 (uh, ph))Th = (ph − p − curlh (uh − u) , R1 (uh, ph))Th ⩽ (∥ph − p∥0,Ω + ∥curlh (uh − u) ∥0,Ω) ∥R1 (uh, ph) ∥0,Ω, which implies (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='63).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' We estimate the second term R2 of the error estimator η(uh, ph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Th).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Let (u, p) ∈ U × Q and (uh, ph) ∈ U h × Qh be the solutions of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='6)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='7) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='16)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='17), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Then we have � τ∈Th h2 τ∥R2 (uh, ph) ∥2 0,τ ≲ C2 � ∥(u, p) − (uh, ph)∥2 DG + � τ∈Th h2 τ ∥f − f h∥2 0,τ � , where C2 is a constant depending on ∥α∥0,∞ and ∥β∥0,∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Let f h ∈ Sconf h and let w := f h − curlh αph − βuh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Hence w is also a polynomial on τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Let wb = bτw, we can observe that wb ∈ H1 0(τ) ⊂ H1 0(Ω) ⊂ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Setting v = wb in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='7) and using Green’s formula, we get (f − curl αp − βu, wb)τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='65) Then by using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='58), the definitons of w and wb, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='65) and Cauchy-Schwarz inequality, we have ∥w∥2 0,τ ≲ ∥b1/2 τ w∥2 0,τ = (w, bτw)τ = ((f h − curlh αph − βuh) , wb)τ = ((f h − f) − curlh α (ph − p) − β (uh − u) , wb)τ ≲ � ∥f h − f∥0,τ + ∥β∥0,∞ ∥uh − u∥0,τ � ∥wb∥0,τ − (curlh α (ph − p) , wb)τ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='66) 14 According to the last term of right hand side (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='66), using Green’s formula with the fact wb = 0 on ∂τ, Cauchy-Schwarz inequality and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='59), we have (curlh α (ph − p) , wb)τ = (α(ph − p), curlh wb)τ ⩽ ∥α∥0,∞ ∥ph − p∥0,τ ∥curlh wb∥0,τ ≲ h−1 τ ∥α∥0,∞ ∥ph − p∥0,τ ∥w∥0,τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='67) Using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='66) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='67), we get ∥w∥0,τ ≲ ∥f h − f∥0,τ + ∥β∥0,∞ ∥uh − u∥0,τ + h−1 τ ∥α∥0,∞ ∥ph − p∥0,τ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='68) By the definition of R2 (uh, ph) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='68), we obtain ∥R2 (uh, ph) ∥0,τ ⩽ ∥w∥0,τ + ∥f h − f∥0,τ ≲ C2 � ∥f h − f∥0,τ + ∥uh − u∥0,τ + h−1 τ ∥ph − p∥0,τ � , which implies � τ∈Th h2 τ∥R2 (uh, ph) ∥2 0,τ ≲ C2 � � τ∈Th � h2 τ ∥f h − f∥2 0,τ + h2 τ ∥uh − u∥2 0,τ + ∥ph − p∥2 0,τ � � ≲ C2 � ∥(u, p) − (uh, ph)∥2 DG + � τ∈Th h2 τ ∥f h − f∥2 0,τ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' In the following, we turn to estimate the third term R3 of the error estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Let (u, p) ∈ U × Q and (uh, ph) ∈ U h × Qh be the solutions of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='6)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='7) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='16)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='17), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Then we have � τ∈Th h2 τ∥R3 (uh) ∥2 0,τ ≲ C1 � ∥u − uh∥2 0,Ω + ∥f − f h∥2 0,Ω + � τ∈Th h2 τ∥∇ · (f − f h) ∥2 0,τ � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='69) where C1 is a constant depending on ∥β∥0,∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Let f h ∈ Sconf h , χ := ∇ · (f h − βuh) and χb = bτχ, where χ is a polynomial on each τ ∈ T and χb ∈ H1 0(τ) ⊂ H1 0(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Setting v = ∇χb ∈ U in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='7) and with the fact χb|∂τ = 0, we get (f − βu, ∇χb)τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='70) Then, using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='52), the Green’s formula and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='70), we obtain ∥χ∥2 0,τ ≲ ∥b1/2 τ χ∥2 0,τ = (χ, bτχ) = (∇ · (f h − βuh) , χb)τ = −((f h − βuh) , ∇χb)τ = ((f − f h) − β (u − uh) , ∇χb)τ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' 15 According to the inverse inequalities, there holds ∥χ∥2 0,τ ⩽ � ∥f h − f∥0,τ + ∥β∥0,∞ ∥uh − u∥0,τ � ∥∇χb∥0,τ ≲ C1h−1 τ � ∥f h − f∥0,τ + ∥uh − u∥0,τ � ∥χ∥0,τ, which imples hτ∥χ∥ ≲ C1 � ∥f h − f∥0,τ + ∥uh − u∥0,τ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='71) Then combing the definition of R3(uh), triangle inequality and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='71), we get hτ∥R3 (uh) ∥0,τ ⩽ hτ∥χ∥0,τ + hτ∥∇ · (f − f h) ∥0,τ ≲ C1 (∥f h − f∥0,τ + ∥uh − u∥0,τ) + hτ∥∇ · (f − f h) ∥0,τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Summing over all elements τ ∈ T , the result (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='69) follows directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Next, we turn to bound the fourth term J1 of the error estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Let (u, p) ∈ U × Q and (uh, ph) ∈ U h × Qh be the solutions of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='6)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='7) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='16)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='17), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Then we have � e∈Eh he∥J1 (ph) ∥2 0,e ≲ C2 � � τ∈Th ∥R1 (uh, ph) ∥2 0,τ + � τ∈Th h2 e∥R2 (uh, ph) ∥2 0,τ + ∥(u, p) − (uh, ph)∥2 DG � , where C2 is a constant depending on ∥α∥0,∞ and ∥β∥0,∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Without loss of generality, let e ∈ Eh be the common edge of τ1 ∈ Th and τ2 ∈ Th, we give the following definition wh = [[αph]]e, ˜wb = bewh ∈ H1 0(τ1 ∪ τ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='72) Applying (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='72), [[αcurl u]]e = 0, Green’s fromula, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='6) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='7), we get ∥b 1 2e wh∥2 0,e = ⟨wh, bewh⟩e = ⟨[[αph]]e, ˜wb⟩e = ⟨[[α(ph − curl u)]]e, ˜wb⟩e = � τ∈ωe [(curl (α(ph − curl u)) , ˜wb)τ − (α(ph − curl u), curl ˜wb)τ] = � τ∈ωe [(curl αph − curl αcurl u + f − f + βu − βu + βuh − βuh, ˜wb)τ − (α(ph − curl u), curl ˜wb)τ] = � τ∈ωe [(curl αph + βuh − f, ˜wb)τ + (βu − βuh, ˜wb)τ 16 − (α(ph − curl u), curl ˜wb)τ] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='73) Using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='60), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='73), Cauchy-Schwarz inequality, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='61) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='62),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' there holds ∥wh∥2 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='e ≲ ∥b 1 2e wh∥2 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='e ≲ � τ∈ωe [(curl αph + βuh − f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' ˜wb)τ + (βu − βuh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' ˜wb)τ − (α(ph − curl u),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' curl ˜wb)τ] ≲ � τ∈ωe � ∥curl αph + βuh − f∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='τ · ∥ ˜wb∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='τ + ∥β∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='∞ ∥u − uh∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='τ · ∥ ˜wb∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='τ +∥α∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='∞ ∥ph − curl u∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='τ · ∥curl ˜wb∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='τ � ≲ C2 � τ∈ωe � ∥curl αph + βuh − f∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='τ · h 1 2e ∥wh∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='e + ∥u − uh∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='τ · h 1 2e ∥wh∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='e + ∥ph − curl u∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='τ · h − 1 2 e ∥wh∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='e � ≲ C2 � τ∈ωe � h 1 2e ∥curl αph + βuh − f∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='τ + h 1 2e ∥u − uh∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='τ +h − 1 2 e ∥ph − curlh uh∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='τ + h − 1 2 e ∥curl u − curlh uh∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='τ � ∥wh∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='e ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' which imples ∥wh∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='e ≲ C2 � τ∈ωe � h 1 2e ∥f − curl αph − βuh∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='τ + h 1 2e ∥u − uh∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='τ +h − 1 2 e ∥ph − curlh uh∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='τ + h − 1 2 e ∥curl u − curlh uh∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='τ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='74) Applying (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='72) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='74) results in he∥J1(ph)∥2 0,e = he∥[[αph]]τ∥2 0,e = he ∥wh∥2 0,e ≲ C2 � � τ∈ωe � ∥R1(uh, ph)∥0,τ + h2 e∥R2(uh, ph)∥0,τ � + ∥ (p − ph, u − uh) ∥2 DG � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' At last, we turn to bound the fifth term J2 of the error estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Let (u, p) ∈ U × Q and (uh, ph) ∈ U h × Qh be the solutions of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='6)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='7) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='16)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='17), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Then we have � e∈Eh he∥J2 (uh) ∥2 0,e ≲ C2 � ∥u − uh∥2 0,Ω + ∥f − f h∥2 0,Ω + � τ∈Th h2 τ∥∇ · (f − f h) ∥2 0,Ω � , where C1 is a constant depending on ∥β∥0,∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' 17 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Without loss of generality, let e ∈ Eh be the common edge of τ1 and τ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Setting φ = [(f h − βuh)]e and ˜φb = beφ, then according to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='54), there holds ∥φ∥2 0,e ≲ ∥b1/2 e φ∥2 0,e = (φ, beφ) = ⟨[(f h − βuh)]e, ˜φb⟩e = � i=1,2 � (∇ · (f h − βuh) , ˜φb)τi + (f h − βuh, ∇˜φb)τi � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='75) where we have used the fact that ˜φb ∈ H1 0 (τ1 ∪ τ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Applying the definition of R3 (uh) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='55) to the first term (∇ · (f h − βuh) , ˜φb)τi of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='75) gives (∇ · (f h − βuh) , ˜φb)τi = (R3 (uh) , ˜φb)τi + (∇ · (f h − f) , ˜φb)τi ⩽ (∥R3 (uh) ∥0,τi + ∥∇ · (f − f h) ∥0,τi) · ∥˜φb∥0,τi ≲ h 1 2e (∥R3 (uh) ∥0,τi + ∥∇ · (f − f h) ∥0,τi) · ∥φ∥0,e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='76) Using the Cauchy-Schwarz inequality and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='56) to the second term of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='75) gives � i=1,2 � f h − βuh, ∇˜φb � τi = � i=1,2 � f h − f − β (uh − u) , ∇˜φb � τi ≲ C1 � i=1,2 � ∥f − f h∥0,τi + ∥u − uh∥0,τi � ∥∇˜φb∥0,τi ≲ C1 � i=1,2 h−1/2 e � ∥f − f h∥0,τi + ∥u − uh∥0,τi � ∥φ∥0,e, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='77) where we have used � f − βu, ∇˜φb � τ1∪τ2 = 0 when we set v = ∇˜φb ∈ U in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' According to the assumptions f ∈ H(div, Ω) and f h ∈ Sconf h , there holds [f − f h]e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='78) Due to the definition of J2 (uh) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='75)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='78), we arrive at ∥J2 (uh) ∥0,e = ∥φ∥0,e + ∥[(f − f h)]e∥0,e ≲ C1 � i=1,2 � h1/2 e (∥R3 (uh) ∥0,τi + ∥∇ · (f − f h) ∥0,τi) + h−1/2 e � ∥f − f h∥0,τi + ∥u − uh∥0,τi � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Hence, combining Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='5, we have � e he∥J2 (uh) ∥2 0,e ≲ C1 � τ � h2 τi � ∥R3 (uh) ∥2 0,τi + ∥∇ · (f − f h) ∥2 0,τi � + ∥f − f h∥2 0,τi + ∥u − uh∥2 0,τi � ≲ C1 � ∥u − uh∥2 0,Ω + ∥f − f h∥2 0,Ω + h2 τ∥∇ · (f − f h) ∥2 0,Ω � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' 18 Next, we present the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' The following result follows immediately by a direct application of Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='3-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='7, κ � e∈Th � e∈∂τ h−1 e ∥J3(uh)∥2 0,e ≲ ∥(u, p) − (uh, ph)∥DG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Numerical experiment In this section, we report some experiments to show the performance of the error indicator and the adaptive algorithm AMIPDG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' We carry out these numerical experiments by using the MATLAB software package iFEM [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' In Experiments 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='1 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='2, we take p = curl u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' In Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='1, we discuss the influence of the penalty parameter κ on the error both in L2 and ∥ · ∥DG norms, and observe the dependency of the condition number of stiffness matrix on κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' At last, we verify the reliability and efficiency of the constructed error indicator (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Let Ω := [−1, 1] × [−1, 1], we construct the following analytical solution of the model (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='1)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='2): u = � cos(πx) sin(πy) − cos(πy) sin(πx) � with coefficients α = 1, β = 1, which corresponds to a right hand source term f(x, y) = � � 2π2 − 1 � cos(πx) sin(πy) − � 2π2 − 1 � cos(πy) sin(πx) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' It is easy to see that the solution u satisfies the boundary condition u · t = 0 on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' In this example, we get an uniform meshes by partitioning the x− and y− into equally distributed M(M ≥ 2) subintervals, and then dividing one square into two triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Let h = 1/M be mesh sizes for different triangular meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Firstly, we fixed mesh with h = 1/32 and report the error estimates in both L2 and ∥ · ∥DG norm for different penalty parameters κ = 1, 50, 100, 150 and 200 in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' We note that ∥u − uh∥0 increases slightly as the penalty parameter κ increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' On the contrary, ∥ (p − ph, u − uh) ∥DG decreases slightly as κ increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Next, we also use fixed mesh with h = 1/32, and observe the influence of different κ on the condition numbers of stiffness matrices in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' It is easy to see that the condition numbers of stiffness matrices increase with the increase of penalty parameters κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' As a way to balance, in the following numerical tests, we always choose κ = 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' 19 Table 1: The errors in both L2 and ∥ · ∥DG norms with h = 1/32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' κ ∥u − uh∥0 ∥ (p − ph, u − uh) ∥DG 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='00399e-02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='51333e-01 50 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='00391e-02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='46739e-01 100 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='00397e-02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='46733e-01 150 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='00400e-02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='46732e-01 200 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='00401e-02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='46732e-01 Table 2: Condition number of stiffness matrices with different κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' κ 1 50 100 150 200 Cond 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='40461e+05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='11946e+07 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='42288e+07 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='73104e+07 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='04046e+07 Now, we verify the reliability and efficiency of the error estimate by comparing η (uh, ph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Th) with ∥(p − ph, u − uh)∥DG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' The numerical results are given in Table 3 which also provide the values for the effectivity index σ = ∥ (p − ph, u − uh) ∥DG/η (uh, ph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Th).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' We observe that the convergence rate of ∥ (p − ph, u − uh) ∥DG is first order and the effectivity index σ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='286.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' This shows that the error indicator is effective and reliable, which is (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Table 3: Rate of convergence of the ∥ (p − ph, u − uh) ∥DG and η (uh, ph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Th) on uniform triangular meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' h ∥ (p − ph, u − uh) ∥DG η (uh, ph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Th) index Error order Error order σ 1/16 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='93259E-01 N/A 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='02621E-00 N/A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='286 1/32 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='46739E-01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='9989 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='13690E-01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='9984 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='286 1/64 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='33830E-02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='9997 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='56920E-01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='9996 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='286 1/128 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='66932E-02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='9999 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='28470E-02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='9999 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='286 1/256 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='83468E-02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='0000 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='42365E-02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='0000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='286 Noting that we only consider uniform meshes and the constant coefficients in Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Next we test adaptive meshes and the jump coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Our adaptive cycle can be implemented by the following algorithm: Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Let Ω := [−1, 1] × [−1, 1], we construct the following analytical solution of the model (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='1)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='2) u = � � y(x2−1)(y2−1) x2+y2+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='02 −x(x2−1)(y2−1) x2+y2+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='02 � � , with the jump coefficients α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='0, β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='0, on Ω1, α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='0, β = 100, on Ω\\Ω1, 20 Algorithm 1 An adaptive mixed interior penalty discontinuous method (AMIPDG) cycle Input initial triangulation T0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' data f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' tolerance tol;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' marking parameter θ ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Output a triangulation TJ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' MIPDG solution (uJ, pJ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' η = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' k = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' while η ≥ tol SOLVE solve discrete varational problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='16)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='17) on Tk to get the solution (uk, pk);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' ESTIMATE compute the posterior error estimator η = η(uk, pk, Tk) by using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='30);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' MARK seek a minimum cardinality Mk ⊂ Tk such that η2 (uk, pk, Mk) ≥ θη2 (uk, pk, Tk) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' REFINE Bisect elements in Mk and the neighboring elements to form a conforming Tk+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' k = k + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' end uJ = uk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' pJ = pk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' TJ = Tk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' where Ω1 = (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='5)2(see the left of Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Note that the solution u satisfies the condition u · t = 0 on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Here, we can observe that u has a relatively large change at (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Figure 1 shows the contours of the exact solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Figure 1: Left: the first component of the analytical solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Right: the second component of the analytical solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' We get an initial mesh T0 by partitioning the x- and y-axes into equally distributed eight subin- tervals and then dividing one square into two triangles, see the middle of Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' The right of Figure 2 shows an adaptively refined mesh with marking parameter θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='5 after k = 8, and we can see that the grid is locally refined near both the origin and at ∂Ω1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' 21 N 5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='5 ： 1 x5 N 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='5 y 1Figure 2: Left: regional diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Middle: the initial mesh with 512 DoFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Right: the adaptive mesh(θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='5) with 20416 DoFs after 8 refinements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Figure 3 shows the curves of ln N − ln ∥(p − pk, u − uk)∥DG for different marking parame- ters θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='5 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='7, where N is the number of degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' The curves indicate the convergence and the quasi-optimality of the adaptive algorithm AMIPDG of the energy error ∥ (p − ph, u − uh) ∥DG, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' ∥(p − pk, u − uk)∥DG ≲ N −1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' From these curves, it seems that the convergence rate are robust for θ changing from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='3 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Figure 3: Quasi optimality of the AMIPDG of the error ∥ (p − ph, u − uh) ∥DG with different marking parameters θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' The examples above are considered convex domain and the jump coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Finally we consider variable coefficients, ‘L-shaped’ domain and unknown exact solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' 22 1 212 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='5 1Rate of convergence is CN-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='5 adaptive refiniment =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='3 adaptive refiniment =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='5 adaptive refiniment =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='7 10 C-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='5 JOJE 100 104 Number of unknownsExample 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Let domain Ω = (−1, 1)2/([0, 1) × [0, 1)) and let the variable coefficients α = 1 1 + x2 + y2 , β = � 1 + x2 xy xy 1 + y2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' We set the homogeneous Dirichlet boundary condition u·t = 0 on ∂Ω, the source f = ( 1 x2+y2+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='01, 1 x2+y2+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='01).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' We get an initial mesh T0 by partitioning the x- and y-axes into equally distributed eight subin- tervals and then dividing one square into two triangles, see the left of Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' The right of Figure 4 shows an adaptively refined mesh with marking parameter- θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='5 after k = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' The grid is locally refined near the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Figure 4: Left: the initial mesh with 384 DoFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Right: the adaptive mesh(θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='5) with 16032 DoFs after 8 refinements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' The Figure 5 shows the curves of ln N − ln η (uk, pk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Tk) for parameters θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' The curves indicate the convergence and the quasi-optimality of the adaptive algorithm AMIPDG of η (uk, pk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Tk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Acknowledgments The authors are supported by the National Natural Science Foundation of China (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' 12071160).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' The second author is also supported by the National Natural Science Foundation of China (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' 11901212).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' References [1] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQfmf09/content/2301.01563v1.pdf'} +page_content=' Arnold, An interior penalty finite element method with discontinuous elements, Siam J.' metadata={'source': 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b/aNE1T4oBgHgl3EQfKQOD/content/tmp_files/2301.02962v1.pdf.txt @@ -0,0 +1,5404 @@ +Bayesian Graphical Entity Resolution using +Exchangeable Random Partition Priors∗ +Neil G. Marchant and Benjamin I. P. Rubinstein +School of Computing and Information Systems +University of Melbourne +Rebecca C. Steorts +Departments of Statistical Science and Computer Science +Duke University +Abstract +Entity resolution (record linkage or de-duplication) is the process of identifying and linking +duplicate records in databases. In this paper, we propose a Bayesian graphical approach for +entity resolution that links records to latent entities, where the prior representation on the linkage +structure is exchangeable. First, we adopt a flexible and tractable set of priors for the linkage +structure, which corresponds to a special class of random partition models. Second, we propose +a more realistic distortion model for categorical/discrete record attributes, which corrects a +logical inconsistency with the standard hit-miss model. Third, we incorporate hyperpriors +to improve flexibility. Fourth, we employ a partially collapsed Gibbs sampler for inferential +speedups. Using a selection of private and non-private data sets, we investigate the impact of +our modeling contributions and compare our model with two alternative Bayesian models. In +addition, we conduct a simulation study for household survey data, where we vary distortion, +duplication rates and data set size. We find that our model performs more consistently than the +alternatives across a variety of scenarios and typically achieves the highest entity resolution +accuracy (F1 score). Open source software is available for our proposed methodology, and we +provide a discussion regarding our work and future directions. +Statement of Significance +In survey statistics, entity resolution (record linkage) is used to identify responses submitted +by the same individual across multiple surveys, even when unique identifiers such as social +∗This is a pre-copyedited, author-produced version of an article accepted for publication in the Journal of Survey +Statistics and Methodology following peer review. The version of record +Marchant, N. G., Rubinstein, B. I. P., and Steorts, R. C. (2023), “Bayesian Graphical Entity Resolution +using Exchangeable Random Partition Priors,” Journal of Survey Statistics and Methodology +is available online at: https://doi.org/10.1093/jssam/smac030. +1 +arXiv:2301.02962v1 [stat.ME] 8 Jan 2023 + +security numbers are not recorded. This paper advances Bayesian methods for entity resolution +by: (i) thoroughly evaluating a general class of priors on links between responses and individuals, +and (ii) proposing a more realistic model for distortions that appear in individuals’ identifying +attributes (name, address, etc.). Both of these contributions are evaluated independently and +jointly on a variety of data sets, one of which is a longitudinal medical survey. The results +show that the general class of priors – the Ewens-Pitman random partitions – achieve similar +accuracy in three previously studied parameter regimes, so long as vague hyperpriors are used. +This is an important insight, as it addresses questions in the literature about which parameter +regime (if any) is most suitable for entity resolution. In addition, the proposed distortion model +is found to significantly improve the accuracy of entity resolution, particularly for string-type +attributes. We provide two simulation studies supporting our work and comparisons. The paper +is complemented by an R package that implements the proposed model using an optimized +partially collapsed Gibbs sampler. +Keywords: +entity resolution, record linkage, Bayesian models, exchangeability, random +partitions, Ewens-Pitman random partitions +1 +Introduction +As commonly known in the literature, entity resolution (ER; record linkage or de-duplication) is the +process of taking large, noisy (dirty) databases and removing duplicate records (often in the absence +of a unique identifier) (Doan et al., 2012; Elmagarmid et al., 2007; Naumann and Herschel, 2010; +Getoor and Machanavajjhala, 2012; Christen, 2012a; Christophides et al., 2020; Ilyas and Chu, 2019; +Papadakis et al., 2021; Binette and Steorts, 2022). This problem has become increasingly important +in many fields, such as survey methodology, official statistics, computer science, political science, +health care, human rights, and others. In this paper, we are motivated by several applications. +For example, we consider a longitudinal health care survey, where information is categorical in +nature due to privacy restrictions on the data set. This may be of interest to those in the survey +methodology community as they may face similar issues. In addition, we consider categorical and +string or textual based data sets (or surveys) such as bibliographic/citation documents, information +from restaurants, and a traditional benchmark (synthetic) study. The goal of analyzing multiple data +sets is to make the survey community more aware of data sets that are relevant for entity resolution +methods. Other overarching goals are to extend recent Bayesian graphical ER methodology for these +data sets, provide comparative analyses, simulation studies, and guidance to researchers. Moreover, +we provide open-source software for the community for our proposed method and two recently +proposed methods in the literature. +The idea of entity resolution dates back to Dunn (1946), who envisioned a “book of life” that +would piece together information about an individual. Newcombe et al. (1959) proposed one of +the first methods for performing ER, based on a heuristic statistical test. This method was later +formalized by Fellegi and Sunter (1969), who developed a model based on agreement patterns +between pairs of records, and a likelihood ratio test for classifying pairs as linked (referring to the +same entity), possibly linked or non-linked. Under some strong assumptions, they showed that their +method – now known as the Fellegi-Sunter (FS) method – is statistically optimal. The FS method +has been advanced over many decades in the ER literature, especially in survey methodology due to +2 + +its scalability, ease of use and simplicity (Winkler, 2006; Christen, 2012a; Sadinle and Fienberg, +2013; Enamorado et al., 2019). However, it has some inherent limitations: it makes inconsistent +(intransitive) predictions, it does not naturally account for uncertainty, it cannot exploit patterns at +the entity-level, and it is incompatible with generative modeling approaches (Tancredi and Liseo, +2011). +Some of these limitations can be addressed by adapting the FS method to a Bayesian setting, while +also imposing consistency constraints on the links between records (Sadinle, 2014, 2017). For +example, Sadinle (2014) proposed a Bayesian extension of the FS model for performing ER within a +single database. It incorporates consistency (transitivity) constraints by requiring that records are +partitioned into groups that are mutually linked. In addition, the model supports multiple levels +of agreement, and incorporates priors on the links and 𝑚/𝑢 probabilities (from the FS model). In +contrast with traditional FS methods, quantification of ER uncertainty is possible by computing the +posterior distribution on the links. However, despite these benefits, the model has not been widely +examined in the literature, perhaps in part due to the lack of a publicly-available implementation. +One of our goals in this paper is to evaluate Sadinle’s model as a representative example of a +Bayesian FS model, and compare it to another class of Bayesian entity resolution models, which we +now describe. +In parallel to developments in Bayesian FS models, others have proposed a new class of generative +models called Bayesian graphical entity resolution models (Tancredi and Liseo, 2011; Steorts, 2015; +Steorts et al., 2016). In contrast with FS methods, these models do not operate on agreement +patterns between pairs of records. Instead, they model a latent population of entities and the process +by which records are generated from entities. They are known as “graphical” models because the +fundamental objects in the model form a bipartite graph – the latent entities correspond to one set of +vertices, the records correspond to another set of vertices, and the links are edges that connect the +vertices. Tancredi and Liseo (2011) proposed one of the first models of this kind for performing +ER across two databases. Subsequently, Steorts et al. (2016) proposed an extension to multiple +databases, while optionally allowing for duplicates within each database. However, Steorts et al. +(2016) discovered a limitation of the their model and the model by Tancredi and Liseo (2011) – the +uniform prior on the linkage structure is highly informative about the number of entities present +in the data. They noted that future work ought to consider more appropriate priors on the linkage +structure for ER. +The models by Tancredi and Liseo (2011) and Steorts et al. (2016) both assume entities are described +by a set of latent categorical attributes (e.g., date of birth, gender) which are distorted in the records. +However, their model of the distortion process is simple and is unable to capture realistic distortions +in string-type attributes, such as names. Steorts (2015) addressed this problem by proposing a string +pseudo-likelihood and an empirically-motivated prior in a model known as blink. The blink model +was later used as a foundation by Marchant et al. (2021) for developing more scalable Bayesian +graphical ER techniques. They proposed an end-to-end method that jointly performs blocking and +ER, where inference can be distributed or parallelized at the block level. Importantly, this enables +propagation of blocking uncertainty to the ER task. They observed a 200× speed-up, which allowed +them to scale blink to a data set containing over one million records. However, the blink model +uses the same uniform prior on the linkage structure as Tancredi and Liseo (2011) and Steorts et al. +3 + +(2016), and suffers from the same limitations. +Motivated by the shortcomings of existing Bayesian graphical ER models, we propose and evaluate +several modeling refinements in this paper. First, we propose a flexible and tractable set of priors for +the linkage structure that are the Ewens-Pitman (EP) family of random partition models (Pitman, +2006). These are the most general family of priors that satisfy exchangeability (an elementary +requirement) and they are more flexible than the uniform priors used in previous work. Second, we +incorporate hyperpriors on the EP parameters to further increase flexibility and reduce the need +for tuning. This is motivated by the informativeness of the uniform prior used in previous work +(Steorts et al., 2016). Third, we propose a more nuanced distortion model for categorical/discrete +attributes, which corrects an inconsistency with the standard hit-miss model used by Tancredi and +Liseo (2011); Steorts et al. (2016); Steorts (2015). Fourth, we design a partially collapsed Gibbs +sampler to fit our model which incorporates computational optimizations. +We evaluate our modeling contributions independently and jointly on a selection of private and +non-private data sets, and compare our model with the Bayesian graphical ER model by Steorts +(2015) and the Bayesian FS model by Sadinle (2014). We also evaluate our model (and the +alternatives) in a controlled simulation study, where we generate synthetic household survey data, +with varying numbers of records, levels of distortion, and rates of duplication. Overall, we find +our model is more robust across the various scenarios tested, and it typically achieves superior ER +accuracies. We provide open source software for all of the ER methods under evaluation, and we +provide a discussion of our contributions and directions for future work. +The rest of the paper proceeds as follows. Section 2 provides background on ER and exchangeable +random partitions and outlines notation used throughout the paper. Section 3 outlines our proposed +Bayesian graphical ER model. Section 4 presents a partially collapsed Gibbs sampling algorithm +for approximating the posterior distribution and other computational speedups. Section 5 presents +an empirical study of our proposed distortion model and linkage structure priors and includes a +comparison to two recent Bayesian ER models. Section 5.5 summarizes a controlled simulation +study that is in the Appendix. Section 6 summarizes our findings. +2 +Background and Notation +In this section, we provide notation, assumptions, and a review of exchangeable random partitions +which are used as a prior in our model. Figure 2 includes an index of symbols used throughout the +paper. +2.1 +Notation and Assumptions +We review notation and assumptions used throughout the paper. We assume the data (from one +or more sources) is structured, meaning that it has been standardized using schema alignment +techniques. For the purposes of our paper, the data is represented in a tabular format, where rows +correspond to records and columns correspond to attributes. This in contrast to unstructured entity +resolution, which deals with textual descriptions (paragraphs) or images. For a full review of these +terms, see Papadakis et al. (2021). +4 + +Let 𝑠 ∈ {1, . . . , 𝑆} be an index over the data sources and 𝑖 ∈ {1, . . . , 𝑁} be an index over the records, +which is unique across all sources. The source of the 𝑖-th record is denoted by 𝜍𝑖 ∈ {1, . . . , 𝑆} and +the record’s attribute values are represented as a tuple x𝑖 = (𝑥𝑖1, . . . , 𝑥𝑖𝐴) indexed by 𝑎 ∈ {1, . . . , 𝐴}. +Assume 𝑥𝑖𝑎 ∈ D𝑎 for all 𝑖 and 𝑎, where the domain of the 𝑎-th attribute D𝑎 is a finite set of +strings. Suppose there exists a (possibly infinite) population of entities indexed by 𝑒 ∈ ℕ, which is +represented in the data. Denote the entity referenced in the 𝑖-th record by 𝜆𝑖 ∈ ℕ and define the +linkage structure as 𝚲 = (𝜆1, . . . , 𝜆𝑁). +We consider the most general case where there are no constraints on 𝚲 – i.e., we permit duplicates +within sources and arbitrary links across sources. The linkage structure 𝚲 induces a partition of +the records into clusters. We label the clusters according to their associated entities, allowing for +empty clusters. The size of cluster 𝑒 is denoted by 𝑁𝑒 = � +𝑖 𝟙[𝜆𝑖 = 𝑒] and the number of non-empty +clusters is denoted by 𝐸 = � +𝑒 𝟙[𝑁𝑒 > 0]. +We are interested in the fully unsupervised setting where no information is known about the linkage +structure or the entities. Our goal is to infer the linkage structure based solely on the observed record +attributes {x1, . . . , x𝑁} and source identifiers {𝜍1, . . . 𝜍𝑁}. Since we are working in a Bayesian +setting, we seek a full posterior (not merely a point estimate) over the linkage structure so that +uncertainty can be propagated to post-ER tasks, which may include regression, multiple systems +estimation, among other examples (Kaplan et al., 2022; Tancredi and Liseo, 2015; Steorts et al., +2018; Tancredi et al., 2020; Sadinle, 2018). While the post-ER task is not a goal of this paper, the +previous references propose recent approaches of such tasks. +2.2 +Exchangeable Random Partitions +The linkage structure is the primary variable of interest for entity resolution, so we pay special +attention to it when designing our model. Since we are working in a Bayesian setting, we must +specify a prior on the linkage structure. We previously noted that the linkage structure can be +interpreted as a partition of the records into subsets, where the records in each subset correspond +to the same entity. This interpretation is convenient, as we can drawn on related work on random +partitions when considering potential priors. In this section, we review a special class of random +partitions called the Ewens-Pitman (EP) family (Pitman, 2006, p. 62), which we use as a prior on +the linkage structure in our model (see Section 3). +Before defining the EP family of random partitions, we define key concepts and notation. Consider +a set of 𝑁 records, where [𝑁] = {1, . . . , 𝑁} denotes the record identifiers. A partition of [𝑁] is +a collection of disjoint non-empty subsets of [𝑁]. For example, {1, 2}, {3} and {1}, {2}, {3} are +partitions of [𝑁] for 𝑁 = 3. We can equivalently define a partition in terms of the linkage structure +𝚲 = (𝜆𝑖)𝑖=1...𝑁 where 𝜆𝑖 labels the subset (entity) record 𝑖 is assigned to.1 Using this notation, the +above examples for 𝑁 = 3 could be written as 𝚲 = (1, 1, 2) and 𝚲 = (1, 2, 3). +Let P[𝑁] denote the set of all partitions of [𝑁]. A random partition of [𝑁] is a random variable +whose values lie in P[𝑁]. The EP family are the most general class of random partitions that satisfy +1One can use any labels to identify the entities. All that matters is that 𝜆𝑖 = 𝜆 𝑗 if records 𝑖 and 𝑗 are assigned to the +same entity and 𝜆𝑖 ≠ 𝜆 𝑗 otherwise. +5 + +the following two desirable properties: +1. Exchangeability. This means the distribution 𝑃𝑁 over the partitions P𝑁 is invariant under +permutations of the record identifiers [𝑁]. +Or equivalently, the distribution over 𝚲 is +exchangeable. This is a reasonable requirement if the records have no natural ordering – e.g., +it is not known whether one record was generated before or after another. +2. Consistency. This is a property of the distribution as 𝑁 varies. It ensures the distribution is not +altered when more records are observed. This is desirable because the model can be learned +sequentially in a consistent manner. We say that the sequence of distributions 𝑃1, 𝑃2, . . . +over P[1], P[2], . . . is consistent if the distribution on P[𝑁] induced by 𝑃𝑀 for 𝑀 > 𝑁 is 𝑃𝑁. +Mathematically, this means +𝑃𝑀({𝜌 : Proj(𝜌, [𝑁]) = 𝜌′}) = 𝑃𝑁 (𝜌′), +where +Proj(𝜌, [𝑁]) := {𝐴𝑒 ∩ [𝑁] : 𝐴𝑒 ∩ [𝑁] ≠ ∅, 𝐴𝑒 ∈ 𝜌} +is the projection of a partition 𝜌 = {𝐴𝑒}𝑒=1...𝐸 of [𝑀] onto [𝑁] and 𝜌′ is a partition of [𝑁]. +In fact, the EP family ensures these properties hold in a limiting sense as 𝑁 → ∞ (Pitman, 2006, p. +62). +We can develop an intuitive understanding of the EP family by examining how a random partition is +generated sequentially, one record at a time. Let (𝜌𝑁)𝑁=1,2,... be a sequence of EP random partitions +where 𝜌𝑁 is a random partition of [𝑁]. We begin at step 1 with 𝜌1 = {1} – i.e., a single record +assigned to a single entity. The random partition 𝜌𝑁 at any later step 𝑁 > 1 is generated conditional +on the random partition 𝜌𝑁−1 at step 𝑁 − 1. Let 𝐸 be the number of subsets (occupied entities) in +𝜌𝑁−1 and 𝑁𝑒 be the size of subset (entity) 𝑒 in 𝜌𝑁−1. Then 𝜌𝑁 is generated by assigning record 𝑁 to: +• an existing subset (entity) with probability 𝑁𝑒−𝜎 +𝑁+𝛼 , or +• a “new” subset (entity) with probability 𝛼+𝐸𝜎 +𝑁+𝛼 , +where 𝜎 and 𝛼 are EP parameters. This construction is known as a two-parameter Chinese Restaurant +Process and is visualized in Figure 1. +The allowable values of the EP parameters fall into two regimes depending on the sign of 𝜎: +• 𝜎 < 0 and 𝛼 = −𝑚𝜎 for some 𝑚 ∈ ℕ. We refer to this regime as the generalized coupon +partitions, since they are closely related to the coupon-collector’s partition (Pitman, 2006, +p. 46). These partitions are generated by sampling with replacement from a finite population +of size 𝑚, where the mixing proportions are drawn from a symmetric Dirichlet distribution +with concentration parameter −𝜎. The coupon-collector’s partition is obtained in the limit +𝜎 → −∞. +• 0 ≤ 𝜎 ≤ 1 with 𝛼 > −𝜎. These are called Pitman-Yor partitions after Pitman and Yor (1997), +and are generated by sampling with replacement from an infinite population. The resulting +6 + +Entity 1 +Entity 2 +Entity 𝑘 +New entity +· · · +· · · +𝑁1 − 𝜎 +𝑁 + 𝛼 +𝑁2 − 𝜎 +𝑁 + 𝛼 +𝑁𝐸 − 𝜎 +𝑁 + 𝛼 +𝛼 + 𝐸𝜎 +𝑁 + 𝛼 +Figure 1: Illustration of a sequential construction of a Ewens-Pitman random partition. At step +𝑁 a record (black circle) is assigned to one of the occupied entities or a new entity (grey circles) +conditioned on the assignments of the previous 𝑁 − 1 records. The probabilities assigned to the +entities are given inside the grey circles and are dependent on the Ewens-Pitman parameters 𝜎 and +𝛼. +partitions demonstrate preferential attachment behavior. The special case 𝜎 = 0 corresponds +to the Ewens partition (Kingman, 1978). +To illustrate the varying behavior of the random partitions as a function of 𝜎, we can examine the +asymptotic number of subsets in the partition (entities) 𝐸𝑁 as 𝑁 → ∞. Pitman (2006, p. 70) shows +𝐸𝑁 +𝑎.𝑠. +≍ +���� +���� +𝑚, +𝜎 < 0 and 𝛼 = −𝑚𝜎 for 𝑚 ∈ ℕ, +𝛼 log 𝑁, +𝜎 = 0 and 𝛼 > 0, +𝑆𝜎𝑁𝜎, +0 < 𝜎 < 1 and 𝛼 > −𝜎, +(1) +where 𝑆𝜎 is a strictly positive random variable. Thus, by varying 𝜎, we can encode a prior belief +that the number of entities 𝐸𝑁 is asymptotically constant, logarithmic, or sub-linear in 𝑁. +3 +Graphical Bayesian ER +In this section, we propose a generative model for entity resolution that incorporates a latent +population of entities, each with a set of unknown attributes. Our model employs the Ewens-Pitman +class of priors on the linkage structure and a modified record distortion model that deviates from +the common “hit-miss” model used by Tancredi and Liseo (2011), Steorts (2015) and Steorts et al. +(2016). We provide an index of the model’s variables and an illustration of their dependence structure +in Figure 2. We close the section with a discussion of our model’s hyperparameters, including +recommendations about how to set these values when limited prior information is available. +3.1 +Model Specification +Entities. +We assume each entity 𝑒 ∈ {1, 2, . . .} is associated with a tuple of attribute values +y𝑒 = (𝑦𝑒1, . . . , 𝑦𝑒𝐴), drawn independent and identically distributed (i.i.d.) from an unknown +distribution G with support on D = � +𝑎 D𝑎. To improve tractability, we assume correlations +7 + +𝑥𝑖𝑎 +𝜆𝑖 +π +𝜎 +𝛼 +𝜁 (1) +𝜁 (0) +𝜒(1) +𝜒(0) +𝑧𝑖𝑎 +𝜔𝑖𝑎 +𝑦𝑒𝑎 +𝜍𝑖 +𝐺𝑎 +𝐻𝑒𝑎 +𝜌𝑎 +𝜏(1) +𝑎 +𝜏(0) +𝑎 +ψ𝑎 +𝜐𝑎 +𝜙𝑎 +𝜃𝑠𝑎 +𝛽(1) +𝑠𝑎 +𝛽(0) +𝑠𝑎 +𝑒 ∈ 1 . . . 𝐸 +𝑖 ∈ 1 . . . 𝑁 +𝑠 ∈ 1 . . . 𝑆 +𝑎 ∈ 1 . . . 𝐴 +𝑖 +index over records +𝑠 +index over sources +𝑎 +index over attributes +𝑒 +index over entities +𝑥𝑖𝑎 +attribute 𝑎 for record 𝑖 +𝑧𝑖𝑎 +distortion indicator for attribute 𝑎 of record 𝑖 +𝜔𝑖𝑎 +distortion propensity for attribute 𝑎 of record 𝑖 +𝜍𝑖 +source of record 𝑖 +𝜆𝑖 +linked entity for record 𝑖 +𝐻𝑒𝑎 +distortion distribution for attribute 𝑎 of entity 𝑒 +𝜌𝑎 +concentration of 𝐻𝑒𝑎 +ψ𝑎 +base distribution for 𝐻𝑒𝑎 +𝜃𝑠𝑎 +distortion probability for attribute 𝑎 in source 𝑠 +𝑦𝑒𝑎 +attribute 𝑎 for entity 𝑒 +𝐺𝑎 +distribution over domain for attribute 𝑎 +π +mixing proportions +𝜎, 𝛼 +Ewens-Pitman parameters +D𝑎 +domain of attribute 𝑎 +dist𝑎 +distance measure for attribute 𝑎 +Figure 2: Plate diagram and index of symbols for our model under a Pitman-Yor prior. +8 + +between attributes are negligible, and place independent Dirichlet Process (DP) priors on each +component of G = (𝐺1, . . . , 𝐺 𝐴): +𝐺𝑎 +ind. +∼ DP(𝜐𝑎, 𝜙𝑎) , +∀𝑎, +𝑦𝑒𝑎 | 𝐺𝑎 +ind. +∼ Discrete(𝐺𝑎), +∀𝑒, 𝑎, +where 𝜐𝑎 > 0 is a concentration parameter and 𝜙𝑎 is a base distribution on domain D𝑎. +Links. +Each record 𝑖 ∈ {1, . . . , 𝑁} is linked to an entity 𝜆𝑖 ∈ {1, 2, . . .} which is assumed +to be drawn from the population with replacement, according to unknown mixing proportions +π = (𝜋1, 𝜋2, . . .). This process induces a partition of the records into clusters according to their +linked entities. Following the discussion about exchangeability in Section 2.2, we assume the +partition is drawn from the Ewens-Pitman (EP) family with parameters (𝜎, 𝛼). The corresponding +distribution on the mixing proportions π depends on the sign of 𝜎, or equivalently, whether the +population of entities is finite or infinite. +For the finite regime (generalized coupon partitions) we let 𝜎 = −𝜅 < 0 and 𝛼 = 𝑚𝜅 for some +𝑚 ∈ ℕ. Our model with hyperpriors on 𝑚 and 𝜅 is as follows: +𝜅 ∼ Gamma(𝜒(0), 𝜒(1)), +𝑚 ∼ NegativeBinomial(𝑟, 𝜈) + 1, +π | 𝜅, 𝑚 ∼ Dirichlet(κ), +𝜆𝑖 | π iid.∼ Categorical(π), +∀𝑖, +(2) +where 𝜒(0), 𝜒(1), 𝑟 > 0 and 0 < 𝜈 ≤ 1 are hyperparameters, and κ is a vector of length 𝑚 with +identical entries 𝜅. The hyperprior on 𝑚 is a shifted negative binomial distribution with density +defined in Appendix A.4.2. +In the infinite regime (Pitman-Yor partitions) the mixing proportions are drawn from a two-parameter +Poisson-Dirichlet distribution (Pitman and Yor, 1997). Our model with hyperpriors on 𝜎 and 𝛼 is +as follows: +𝜎 ∼ Beta(𝜁 (0), 𝜁 (1)), +𝛼 ∼ Gamma(𝜒(0), 𝜒(1)), +π | 𝜎, 𝛼 ∼ PoissonDirichlet(𝜎, 𝛼), +𝜆𝑖 | π iid.∼ Categorical(π), +∀𝑖, +(3) +where 𝜒(0), 𝜒(1), 𝜁 (0), 𝜁 (1) > 0 are hyperparameters. Here we assume 𝛼 > 0 and 0 < 𝜎 < 1, which +is a subset of the admissible parameter space: 0 ≤ 𝜎 ≤ 1 and 𝛼 > −𝜎. We also consider the case +where 𝜎 = 0, which corresponds to the Ewens partition. +Remark. By placing hyperpriors on the EP parameters, we can improve robustness to misspecified +hyperparameters which are difficult to set in a non-informative manner. Special cases of the above +priors have been used in other ER models, albeit with fixed hyperparameters. Tancredi and Liseo +(2011), Steorts (2015) and Steorts et al. (2016) used a coupon-collector’s partition with 𝜅 → ∞ and +9 + +𝑚 fixed, which was shown to be highly informative for the observed population size. Steorts et al. +(2018) used a Pitman-Yor partition with 𝜎 and 𝛼 fixed. +Sources. +We assume the data source 𝜍𝑖 ∈ {1, . . . , 𝑆} associated with record 𝑖 is drawn i.i.d. from +a discrete distribution ξ over the sources {1, . . . , 𝑆}. There is no need to specify ξ since it is +independent of the other model parameters, and the source indicators 𝜍𝑖 are assumed to be fully +observed. +Distortion. +We assume the attributes x𝑖 for record 𝑖 are generated by distorting the associated +entity attributes y𝜆𝑖. For simplicity, we assume the distortion process occurs independently for each +attribute. To decide whether the 𝑎-th attribute is distorted, a binary indicator 𝑧𝑖𝑎 is drawn which +depends on the distortion propensity 𝜔𝑖𝑎 scaled by a source/attribute-level factor 𝜃𝜍𝑖𝑎. We place a +Beta prior on 𝜃𝜍𝑖𝑎 and assume the distortion propensity 𝜔𝑖𝑎 is deterministic given the true attribute +value 𝑦𝜆𝑖𝑎. Concretely, we have +𝜃𝑠𝑎 +ind. +∼ Beta +� +𝛽(0) +𝑠𝑎 , 𝛽(1) +𝑠𝑎 +� +∀𝑠, 𝑎 +(4) +𝜔𝑖𝑎 | 𝑦𝜆𝑖𝑎 = propensity +� +min +𝑥∈D𝑎\{𝑦𝜆𝑖 𝑎} dist𝑎(𝑦𝜆𝑖𝑎, 𝑥), max +𝑥,𝑦∈D𝑎 dist𝑎(𝑦, 𝑥) +� +∀𝑖, 𝑎 +𝑧𝑖𝑎 | 𝜃𝜍𝑖𝑎, 𝜔𝑖𝑎 +ind. +∼ Bernoulli�𝜃𝜍𝑖𝑎𝜔𝑖𝑎 +� +∀𝑖, 𝑎 +(5) +where 𝛽(0) +𝑠𝑎 , 𝛽(0) +𝑠𝑎 > 0 are hyperparameters and +propensity(𝑑min, 𝑑max) := +���� +���� +0, +𝑑min = ∞ and 𝑑max = ∞, +1, +𝑑min = 0 and 𝑑max = 0, +e− 𝑑min +2𝑑max , +otherwise. +(6) +The distortion propensity 𝜔𝑖𝑎 accounts for the fact that some entity attribute values 𝑦𝜆𝑖𝑎 ∈ D𝑎 are +more likely to be distorted than others. It makes use of prior information in the attribute distance +measure dist𝑎(𝑦, 𝑥) (see Section 3.2). If 𝑦𝜆𝑖𝑎 is not close to any other values in the domain, it is +unlikely to be distorted and 𝜔𝑖𝑎 approaches zero. On the other hand, if 𝑦𝜆𝑖𝑎 is close to at least one +other value in the domain, distortion can occur and 𝜔𝑖𝑎 approaches one. This logic is not included +in a similar model by Steorts (2015), which effectively assumes 𝜔𝑖𝑎 = 1. +After drawing the distortion indicator 𝑧𝑖𝑎, record attribute 𝑥𝑖𝑎 is generated by copying the linked +entity attribute 𝑦𝜆𝑖𝑎 directly (if 𝑧𝑖𝑎 = 0) or subject to distortion (if 𝑧𝑖𝑎 = 1). If 𝑧𝑖𝑎 = 1, the distorted +value is drawn from a distortion distribution 𝐻𝜆𝑖𝑎 associated with the linked entity 𝜆𝑖. We assume +𝐻𝜆𝑖𝑎 itself is drawn from a Dirichlet Process: +𝜌𝑎 ∼ Gamma +� +𝜏(0) +𝑎 , 𝜏(1) +𝑎 +� +∀𝑎, +(7) +𝐻𝑒𝑎 | 𝑦𝑒𝑎, 𝜌𝑎 +ind. +∼ DP(𝜌𝑎; ψ𝑎(𝑦𝑒𝑎)) +∀𝑒, 𝑎, +(8) +10 + +where 𝜏(0) +𝑎 , 𝜏(1) +𝑎 +> 0 are hyperparameters, and ψ𝑎(𝑦𝑒𝑎) is a prior base distribution with support on a +subset of D𝑎 \ {𝑦𝜆𝑖𝑎}. This differs from models by Tancredi and Liseo (2011), Steorts (2015) and +Steorts et al. (2016, 2018), which assume 𝐻𝑒𝑎 is deterministic conditional on 𝑦𝑒𝑎. +Summarizing this symbolically, we have +𝑥𝑖𝑎 | 𝑧𝑖𝑎, 𝑦𝜆𝑖𝑎, 𝐻𝜆𝑖𝑎 +ind. +∼ +� +𝛿(𝑦𝜆𝑖𝑎), +if 𝑧𝑖𝑎 = 0, +𝐻𝜆𝑖𝑎, +if 𝑧𝑖𝑎 = 1, +∀𝑖, 𝑎 +(9) +where 𝛿(𝑦) denotes a point mass at 𝑦. This is reminiscent of a hit-miss model (Copas and Hilton, +1990). However, our construction differs, in that the record and entity attributes are forbidden from +matching (𝑥𝑖𝑎 ≠ 𝑦𝜆𝑖𝑎) if the record attribute is distorted (𝑧𝑖𝑎 = 1). +Remark. The hit-miss model of Copas and Hilton (1990) was designed for modeling distortion of +continuous attributes. For continuous attributes, the probability of drawing the non-distorted value +(𝑦𝜆𝑖𝑎) from the miss component 𝐻𝜆𝑖𝑎 is zero, assuming 𝐻𝜆𝑖𝑎 is described by a continuous density +function. This ensures the record value 𝑥𝑖𝑎 is always distorted (𝑥𝑖𝑎 ≠ 𝑦𝜆𝑖𝑎) if 𝑧𝑖𝑎 = 1. Our proposal +replicates the same behavior for discrete attributes by ensuring 𝐻𝑒𝑎 has no mass on 𝑦𝑒𝑎. This is +especially important if 𝐻𝑒𝑎 were to place significant mass on 𝑦𝑒𝑎, as the line between distorted +and non-distorted values would become blurred. Apart from the modeling advantages, excluding +𝑦𝑒𝑎 from the support of 𝐻𝑒𝑎 also makes inference more tractable as we can collapse 𝐻𝑒𝑎 (see +Appendices A.2 and A.3). +3.2 +Choice of Hyperparameters +In this section, we provide recommendations for setting the hyperparameters in our model. +Distance Measures. +Our proposed distortion model is parameterized by a set of distance measures +{dist𝑎}, one for each attribute 𝑎 ∈ {1, . . . , 𝐴}. They encode prior knowledge about the likelihood +that a record attribute value 𝑥 appears as a distorted alternative to an entity attribute value 𝑦. The +larger the distance dist𝑎(𝑦, 𝑥), the less likely 𝑥 is a distortion of 𝑦. Since the likelihood of distorting +𝑥 to 𝑦 may not be the same as the likelihood of distorting 𝑦 to 𝑥, we do not require that the distance +measures are symmetric. We recommend selecting the distance measures carefully, leveraging prior +knowledge about the distortion process where possible. For instance, one might select edit distance +to model typographic distortion in a generic string-type attribute. For categorical attributes, one +could select a constant distance function dist𝑎(𝑦, 𝑥) ≡ 0, which encodes the prior belief that all +values in the domain are equally likely as a distorted alternative to 𝑦. +Distortion Base Distribution. +We recommend using the distance measures to set the base +distribution ψ𝑎(𝑦𝑒𝑎) in Equation (8). Specifically, we recommend a softmax distribution +𝜓𝑎(𝑥 | 𝑦𝑒𝑎) ∝ 𝟙[𝑥 ≠ 𝑦𝑒𝑎] exp(− dist𝑎(𝑦𝑒𝑎, 𝑥)), +(10) +where the temperature parameter is absorbed in the definition of the distance measure, and the +indicator function excludes 𝑦𝑒𝑎 from the support. This places more weight on values in the domain +11 + +closer to 𝑦𝑒𝑎 and less weight on values further away. Unlike Steorts (2015), we do not include a +factor proportional to the empirical frequency of 𝑥, as distorted values (e.g., typographical errors) +tend to be infrequent for the applications we consider. For a categorical attribute with dist𝑎(𝑦, 𝑥) ≡ 0, +Equation (10) reduces to the uniform distribution. In this case, it may be appropriate to incorporate a +factor proportional to the empirical frequencies by setting 𝜓𝑎(𝑥 | 𝑦𝑒𝑎) ∝ 𝟙[𝑥 ≠ 𝑦𝑒𝑎] �𝑁 +𝑖=1 𝟙[𝑥𝑖𝑎 = 𝑥]. +Other Hyperparameters. +In the absence of prior knowledge, we recommend setting the remaining +hyperparameters to yield vague priors – i.e., priors that provide little information relative to the +experiment (Tiao and Box, 1973; Bernardo and Smith, 2009). We note that there are different views +in the Bayesian community about how to specify vague and/or uninformative priors. For more on +this, we refer to Syversveen (1998) and Irony and Singpurwalla (1997). Putting aside such debates, +our recommendations are as follows: +• For the shifted negative binomial prior on 𝑚, we set 𝑟 and 𝜈 so that the prior mean is 𝑁 and +the prior variance is 𝑁2. +• For the gamma prior on 𝛼, we set 𝜒(0) = 1 and 𝜒(1) to be small (e.g., 10−2). +• For the beta prior on 𝜎, we set 𝜁 (0) = 𝜁 (1) = 1 to yield a flat prior. +• For the Dirichlet prior on the entity attribute distribution, we recommend setting 𝜐𝑎 = 1 and +using a uniform base distribution 𝜙𝑎 for all 𝑎. +• For the gamma prior on the concentration parameter 𝜌𝑎, we recommend setting 𝜏(0) = 2 and +𝜏(1) small (e.g., 10−4) for all 𝑎. +• For the beta priors on 𝜃𝑠,𝑎, we encode a weak prior belief of low distortion by setting 𝛽(0) +𝑠𝑎 = 1 +and 𝛽(1) +𝑠𝑎 = 4 for all 𝑠, 𝑎. +Remark. The hyperparameters can be varied to encourage over-linkage (linking records that do +not correspond to the same entity) or under-linkage (failing to link records that correspond to the +same entity). Since perfect linkage is not always possible, practitioners may have to decide whether +over-linkage or under-linkage is preferred for a given application. We can encourage over-linkage in +our model by setting: +• 𝛽(0) +𝑠𝑎 ≫ 𝛽(1) +𝑠𝑎 (prior belief of high distortion), +• 𝜐𝑎 ≪ 1 (prior belief of low diversity in attribute 𝑎 among entities), +• 𝜁 (0) ≪ 𝜁 (1) and 𝜒(0) ≪ 𝜒(1) (prior belief of more links for Pitman-Yor prior), or +• 𝑟 close to 0 and 𝜈 close to 1 (prior belief of more links for generalized coupon prior). +Similarly, we can encourage under-linkage by reversing the inequalities above. We measure the +extent to which our model over- or under-links in our empirical evaluation (Section 5) using precision +and recall metrics defined in Equations (12) and (13). +12 + +4 +Inference +To perform entity resolution using our model, we must find the posterior distribution over the linkage +structure conditional on the observed record attributes and their sources. Since the posterior is not +analytically tractable, we propose an approximate inference scheme based on Markov chain Monte +Carlo (MCMC). +MCMC produces approximate samples from the posterior distribution by constructing a Markov +chain whose equilibrium distribution matches the posterior distribution. The samples produced by +MCMC are approximate in the sense that they are autocorrelated, and they may only match the +equilibrium (posterior) distribution asymptotically. Various algorithms exist within the MCMC +framework – we refer the reader to Gamerman and Lopes (2006) or Brooks et al. (2011) for an +introduction to the field. +In this paper, we use an MCMC algorithm called partially collapsed Gibbs (PCG) sampling (van +Dyk and Park, 2008). It is a generalization of Gibbs sampling that reduces the extent of conditioning +in the variable updates by collapsing (marginalizing out) variables and/or updating variables in +groups. This can significantly improve convergence and reduce autocorrelation, so long as prescribed +rules are followed to ensure the equilibrium distribution of the Markov chain is preserved. +Ideally, we would like to reduce the extent of conditioning as much as possible, however this +must be balanced with computational and mathematical constraints. In our proposed sampling +scheme, we fully collapse the entity mixing proportions π and the distortion distributions 𝐻𝑒𝑎. We +partially-collapse the distortion indicators 𝑧𝑖𝑎 in a joint update for the entity attributes 𝑦𝑒𝑎 and for +the distortion distribution concentration 𝜌𝑎. By collapsing the mixing proportions, we obtain an +urn-based scheme for updating the linkage structure similar to those used for nonparametric mixture +models (Neal, 2000). In the remainder of this section, we highlight some less trivial aspects of +inference – full details are provided in Appendix A. +4.1 +Nonconjugacy +While we attempted to maintain conjugacy in our model, we were unable to avoid nonconjugate +priors in some cases. This complicates inference, as the posterior conditional distributions used in +Gibbs sampling are no longer of a standard form. There are several well-established methods for +dealing with nonconjugacy, including Metropolis-Hastings algorithms (Chib and Greenberg, 1995), +rejection sampling (Gilks and Wild, 1992) and auxiliary variable methods (Damlen et al., 1999). +We opt to use auxiliary variable methods owing to their simplicity, as there is no need to design +proposals or monitor acceptance rates. +There are three sets of parameters in our model for which nonconjugacy is an issue: +1. The distortion probabilities 𝜃𝑠𝑎 defined in Equation (4), where the incorporation of the +distortion propensities 𝜔𝑖𝑎 breaks the conjugacy of the beta prior. We propose an auxiliary +variable sampling scheme to update 𝜃𝑠𝑎 in Appendix A.1. +2. The EP parameters: 𝜅 and 𝑚 defined in Equation (2) or 𝛼 and 𝜎 defined in Equation (3), +13 + +depending on the regime. We use an auxiliary variable scheme proposed by Teh (2006), to +update 𝛼 and 𝜎 under a gamma and beta prior, as summarized in Appendix A.4.1. We design +an auxiliary variable update for 𝜅 and 𝑚 under a gamma and shifted negative binomial prior +in Appendix A.4.2. +3. The distortion distribution concentration 𝜌𝑎 defined in Equation (7). We design an auxiliary +variable update for 𝜌𝑎 in Appendix A.5. +4.2 +Collapsing the Distortion Indicators +Marchant et al. (2021) demonstrated the importance of collapsing the distortion indicators {𝑧𝑖𝑎} to +improve convergence/mixing for a hit-miss model similar to Equation (9). The posterior factors +involving 𝑧𝑖𝑎 factorize over 𝑖 and 𝑎, so that collapsing 𝑧𝑖𝑎 yields: +𝑃(𝑥𝑖𝑎 | 𝜃𝜍𝑖𝑎, 𝜔𝑖𝑎, 𝑦𝜆𝑖𝑎, 𝐻𝜆𝑖𝑎) ∝ +1 +∑︁ +𝑧𝑖𝑎=0 +𝑃(𝑥𝑖𝑎 | 𝑧𝑖𝑎, 𝑦𝜆𝑖𝑎, 𝐻𝜆𝑖𝑎)𝑃(𝑧𝑖𝑎 | 𝜃𝜍𝑖𝑎, 𝜔𝑖𝑎) +∝ (1 − 𝜃𝜍𝑖𝑎𝜔𝑖𝑎)𝟙[𝑥𝑖𝑎 = 𝑦𝜆𝑖𝑎] + 𝜃𝜍𝑖𝑎𝜔𝑖𝑎𝐻𝜆𝑖𝑎(𝑥𝑖𝑎). +(11) +We use this result to implement a collapsed update for the entity attributes {𝑦𝑒𝑎}. While it is +possible to implement a collapsed update for the linkage structure {𝜆𝑖}, we opt not to do so, since +conditioning on the distortion indicators allows us to reduce computational complexity via indexing +(see Section 4.3). This seems to be more efficient empirically (Marchant et al., 2021), so long as the +level of distortion is not too high. +4.3 +Computational Considerations +We now discuss ways of improving the computational complexity. The main bottleneck is the update +for the linkage structure which scales naïvely as 𝑂(𝑁 · 𝐸) where 𝐸 is the number of instantiated +entities.2 The update for the entity attributes may also be problematic for large domains D𝑎 as it +scales as 𝑂(𝐸 · |D𝑎|) for the 𝑎-th attribute. +We are able to reduce the computational complexity of the linkage structure update by exploiting +constraints imposed by the distortion model. Close inspection of the update for the entity linked +to record 𝑖 (see Appendix A.3) reveals that some entities can be immediately excluded from +consideration. Specifically, only those entities whose attributes match the corresponding non- +distorted record attributes (𝑥𝑖𝑎 with 𝑧𝑖𝑎 = 0) may be linked to record 𝑖. In order to efficiently query +this set of entities, we maintain inverted indices that map an attribute value 𝑥 ∈ D𝑎 to the set of +entities instantiated with that value {𝑒 : 𝑥 = 𝑦𝑒𝑎}. This approach is considerably more efficient than +iterating over all entities sequentially, so long as the level of distortion is relatively low. However it +is important to note that it relies crucially on not collapsing the distortion indicators. +To improve the complexity of the entity attribute update, we can impose a cut-off on the distance +2When stating time complexities in this section, we assume a categorical random variate can be drawn in Θ(𝐶) time +where 𝐶 is the number of categories. The algorithm proposed by Vose (1991) satisfies this constraint. +14 + +measures. Concretely, for attribute 𝑎 we replace the “raw” distance measure dist𝑎 by +dist𝑎(𝑦, 𝑥) = +� +dist𝑎(𝑦, 𝑥), +if dist𝑎(𝑦, 𝑥) ≤ 𝑑(cut) +𝑎 +, +∞, +otherwise, +where 𝑑(cut) +𝑎 +∈ (0, ∞) is a configurable cut-off. This approximation eliminates the need to consider +unlikely distortions from entity attribute 𝑦 to record attribute 𝑥, for which dist(𝑦, 𝑥) > 𝑑(cut) +𝑎 +. It +plays a similar role to blocking in the record linkage literature (Christen, 2012b) and resembles an +approach proposed by Marchant et al. (2021). To make use of this approximation, we build indices +that can efficiently answer range queries – one for each attribute. The index for the 𝑎-th attribute +takes a query value 𝑥 ∈ D𝑎 and returns the set of entity attribute values that fall below the cut-off: +{𝑦 ∈ D𝑎 : dist𝑎(𝑦, 𝑥) ≤ 𝑑(cut) +𝑎 +}. +5 +Model Comparisons +We conduct an empirical study of our ER model using data sets for which the true linkage structure is +known. Section 5.1 describes the data sets used in the study, which are motivated by ER applications +in private and non-private settings. We explain how our model (and baseline models) are evaluated +in Section 5.2, by computing metrics that assess how well the posterior predictions align with the +true linkage structure. Section 5.3 assesses the impact of our modeling contributions by varying +the distortion model and the prior on the linkage structure. Section 5.4 compares our ER model +against baselines proposed by Sadinle (2014) and Steorts (2015). Finally, Section 5.5 summarizes a +controlled simulation study that can be found in Appendix C. +5.1 +Data Sets +We study entity resolution in private and non-private settings, both of which are encountered by +practitioners. The data sets we use in our study are summarized in Table 1. +Private Setting. +In this setting the practitioner has access to de-identified data, where sensitive +attributes such as names, addresses, phone numbers, etc. are removed. This can make ER quite +challenging, as the remaining non-sensitive attributes may carry limited information about the +identity of records. To study ER in this setting, we use data extracted from the National Long Term +Care Survey (Manton, 2010), which we refer to as nltcs. +Our extract contains de-identified respondent records from the 1982, 1989 and 1994 waves of the +survey in the U.S. state of Alabama. We use all of the available attributes for ER, which include +date of birth (DOB_YEAR, DOB_MONTH, DOB_DAY), registration office (REGOFF) and sex (SEX). Since +the data is well-curated, the only distortion that can occur is when a valid attribute value is replaced +by another valid attribute value. We, therefore, model the attributes as categorical by employing a +constant distance function. +Non-private Setting. +In this setting, the practitioner has access to data with sensitive attributes, +such as names. We assume unique identifiers such as social security numbers are not available, as +15 + +Data set +Setting +Entity type +# records (𝑁) +# entities +nltcs +Private +People +5,359 +3,307 +RLdata +Non-private +People +10,000 +9,000 +cora +Non-private +Citations +1,295 +125 +rest +Non-private +Restaurants +864 +752 +Table 1: Summary of data sets. +ER would otherwise be trivial. Obtaining real survey data for a non-private setting with ground +truth is challenging, so we use three publicly-available data sets from the ER literature. Although +these data sets cover other domains, they exhibit characteristics one would expect to encounter in +real survey data. Namely, we observe the presence of multiple “name-like” attributes, as well as +different levels of variation and distortion. Below, we provide a brief description of each data set +and the attributes used for ER: +• RLdata is a synthetic person data set, where 10% of the records are duplicates with random +errors (Sariyar and Borg, 2010). We model the name attributes (fname_c1 and lname_c1) +using the normalized Levenshtein distance measure. The attributes related to date of birth – +bd, bm and by – are modeled as categorical attributes with a constant distance measure.3 +• cora is a collection of computer science citation records hosted on the RIDDLE repository +(Bilenko, 2003). It is the “dirtiest” of all the data sets we consider, as it was extracted from +various online sources with different citation styles. As a pre-processing step, we separate +hyphenated words and remove punctuation. We also correct several erroneous ground truth +labels. The title, venue and authors attributes generally contain multiple words with +semantic and character-level variations, and are therefore modeled using a hybrid token/edit +distance measure described in Appendix B. The year attribute is modeled using normalized +Levenshtein distance. +• rest is a collection of restaurant records from the Fodor and Zagat restaurant guides hosted on +the RIDDLE repository (Bilenko, 2003). It is not as “dirty” as cora as there are fewer sources +and less variation between them. We applied the same pre-processing steps as for cora. The +name and addr attributes generally contain multiple words and are therefore modeled using +the same hybrid distance measure as for cora. The city and type (cuisine) attributes are +modeled as categorical with a constant distance measure. +5.2 +Model Evaluation +We evaluate an ER model on a data set by comparing the inferred linkage structure ˆΛ to the true +linkage structure Λ. Recall that Λ = (𝜆1, . . . , 𝜆𝑁) specifies the corresponding entity 𝜆𝑖 for each +record 𝑖 in the data set. The agreement between ˆΛ and Λ can be measured using pairwise precision +and recall. The pairwise precision is the proportion of record pairs linked in ˆΛ that are also linked +3This is a benchmark data set that is widely used in the literature. +16 + +in Λ: +Pr( ˆΛ, Λ) = +�𝑁 +𝑖≠𝑗=1 𝟙[ ˆ𝜆𝑖 = ˆ𝜆 𝑗]𝟙[𝜆𝑖 = 𝜆 𝑗] +�𝑁 +𝑖≠𝑗=1 𝟙[ ˆ𝜆𝑖 = ˆ𝜆 𝑗] +. +(12) +It takes on values from 0 to 1, where larger values indicate fewer false positive errors. The pairwise +recall is the proportion of record pairs linked in Λ that are also linked in ˆΛ: +Re( ˆΛ, Λ) = +�𝑁 +𝑖≠𝑗=1 𝟙[ ˆ𝜆𝑖 = ˆ𝜆 𝑗]𝟙[𝜆𝑖 = 𝜆 𝑗] +�𝑁 +𝑖≠𝑗=1 𝟙[𝜆𝑖 = 𝜆 𝑗] +. +(13) +It takes on values from 0 to 1, where larger values indicate fewer false negative errors. It is rarely +possible to achieve high precision and recall – one must usually make a trade-off depending on +which types of errors are more costly in a given application. If precision and recall are equally +important, then one can measure the agreement between ˆΛ and Λ using the pairwise F1 score which +is the harmonic mean of precision and recall: +F1( ˆΛ, Λ) = 2 Pr( ˆΛ, Λ) · Re( ˆΛ, Λ) +Pr( ˆΛ, Λ) + Re( ˆΛ, Λ) +. +(14) +Since the models in our study are Bayesian, the inferred (posterior) linkage structure ˆΛ is a random +variable and the metrics in Equations (12)–(14) can be regarded as random variables. We estimate +the distribution of the metrics under the posterior using samples generated via MCMC. In doing so, +we are able to account for posterior uncertainty in our evaluation. For each metric, we report a point +estimate using the median, along with a 95% equi-tailed credible interval. Further details about the +MCMC implementation and configuration for each model are provided in Appendix E and MCMC +diagnostics are in Appendix G. +5.3 +Study of Linkage Structure Priors and Distortion Model +In this section, we study the effect of two modeling contributions proposed in Section 3.1 – the +Ewens-Pitman (EP) linkage structure priors and the refined distortion model. Our objective is +to determine the impact of each modeling contribution in isolation, using the blink model as a +baseline. We summarize the results here and refer the reader to Appendix D for comprehensive +results covering eight combinations of linkage structure priors and distortion models. +Linkage Structure Priors. +We consider four priors on the linkage structure, which correspond to +distinct EP parameter regimes (see Section 2.2): +1. PY: Pitman-Yor regime with 𝜎 ∈ (0, 1) and hyperpriors on 𝜎, 𝛼 as detailed in Equation (3). +2. Ewens: Ewens regime with 𝜎 = 0 and a hyperprior on 𝛼 as detailed in Equation (3). +3. GenCoupon: generalized coupon regime with 𝜎 = −𝜅 < 0 and hyperpriors on 𝜅, 𝑚 as detailed +in Equation (2). +17 + +RLdata +nltcs +cora +rest +−10 +−5 +0 +−2 +−1 +0 +1 +2 +50 +100 +150 +200 +−4 +−2 +0 +Coupon +GenCoupon +Ewens +PY +Relative error (%) +Prior +Figure 3: Posterior relative error in the predicted number of entities for all data sets and linkage +structure priors. Under-linkage is observed for cora, which is likely due to significant noise that is +not well-captured by the distortion model. +4. Coupon: coupon collector’s partition used by with 𝜅 → ∞ and 𝑚 = 𝑁. +The first three priors are flexible, in the sense that hyperpriors are placed on the EP parameters. +The last prior is a particular instance of GenCoupon where the EP parameters are fixed. It is used +in models by Tancredi and Liseo (2011), Steorts (2015) and Steorts et al. (2016), and serves as a +baseline here. +ER evaluation metrics are presented in Table 2 for the four linkage structure priors, assuming the +rest of the model follows the specification in Section 3.1. Another perspective on ER accuracy is +provided in Figure 3, which plots the relative error in the inferred number of entities. Both results +demonstrate the benefit of placing hyperpriors on the EP parameters, as is done for PY, Ewens, and +GenCoupon. These linkage structure priors achieve superior F1 scores compared to Coupon, where +the EP parameters are fixed. Figure S8 (Appendix D) is consistent with this finding, demonstrating +that vastly different values of the EP parameters are inferred for each data set when hyperpriors are +used. Another interesting observation is the fact that the ER accuracy is relatively similar among +PY, Ewens and GenCoupon. This was unexpected at first, given the three parameter regimes are +known to exhibit distinct asymptotic behavior (see equation 1). This suggests all three regimes +may be flexible enough to model the linkage structure of the data sets we consider here. It would +be interesting to see if these observations translate to much larger data sets, where the asymptotic +behavior of the three regimes would become more apparent. +Distortion Model. +We compare our proposed distortion model (specified in the latter part of +Section 3.1) to the distortion model proposed by Steorts (2015). For brevity, we refer to our distortion +model as Ours and that of Steorts (2015) as blink. Here, we report results for the GenCoupon linkage +structure prior – the results for the other linkage structure priors are reported in Appendix D and +exhibit similar trends. +Figure 4 plots the inferred level of distortion for each attribute under both distortion models. It +shows that blink tends to encourage high distortion, particularly for attributes with non-constant +distance measures.4 For example, the fname_c1 and lname_c1 attributes for RLdata are predicted +4The attributes modeled with non-constant distance measures are: all attributes for cora, name and addr for rest, +and fname_c1 and lname_c1 for RLdata. +18 + +Evaluation metric +Data set +EP regime +Precision +Recall +F1 score +RLdata +PY +0.896 (0.879, 0.917) +0.961 (0.952, 0.972) +0.928 (0.918, 0.939) +Ewens +0.870 (0.853, 0.893) +0.970 (0.961, 0.978) +0.917 (0.908, 0.931) +GenCoupon +0.903 (0.886, 0.920) +0.966 (0.955, 0.975) +0.933 (0.923, 0.941) +Coupon +0.402 (0.396, 0.410) +0.987 (0.982, 0.993) +0.572 (0.565, 0.580) +nltcs +PY +0.921 (0.908, 0.933) +0.924 (0.915, 0.934) +0.923 (0.915, 0.930) +Ewens +0.921 (0.910, 0.932) +0.925 (0.915, 0.934) +0.923 (0.916, 0.930) +GenCoupon +0.902 (0.879, 0.918) +0.935 (0.926, 0.944) +0.918 (0.906, 0.927) +Coupon +0.919 (0.908, 0.930) +0.926 (0.916, 0.935) +0.923 (0.915, 0.930) +cora +PY +0.971 (0.963, 0.979) +0.671 (0.647, 0.696) +0.794 (0.776, 0.813) +Ewens +0.974 (0.965, 0.981) +0.673 (0.645, 0.697) +0.796 (0.775, 0.813) +GenCoupon +0.973 (0.965, 0.981) +0.657 (0.632, 0.683) +0.784 (0.766, 0.804) +Coupon +0.978 (0.971, 0.986) +0.173 (0.164, 0.181) +0.294 (0.281, 0.306) +rest +PY +0.770 (0.735, 0.824) +0.812 (0.759, 0.884) +0.795 (0.755, 0.828) +Ewens +0.770 (0.711, 0.823) +0.830 (0.781, 0.875) +0.798 (0.760, 0.838) +GenCoupon +0.794 (0.742, 0.850) +0.821 (0.777, 0.875) +0.807 (0.773, 0.849) +Coupon +0.637 (0.602, 0.674) +0.911 (0.893, 0.938) +0.750 (0.722, 0.781) +Table 2: Posterior evaluation metrics for our model under four linkage structure priors corresponding +to distinct Ewens-Pitman (EP) parameter regimes. A point estimate for each evaluation metric is +reported based on the median, along with a 95% equi-tailed credible interval. Similar performance +is observed for the three regimes where the EP parameters are permitted to vary (PY, Ewens and +GenCoupon). A significant drop in performance is observed for the Coupon regime on RLdata and +cora. +19 + +RLdata +nltcs +cora +rest +0 +25 +50 +75 +100 +bd +bm +by +fname_c1 +lname_c1 +dob_day +dob_month +dob_year +regoff +sex +authors +title +venue +year +addr +city +name +type +Distortion level (%) +Attribute +Distortion model +Ours +blink +Figure 4: Comparison of the posterior attribute-level distortion under two distortion models: Ours +(red top-most intervals) and blink (teal bottom-most intervals). The blink distortion model tends to +favor higher levels of distortion – in some cases approaching 100 percent – which is not consistent +with expectations. +to be almost 100% distorted under the blink distortion model, which is inconsistent with expectations +for this data set. Our distortion model does not appear to suffer from this problem, as it requires +disagreement between entity and record attributes in order to classify them as “distorted”. Since +high distortion makes reliable linkage more challenging, we expect that our distortion model is +likely to perform better in practice. Indeed, it achieves a better balance between precision and recall +in our full results (see Figure S7 in Appendix D). +Summary. +We return to the original goal of this section and summarize what we have learned +in this study. First, we have learned that our proposed linkage structure prior is generally more +robust due to the use of hyperpriors. In addition, our inferences are relatively insensitive to the +EP parameter regime, which may be due to the fact that the data sets are relatively small in size. +This behavior also holds for the blink distortion model when combined with our proposed linkage +structure priors. Second, when studying the performance of the distortion models (blink versus our +proposed distortion model), we find that ours predicts more reasonable distortion rates and improves +the linkage accuracy as measured by F1 score. Thus, based on this study, we would recommend +our distortion model and linkage structure priors moving forward for data sets similar to those we +have considered. However, we stress that further exploration is needed to provide more general +recommendations for other data sets. +20 + +5.4 +Comparison with Baseline Models +In this section, we study how our entity resolution model performs in comparison with models +by Steorts (2015) and (Sadinle, 2014). The blink model by Steorts (2015) is a natural baseline to +consider, as it served as inspiration for our model. Compared to our model, blink is less Bayesian +in its design, as many of the parameters are set empirically or arbitrarily. Both our model and +blink, are examples of direct modeling approaches to ER – i.e., they model how the observed +records are generated, incorporating the linkage structure as a latent variable. In contrast, the +model by Sadinle (2014) (which we refer to as Sadinle) adopts a comparison-based approach to ER. +Instead of modeling the data directly, it models attribute-level comparisons between pairs of records, +incorporating the presence/absence of a link between the pair as a latent variable. Sadinle (2018) +compares direct and comparison-based approaches from a methodological perspective, however we +are not aware of any empirical comparisons in the literature. Our goal in this section is to provide a +comparison for the first time on a variety of data sets, where we make no strong claims that our +results generalize to all applications or data sets. +In order to make the comparison as fair as possible, we use the same distance functions to model the +distortion in our model, blink, and Sadinle. For instance, if we use edit distance to model distortion +for a name attribute in our model and blink, then we also use edit distance to compare the same +name attribute in Sadinle. We set the distance cut-offs for our model and blink (see Section 4.3) to +align with the blocking design used for Sadinle. Further information about our experimental setup +is provided in Appendix E. +ER evaluation metrics are presented in Table 3 for all three models. For simplicity we only +provide results for our model under the GenCoupon prior, which is denoted Ours in the table. Our +model achieves the highest (or equal-highest) F1 score for all four data sets. We expect the poorer +performance of blink is due to its use of subjective (inflexible) priors and its distortion model, which +tends to favour high distortion and over-linkage. Sadinle achieves the second highest F1 score in the +non-private setting (RLdata, cora, rest), and the lowest F1 score in the private setting (nltcs). The +poorer performance for nltcs may be partly related to the blocking scheme, which is less aggressive, +leaving the model more susceptible to over-linkage. Another important factor is the sensitivity of +Sadinle to the truncation points for the priors on the 𝑚-probabilities. We perform coarse-grained +tuning of the truncation points in Appendix F, however fine-grained tuning could result in additional +performance gains. +Summary. +This study provides evidence that our model achieves a better balance between precision +and recall than blink and Sadinle. We stress that our results are based on four data sets – further +experimentation is required to determine whether our results generalize to other data sets and +applications. We speculate that the better performance of our model is mainly due to improved +flexibility resulting from the addition of priors and hyperpriors, which can be viewed as performing +model selection. +21 + +Evaluation metric +Data set +Model +Precision +Recall +F1 score +RLdata +Ours +0.917 (0.902, 0.932) +0.966 (0.953, 0.973) +0.934 (0.922, 0.942) +blink +0.336 (0.327, 0.344) +0.992 (0.988, 0.996) +0.502 (0.492, 0.511) +Sadinle +0.534 (0.524, 0.546) +0.964 (0.962, 0.966) +0.687 (0.679, 0.697) +nltcs +Ours +0.901 (0.878, 0.917) +0.934 (0.923, 0.943) +0.916 (0.903, 0.925) +blink +0.904 (0.890, 0.918) +0.918 (0.904, 0.925) +0.910 (0.903, 0.918) +Sadinle +0.312 (0.304, 0.319) +0.975 (0.969, 0.979) +0.473 (0.464, 0.480) +cora +Ours +0.973 (0.965, 0.980) +0.657 (0.627, 0.683) +0.784 (0.764, 0.803) +blink +0.978 (0.970, 0.985) +0.219 (0.207, 0.234) +0.358 (0.341, 0.378) +Sadinle +0.982 (0.981, 0.983) +0.359 (0.357, 0.362) +0.526 (0.524, 0.529) +rest +Ours +0.795 (0.749, 0.836) +0.830 (0.776, 0.871) +0.811 (0.774, 0.848) +blink +0.635 (0.586, 0.671) +0.920 (0.893, 0.946) +0.751 (0.713, 0.780) +Sadinle +0.993 (0.985, 1.000) +0.603 (0.598, 0.607) +0.750 (0.744, 0.756) +Table 3: Posterior performance of our model against two baselines: blink (Steorts, 2015) and Sadinle +(Sadinle, 2014). A point estimate for each evaluation metric is reported based on the median, along +with a 95% equi-tailed credible interval. Our model achieves the highest (or equal-highest) F1 score +within the credible intervals for all data sets. +5.5 +Controlled Simulation Study +We conduct a simulation study to evaluate our model under controlled conditions, where we vary +the size of the data set, the level of distortion, and the level of duplication. Due to space constraints, +we summarize the study here – full details can be found in Appendix C. We design a simulator for +household survey data sets, where responses are collected for individuals within households. Since +the attributes of individuals within a household are dependent – e.g., the address is the same, family +members may share the same last name, the age of individuals may be correlated – the simulated +data follows a more complex generative process than our entity resolution model. This is intentional, +as it allows us to evaluate our model in a more realistic setting where it is misspecified for the data. +The dataset simulator also incorporates a record generation process which is misspecified for our +model. Rather than sampling individuals from the population, our data set simulator iterates over all +individuals, randomly deciding whether to include the individual, and if so, how many distorted +records to create. +We run entity resolution using our model on 16 simulated data sets, using the blink and Sadinle +models as baselines. We summarize the results across three factors below: +• Duplication level. The level of duplication has minimal impact on the performance of our +model. blink performs well for moderate to high levels of duplication, however it over-links +severely when the level of duplication is low. The performance of Sadinle does not seem +to follow a consistent trend as the duplication varies – it achieves a lower F1 score than our +model and blink in all cases. +22 + +• Distortion level. We find the more distorted data sets are more difficult to link. Specifically, +we observe a drop in recall of around 10 percentage points for our model and blink when +compared to the data sets with low distortion. Larger drops in recall of 15–20 percentage +points are observed for Sadinle. +• Data set size. We find that our model performs similarly for both data set sizes (1000 and +10000 records). blink also performs similarly in most scenarios, however, we observe a drop +in precision for the larger data set when the level of duplication is low. Sadinle performs +significantly worse for the larger data sets in terms of precision, however, the recall is relatively +stable. +In summary, the simulation study shows that our model achieves the most consistent performance +across all scenarios tested. blink is also competitive, but it is has poor performance in the low +duplication scenario. +Sadinle achieves the lowest F1 score when the level of duplication is +non-negligible, and is somewhat competitive when the level of distortion is low. +6 +Discussion +In this section, we summarize our contributions and provide a discussion regarding future work. We +have proposed a Bayesian model for entity resolution that addresses limitations of previous work +(Steorts et al., 2016; Steorts, 2015; Marchant et al., 2021). Our model can be viewed as performing +graphical entity resolution, where observed records are clustered to (unobserved) latent entities. To +improve upon the scalability of previous work, we designed a partially collapsed Gibbs sampler +with an optimized implementation that can handle data sets of around 10,000 records. This allowed +us to provide comparisons with models by Steorts (2015) and Sadinle (2014), which was previously +only possible for toy-sized data sets. We provided comparisons to real and synthetic data sets and a +controlled simulation study. We observed that our model tends to be less sensitive to changes in the +hyperparameters than competing models for the data sets considered. Further analysis is required to +make more general conclusions beyond the data sets and simulations considered in this paper. +There are many potential avenues for future work. First, it would be of interest to explore scaling +for our proposed model and the model by Sadinle (2014). This could be achieved by designing +parallel/distributed inference algorithms, by investigating more efficient MCMC algorithms, or +by resorting to blocking techniques. Another area of interest, is exploring more diverse data +sets to understand the strengths and weaknesses of each method in practice. Although we made +recommendations based on four data sets and a simulation study, more comparisons would help to +alleviate any selection bias regarding data sets and provide guidance to users. Finally, future work +could consider microclustering priors (Miller et al., 2015) to assess their effectiveness compared to +the infinitely-exchangeable linkage structure priors considered here. 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Some of the updates are non-trivial +due to non-conjugacy of the proposed model. +A.1 +Update for the Distortion Probabilities +In this section, we provide the update for the distortion probability 𝜃𝑠𝑎 (source 𝑠 and attribute 𝑎). +This update is complicated by the presence of the distortion propensity variables 𝜔𝑖𝑎, which breaks +the conjugacy of the beta prior. To overcome this problem, we introduce the following auxiliary +variables: +𝑞𝑖𝑎 | 𝜔𝑖𝑎 ∼ Bernoulli(𝜔𝑖𝑎) +∀𝑖, 𝑎 +and modify the conditional distribution for the distortion indicators as follows: +𝑧𝑖𝑎 | 𝜃𝜍𝑖𝑎, 𝑞𝑖𝑎 ∼ Bernoulli(𝜃𝜍𝑖𝑎𝑞𝑖𝑎) +∀𝑖, 𝑎. +It is straightforward to show that one recovers the original model in Equation (5) when the auxiliary +variables are marginalized out. +Observe that the contribution to the posterior involving 𝑞𝑖𝑎 is +(𝜃𝜍𝑖𝑎𝑞𝑖𝑎)𝑧𝑖𝑎(1 − 𝜃𝜍𝑖𝑎𝑞𝑖𝑎)1−𝑧𝑖𝑎𝜔𝑞𝑖𝑎 +𝑖𝑎 (1 − 𝜔𝑖𝑎)1−𝑞𝑖𝑎 = +� +𝜃𝑧𝑖𝑎 +𝜍𝑖𝑎(1 − 𝜃𝜍𝑖𝑎)1−𝑧𝑖𝑎𝜔𝑖𝑎 +�𝑞𝑖𝑎 [1 − 𝜔𝑖𝑎]1−𝑞𝑖𝑎 . +Thus, the distribution of 𝑞𝑖𝑎 conditional on the other variables is: +𝑞𝑖𝑎 | 𝜔𝑖𝑎, 𝜃𝜍𝑖𝑎, 𝑧𝑖𝑎 ∼ Bernoulli +� +𝜔𝑖𝑎𝜃𝑧𝑖𝑎 +𝜍𝑖𝑎(1 − 𝜃𝜍𝑖𝑎)1−𝑧𝑖𝑎 +𝜔𝑖𝑎𝜃𝑧𝑖𝑎 +𝜍𝑖𝑎(1 − 𝜃𝜍𝑖𝑎)1−𝑧𝑖𝑎 + 1 − 𝜔𝑖𝑎 +� +∀𝑖, 𝑎. +(S15) +Next, observe that the contribution to the posterior involving 𝜃𝑠𝑎 is +𝜃𝛽(0) +𝑠𝑎 −1 +𝑠𝑎 +(1 − 𝜃𝑠𝑎)𝛽(1) +𝑠𝑎 −1 � +𝑖:𝜍𝑖=𝑠 +(𝜃𝑠𝑎𝑞𝑖𝑎)𝑧𝑖𝑎(1 − 𝜃𝑠𝑎𝑞𝑖𝑎)1−𝑧𝑖𝑎. +Hence, the distribution of 𝜃𝑠𝑎 conditional on the other variables is: +𝜃𝑠𝑎 | Q, Z, S ∼ Beta +� +𝛽(0) +𝑠𝑎 + +∑︁ +𝑖:𝜍𝑖=𝑠 +𝑧𝑖𝑎, 𝛽(1) +𝑠𝑎 + +∑︁ +𝑖:𝜍𝑖=𝑠 +𝑞𝑖𝑎(1 − 𝑧𝑖𝑎) +� +∀𝑠, 𝑎. +(S16) +It is also straightforward to see that the distribution of 𝑧𝑖𝑎 conditional on the other variables is a +point mass. In particular, we have +𝑧𝑖𝑎 | 𝑥𝑖𝑎, 𝜆𝑖, Y = +� +1, +if 𝑥𝑖𝑎 ≠ 𝑦𝜆𝑖𝑎 +0, +otherwise +(S17) +28 + +In summary, to update the distortion probabilities, one would first compute the distortion indicators +{𝑧𝑖𝑎} using Equation (S17). Then, conditional on the other variables, one would draw auxiliary +variables {𝑞𝑖𝑎} using Equation (S15). Finally, one can update the distortion probabilities {𝜃𝑠𝑎} +using Equation (S16). The updates for the other model parameters are unaffected by the introduction +of the auxiliary variables {𝑞𝑖𝑎}. +A.2 +Update for the Entity Attributes +In this section, we provide the update for the entity attributes. When updating entity attribute 𝑦𝑒𝑎, +we collapse the base distribution 𝐻𝑒𝑎 and distortion indicators Z. +The posterior factors involving 𝑦𝑒𝑎 after collapsing 𝐻𝑒𝑎 are as follows: +𝑃(𝑦𝑒𝑎 | Z, 𝛀, 𝚯, S, 𝐺𝑎, 𝜌𝑎) ∝ 𝑃(𝑦𝑒𝑎 | 𝐺𝑎) +× +∫ +� +𝑖:𝜆𝑖=𝑒 +𝑃(𝑥𝑖𝑎 | 𝜃𝜍𝑖𝑎, 𝜔𝑖𝑎, 𝑦𝑒𝑎, 𝐻𝑒𝑎)𝑃(𝐻𝑒𝑎 | 𝑦𝑒𝑎, 𝜌𝑎) d𝐻𝑒𝑎 +∝ 𝑃(𝑦𝑒𝑎 | 𝐺𝑎) +� +𝑖:𝜆𝑖=𝑒 +𝑥𝑖𝑎=𝑦𝑒𝑎 +(1 − 𝜃𝜍𝑖𝑎𝜔𝑖𝑎) +� +𝑖:𝜆𝑖=𝑒 +𝑥𝑖𝑎≠𝑦𝑒𝑎 +(𝜃𝜍𝑖𝑎𝜔𝑖𝑎) +× +∫ +� +𝑖:𝜆𝑖=𝑒 +𝑥𝑖𝑎≠𝑦𝑒𝑎 +𝐻𝑒𝑎(𝑥𝑖𝑎) +� +𝑣∈D𝑎\{𝑦𝑒𝑎} 𝐻𝑒𝑎(𝑣)𝜌𝑎𝜓𝑎(𝑣|𝑦𝑒𝑎)−1 +B(𝜌𝑎ψ𝑎(𝑦𝑒𝑎)) +d𝐻𝑒𝑎 +∝ 𝐺𝑎(𝑦𝑒𝑎) +� +𝑖:𝜆𝑖=𝑒 +𝑥𝑖𝑎=𝑦𝑒𝑎 +(1 − 𝜃𝜍𝑖𝑎𝜔𝑖𝑎) +� +𝑖:𝜆𝑖=𝑒 +𝑥𝑖𝑎≠𝑦𝑒𝑎 +(𝜃𝜍𝑖𝑎𝜔𝑖𝑎) +× +Γ(𝜌𝑎) +Γ( ¯𝑛𝑒𝑎(𝑦𝑒𝑎) + 𝜌𝑎) +� +𝑣∈V𝑒𝑎 +Γ(𝑛𝑒𝑎(𝑣) + 𝜌𝑎𝜓𝑎(𝑣 | 𝑦𝑒𝑎)) +Γ(𝜌𝑎𝜓𝑎(𝑣 | 𝑦𝑒𝑎)) +, +where B(·) is the multivariate beta function, V𝑒𝑎 = �� +𝑖:𝜆𝑖=𝑒{𝑥𝑖𝑎}� \ {𝑦𝑒𝑎} are the distorted record +values for the 𝑎-th attribute associated with entity 𝑒, 𝑛𝑒𝑎(𝑣) = � +𝑖:𝜆𝑖=𝑒 𝟙[𝑥𝑖𝑎 = 𝑣] is the number of +records linked to entity 𝑒 whose 𝑎-th attribute is equal to 𝑣 and ¯𝑛𝑒𝑎(𝑣) = � +𝑖:𝜆𝑖=𝑒 𝟙[𝑥𝑖𝑎 ≠ 𝑣] is the +number of records linked to entity 𝑒 whose 𝑎-th attribute is not equal to 𝑣. +We can rewrite the above expression in a more computationally convenient form by repeatedly +29 + +applying the recurrence relation for the Gamma functions5 to yield: +𝑃(𝑦𝑒𝑎 | Z, 𝛀, 𝚯, S, 𝐺𝑎, 𝜌𝑎) ∝ 𝐺𝑎(𝑦𝑒𝑎) +� +𝑣∈V𝑒𝑎 +�𝑛𝑒𝑎(𝑣) +𝑗=1 +{ 𝑗 − 1 + 𝜌𝑎𝜓𝑎(𝑣 | 𝑦𝑒𝑎)} +�¯𝑛𝑒𝑎(𝑦𝑒𝑎) +𝑗=1 +{ 𝑗 − 1 + 𝜌𝑎} +× +� +𝑖:𝜆𝑖=𝑒 +𝑥𝑖𝑎=𝑦𝑒𝑎 +(1 − 𝜃𝜍𝑖𝑎𝜔𝑖𝑎) +� +𝑖:𝜆𝑖=𝑒 +𝑥𝑖𝑎≠𝑦𝑒𝑎 +(𝜃𝜍𝑖𝑎𝜔𝑖𝑎). +Observe that the above distribution may only have support on a subset of the full domain D𝑎 when +distance thresholds are applied, as discussed in Section 4.3. In particular, one can show that the +support is a subset of +� +𝑖:𝜆𝑖=𝑒 +{𝑦 ∈ D𝑎 : dist𝑎(𝑦, 𝑥𝑖𝑎) ≤ 𝑑(cut) +𝑎 +}. +This fact can be used to implement the update more efficiently, since it is not necessary to construct +a pmf over the entire domain D𝑎. +A.3 +Update for the Linkage Structure +In this section, we provide the update for the linkage structure. When updating the linkage structure, +we use an urn-based scheme as described by Neal (2000). In doing so, we only need to keep track +of entities in the population that are linked to records – any isolated entities not linked to records +are ignored. This is important, as the population may be infinite in size for some Ewens-Pitman +parameter regimes (when 𝜎 ≥ 0). +To update the linked entity 𝜆𝑖 for record 𝑖, we remove the current link and allow the record to either +join one of the remaining instantiated entities (with at least one other record) or instantiate a “new” +entity. The conditional distribution has the following form: +𝑃(𝜆𝑖 = 𝑒 | Z, X, Y , 𝚲−𝑖, {𝜌𝑎}) ∝ +������ +������ +𝐶 |𝑒|−𝜎 +𝛼+𝑁−1 +� +𝑎 +∫ +𝑃(𝑥𝑖𝑎 | 𝑧𝑖𝑎, 𝑦𝑒𝑎, 𝐻𝑒𝑎)dℍ−𝑖,𝑒𝑎, +if 𝑒 is instantiated and |𝑒| > 0, +𝐶 𝛼+𝜎𝐸 +𝛼+𝑁−1 +� +𝑎 +� +𝑦𝑒𝑎∈D𝑎 𝑃(𝑦𝑒𝑎 | 𝐺𝑎) +∫ +𝑃(𝑥𝑖𝑎 | 𝑧𝑖𝑎, 𝑦𝑒𝑎, 𝐻𝑒𝑎)dℍ0,𝑒𝑎, +if 𝑒 is “new”, +(S18) +where +• 𝐶 is a normalization constant; +• 𝚲−𝑖 = (𝜆1, . . . , 𝜆𝑖−1, 𝜆𝑖+1, . . . , 𝜆𝑁) are the linked entities for all records excluding 𝑖; +5Repeated application of the recurrence relation for the Gamma function yields +Γ(𝑧) = +Γ(𝑧 + 𝑛 + 1) +𝑧(𝑧 + 1) · · · (𝑧 + 𝑛) +for complex 𝑧 (excluding zero and the negative integers) and non-negative integer 𝑛. +30 + +• |𝑒| = � +𝑖′≠𝑖 𝟙[𝜆𝑖′ = 𝑒] is the number of records (excluding 𝑖) linked to entity 𝑒; +• 𝐸 = � +𝑒′≠𝑒 𝟙[|𝑒| > 0] is the number of instantiated entities with at least one linked record; +• ℍ0,𝑒𝑎 is the prior for 𝐻𝑒𝑎; and +• ℍ−𝑖,𝑒𝑎 is the posterior for 𝐻𝑒𝑎 given the observed distorted record attributes 𝑥𝑖′𝑎 for which +𝑖′ ≠ 𝑖 and 𝜆𝑖′ = 𝑒 (also conditioned on 𝑦𝑒𝑎 and 𝜌𝑎). +Recall that the prior ℍ0,𝑒𝑎 for 𝐻𝑒𝑎 conditioned on 𝑦𝑒𝑎 and 𝜌𝑎 is Dirichlet(𝜌𝑎ψ𝑎(𝑦𝑒𝑎)). Since 𝑥𝑖𝑎 is +Categorical(𝐻𝑒𝑎) if 𝑧𝑖𝑎 = 1 (and a point mass at 𝑦𝑒𝑎 if 𝑧𝑖𝑎 = 0), the posterior ℍ−𝑖,𝑒𝑎 is also Dirichlet +by conjugacy. In particular, one can show that ℍ−𝑖,𝑒𝑎 is Dirichlet(α−𝑖,𝑒𝑎) where +𝛼−𝑖,𝑒𝑎(𝑣) = 𝜌𝑎𝜓𝑎(𝑣 | 𝑦𝑒𝑎) + +∑︁ +𝑖′≠𝑖:𝜆𝑖′=𝑒 +𝑧𝑖′𝑎𝟙[𝑥𝑖′𝑎 = 𝑣] +for 𝑣 ∈ D𝑎 \ {𝑦𝑒𝑎}. We can therefore simplify the integral in Equation (S18) with respect to ℍ−𝑖,𝑒𝑎 +as follows: ∫ +𝑃(𝑥𝑖𝑎 | 𝑧𝑖𝑎, 𝑦𝑒𝑎, 𝐻𝑒𝑎) 𝑑ℍ−𝑖,𝑒𝑎 += 𝟙[𝑥𝑖𝑎 = 𝑦𝑒𝑎]1−𝑧𝑖𝑎 +∫ +𝐻𝑒𝑎(𝑥𝑖𝑎)𝑧𝑖𝑎Γ(𝜌𝑎) +� +𝑣∈D𝑎\{𝑦𝑒𝑎} +𝐻𝑒𝑎(𝑣)𝜌𝑎𝜓𝑎(𝑣|𝑦𝑒𝑎)−1 +Γ(𝜌𝑎𝜓𝑎(𝑣 | 𝑦𝑒𝑎)) d𝐻𝑒𝑎 += +� +𝛼−𝑖,𝑒𝑎(𝑥𝑖𝑎) +� +𝑣 ∈D𝑎\{𝑦𝑒𝑎 } 𝛼−𝑖,𝑒𝑎(𝑣), +𝑧𝑖𝑎 = 1 +𝟙[𝑥𝑖𝑎 = 𝑦𝑒𝑎], +𝑧𝑖𝑎 = 0 +(S19) +By a similar argument, the integral in Equation (S18) with respect to ℍ0,𝑒𝑎 can be simplified to: +∫ +𝑃(𝑥𝑖𝑎 | 𝑧𝑖𝑎, 𝑦𝑒𝑎, 𝐻𝑒𝑎) 𝑑ℍ0,𝑒𝑎 = +� +𝜓𝑎(𝑥𝑖𝑎 | 𝑦𝑒𝑎), +𝑧𝑖𝑎 = 1, +𝟙[𝑥𝑖𝑎 = 𝑦𝑒𝑎], +𝑧𝑖𝑎 = 0. +(S20) +Putting Equations (S19) and (S20) in (S18), then gives: +𝑃(𝜆𝑖 = 𝑒 | Z, X, Y , 𝚲−𝑖, {𝜌𝑎}) ∝ +������� +������� +𝐶 |𝑒|−𝜎 +𝛼+𝑁−1 +� +𝑎:𝑧𝑖𝑎=1 +𝛼−𝑖,𝑒𝑎(𝑥𝑖𝑎) +� +𝑣 ∈D𝑎\{𝑦𝑒𝑎 } 𝛼−𝑖,𝑒𝑎(𝑣), +if 𝑒 is instantiated and |𝑒| > 0, +𝐶 𝛼+𝜎𝐸 +𝛼+𝑁−1 +� +𝑎:𝑧𝑖𝑎=1 +� +𝑦∈D𝑎 𝐺𝑎(𝑦)𝜓𝑎(𝑥𝑖𝑎 | 𝑦)𝑧𝑖𝑎𝟙[𝑥𝑖𝑎 = 𝑦]1−𝑧𝑖𝑎, +if 𝑒 is “new”. +A.4 +Update for the Ewens-Pitman Parameters +In this section, we describe the update for the Ewens-Pitman parameters. Since the priors on +the Ewens-Pitman parameters 𝛼 and 𝜎 are non-conjugate, we cannot perform a direct Gibbs +update. Thus, we describe tractable updates which require the introduction of auxiliary variables. +The updates (and priors) differ depending on the range of 𝜎. Teh (2006) proposed a scheme +for beta/gamma priors when 0 ≤ 𝜎 < 1 and 𝛼 > 0, which is summarized in Section A.4.1. In +Section A.4.2 we propose a similar scheme for gamma/shifted negative binomial priors when 𝜎 < 0. +31 + +A.4.1 +Case 0 ≤ 𝜎 < 1 and 𝛼 > 0 +Teh (2006) proposed an auxiliary variable scheme for the regime 0 ≤ 𝜎 < 1 and 𝛼 > 0 such that the +priors +𝜎 ∼ Beta +� +𝜁 (0), 𝜁 (1)� +and +𝛼 ∼ Gamma +� +𝜒(0), 𝜒(1)� +are conjugate. We provide a summary of the scheme here, but refer the reader to (Teh, 2006) for +further details. The scheme introduces the following sets of auxiliary variables conditional on the +two parameters 𝛼 and 𝜎: +𝑤 | 𝑁, 𝛼 ∼ Beta(𝛼 + 1, 𝑁 − 1), +𝑢𝑘 | 𝜎, 𝛼, 𝐸 ∼ Bernoulli +� +𝛼 +𝛼 + 𝜎𝑘 +� +, +𝑘 ∈ {1, . . . , 𝐸 − 1} +𝑣𝑒 𝑗 | 𝜎, 𝚲 ∼ Bernoulli +� 𝑗 − 1 +𝑗 − 𝜎 +� +, +∀𝑒, 𝑗 ∈ {1, . . . , 𝑁𝑒 − 1}. +(S21) +Here 𝑁𝑒 = |{𝑖 : 𝜆𝑖 = 𝑒}| denotes the number of records linked to entity 𝑒, and 𝐸 = � +𝑒 𝟙[𝑁𝑒 > 1] +denotes the number of entities linked to at least one record. +It follows that the posterior distributions of 𝛼 and 𝜎 conditional on the auxiliary variables and other +model parameter are given by: +𝜎 | {𝑢𝑘}, {𝑣𝑒 𝑗}, 𝚲 ∼ Beta�� +� +𝜁 (0) + +𝐸−1 +∑︁ +𝑘=1 +(1 − 𝑢𝑘), 𝜁 (1) + +∑︁ +𝑒:𝑁𝑒>1 +𝑁𝑒−1 +∑︁ +𝑗=1 +(1 − 𝑣𝑒 𝑗)�� +� +, +𝛼 | {𝑢𝑘}, 𝑤, 𝚲 ∼ Gamma +� +𝜒(0) + +𝐸−1 +∑︁ +𝑘=1 +𝑢𝑘, 𝜒(1) − log 𝑤 +� +. +(S22) +Thus, to update 𝛼 and 𝜎, one would first draw auxiliary variables 𝑤, {𝑢𝑘} and {𝑣𝑒 𝑗} conditional +on the linkage structure 𝚲 and the old values of 𝛼 and 𝜎 using Equation (S21). Then, conditional +on the auxiliary variables and the linkage structure, one would draw new values for 𝛼 and 𝜎 using +Equation (S22). +A.4.2 +Case 𝜎 < 0 and 𝛼 = 𝑚𝜅 for 𝑚 ∈ ℕ +We describe an auxiliary variable scheme for updating the Ewens-Pitman parameters in the regime +𝜎 < 0 and 𝛼 = 𝑚𝜅, where 𝑚 ∈ ℕ and 𝜅 > 0. This scheme is inspired by Teh (2006). The likelihood +factor associated with the partition of 𝑁 records into 𝐸 entities is as follows (Pitman, 2006): +𝑃(partition config) = (𝑚)𝐸↓ +(𝑚𝜅)𝑁↑ +𝐸 +� +𝑒=1 +(𝜅)𝑁𝑒↑ = 𝜅𝐸−1(𝑚 − 1)𝐸−1↓ +(𝑚𝜅 − 1)𝑁−1↑ +𝐸 +� +𝑒=1 +(𝜅 − 1)𝑁𝑒−1↑, +(S23) +where “partition config” is a representation of the linkage structure 𝚲 as a partition6, 𝑁𝑒 is the number +of records linked to the 𝑒-th entity, (𝑥)𝑛↑ = �𝑛−1 +𝑖=0 (𝑥+𝑖) is the rising factorial, and (𝑥)𝑛↓ = �𝑛−1 +𝑖=0 (𝑥−𝑖) +6Records 𝑖 and 𝑗 belong to the same subset of the partition if 𝜆𝑖 = 𝜆 𝑗, and otherwise belong to different subsets. +32 + +is the falling factorial. We begin by expressing the denominator in this equation as +1 +(𝑚𝜅 − 1)𝑁−1↑ += Γ(𝑚𝜅 + 1) +Γ(𝑚𝜅 + 𝑁) = B(𝑚𝜅 + 1, 𝑁 − 1) +Γ(𝑁 − 1) += +1 +Γ(𝑁 − 1) +∫ 1 +0 +𝑤𝑚𝜅(1 − 𝑤)𝑁−2 d𝑤, +which allows us to introduce the following auxiliary variable: +𝑤 | 𝑚, 𝜅, 𝑁 ∼ Beta(𝑚𝜅 + 1, 𝑁 − 1). +(S24) +Expressing each of the latter factors in Equation (S23) as +(𝜅 − 1)𝑁𝑒−1↑ = +𝑁𝑒−1 +� +𝑗=1 +(𝜅 + 𝑗) = +𝑁𝑒−1 +� +𝑗=1 +∑︁ +𝑣𝑒 𝑗∈{0,1} +𝜅𝑣𝑒 𝑗 𝑗1−𝑣𝑒 𝑗 +permits us to introduce the following additional auxiliary variables: +𝑣𝑒 𝑗 | 𝜅 ∼ Bernoulli +� +𝜅 +𝜅 + 𝑗 +� +, +∀𝑒, 𝑗 ∈ {1, . . . , 𝑁𝑒 − 1}. +(S25) +With this representation, we can place conjugate priors on 𝜅 and 𝑚, namely: +𝜅 ∼ Gamma(𝜒(0), 𝜒(1)) and 𝑚 ∼ NegativeBinomial(𝑟, 𝜈) + 1. +(S26) +The distribution on 𝑚 is a shifted negative binomial with support on the positive integers. The +parameterization we adopt for the negative binomial is in terms of the number of failures 𝑥 ∈ +{0, 1, 2, . . .} in a sequence of trials before a given number of successes 𝑟 > 0 occur. Each trial is an +i.i.d. draw from a Bernoulli distribution with success probability 𝜈. The density of 𝑥 is given by +𝑃(𝑥 | 𝑟, 𝜈) = (𝑥 + 𝑟 − 1)! +(𝑟 − 1)!𝑥! 𝜈𝑟(1 − 𝜈)𝑥. +Finally, we combine the priors in Equation (S26) with the likelihood factors to obtain the following +posterior distributions for the 𝑚 and 𝜅, conditional on the other model parameters: +𝑚 | 𝑤, 𝜅, 𝚲 ∼ NegBinomial(𝑟 + 𝐸 − 1, 1 − (1 − 𝜈)𝑤𝜅) + 𝐸, +𝜅 | {𝑣𝑒 𝑗}, 𝑤, 𝑚, 𝚲 ∼ Gamma�� +� +𝜒(0) + 𝐸 − 1 + +𝐸 +∑︁ +𝑒=1 +𝑁𝑒−1 +∑︁ +𝑗=1 +𝑣𝑒 𝑗, 𝜒(1) − 𝑚 log 𝑤�� +� +. +(S27) +Thus, to update 𝜅 and 𝑚, one would first draw auxiliary variables 𝑤 and {𝑣𝑒 𝑗} conditional on +the linkage structure 𝚲 and the old values of 𝛼 and 𝜎 using Equations (S24) and (S25). Then, +conditional on the auxiliary variables and the linkage structure, one would draw new values for 𝜅 +and 𝑚 using Equation (S27). +33 + +A.5 +Update for the Distortion Distribution Concentration +In this section, we provide the update for the distortion distribution concentration 𝜌𝑎. Since we +cannot rely on conjugacy for the update, we propose an auxiliary variable scheme. When updating +𝜌𝑎, we condition on the entity attribute values {𝑦𝑒𝑎}𝑒=1...𝐸, the record attribute values {𝑥𝑖𝑎}𝑖=1...𝑁 +and the links 𝚲 = {𝜆𝑖}𝑖=1...𝑁. We collapse the distortion distributions {𝐻𝑒𝑎}𝑒=1...𝐸. The contribution +to the likelihood involving 𝜌𝑎 is: +� +𝑒 +∫ +� +𝑖:𝜆𝑖=𝑒 +𝑃(𝑥𝑖𝑎 | 𝜃𝜍𝑖𝑎, 𝜔𝑖𝑎, 𝑦𝑒𝑎, 𝐻𝑒𝑎)𝑃(𝐻𝑒𝑎 | 𝜌𝑎) d𝐻𝑒𝑎 +∝ +� +𝑒 +Γ(𝜌𝑎) +Γ( ¯𝑛𝑒𝑎(𝑦𝑒𝑎) + 𝜌𝑎) +� +𝑣∈V𝑒𝑎 +Γ(𝑛𝑒𝑎(𝑣) + 𝜌𝑎𝜓𝑎(𝑣 | 𝑦𝑒𝑎)) +Γ(𝜌𝑎𝜓𝑎(𝑣 | 𝑦𝑒𝑎)) += +� +𝑒 +B(𝜌𝑎, ¯𝑛𝑒𝑎(𝑦𝑒𝑎)) +Γ( ¯𝑛𝑒𝑎(𝑦𝑒𝑎)) +� +𝑣∈V𝑒𝑎 +𝑛𝑒𝑎(𝑣) +� +𝑗=1 +{ 𝑗 − 1 + 𝜌𝑎𝜓𝑎(𝑣 | 𝑦𝑒𝑎)} +(S28) +where B(·, ·) is the beta function, V𝑒𝑎 := �� +𝑖:𝜆𝑖=𝑒{𝑥𝑖𝑎}� \ {𝑦𝑒𝑎}, 𝑛𝑒𝑎(𝑣) = � +𝑖:𝜆𝑖=𝑒 𝟙[𝑥𝑖𝑎 = 𝑣] and +¯𝑛𝑒𝑎(𝑣) = � +𝑖:𝜆𝑖=𝑒 𝟙[𝑥𝑖𝑎 ≠ 𝑣]. +Expressing the beta function in Equation (S28) as +B(𝜌𝑎, ¯𝑛𝑒𝑎(𝑦𝑒𝑎)) = +∫ 1 +0 +𝑤𝜌𝑎−1 +𝑒 +(1 − 𝑤𝑒) ¯𝑛𝑒𝑎(𝑦𝑒𝑎)−1 d𝑤𝑒 +permits us to introduce the following auxiliary variables: +𝑤𝑒 | 𝜌𝑎, X, Y , 𝚲 ∼ Beta +� +𝜌𝑎, +∑︁ +𝑒 +∑︁ +𝑖 +𝟙[𝑥𝑖𝑎 ≠ 𝑦𝜆𝑖𝑎] +� +, +for all 𝑒. We can also express each of the latter factors in Equation (S28) as +𝑗 − 1 + 𝜌𝑎𝜓𝑎(𝑣 | 𝑦𝑒𝑎) = +1 +∑︁ +𝑢𝑒𝑣 𝑗=0 +𝜌𝑎𝜓𝑎(𝑣 | 𝑦𝑒𝑎)𝑢𝑒𝑣 𝑗 ( 𝑗 − 1)1−𝑢𝑒𝑣 𝑗 +which permits us to introduce the following auxiliary variables: +𝑢𝑒𝑣 𝑗 | 𝜌𝑎, X, Y , 𝚲 ∼ Bernoulli +� +𝜌𝑎𝜓𝑎(𝑣 | 𝑦𝑒𝑎) +𝑗 − 1 + 𝜌𝑎𝜓𝑎(𝑣 | 𝑦𝑒𝑎) +� +(S29) +for all 𝑒, 𝑣 ∈ V𝑒𝑎 and 𝑗 ∈ {1, . . . , 𝑛𝑒𝑎(𝑣)}. +Now since the prior on 𝜌𝑎 is Gamma(𝜏(0) +𝑎 , 𝜏(1) +𝑎 ), we obtain the following posterior distribution for +𝜌𝑎 conditional on the other parameters: +𝜌𝑎 | {𝑤𝑒}, {𝑢𝑒𝑣 𝑗}, X, Y , 𝚲 ∼ Gamma�� +� +𝜏(0) +𝑎 ++ +∑︁ +𝑒 +∑︁ +𝑣∈V𝑒𝑎 +𝑛𝑒𝑎(𝑣) +∑︁ +𝑗=1 +𝑢𝑒𝑣 𝑗, 𝜏(1) +𝑎 +− +∑︁ +𝑒 +log 𝑤𝑒�� +� +. +(S30) +34 + +Thus to update 𝜌𝑎, one would first draw auxiliary variables {𝑤𝑒} and {𝑢𝑒𝑣 𝑗} conditional on the +record attributes X, entity attributes Y , linkage structure 𝚲, and the previous value of 𝜌𝑎 using +Equations (A.5) and (S29). Then, conditional on the auxiliary variables and X, Y , 𝚲, one would +draw a new value for 𝜌𝑎 using Equation (S30). +B +Hybrid Distance Measure +In this appendix, we describe a hybrid distance measure that is useful for comparing text strings +containing multiple tokens (words), where individual tokens may be subject to distortion. We use +the measure in this paper for comparing name and address attributes in the cora and rest data sets +(see Section 5.1), however it may have wider applications beyond this paper. Our measure draws +inspiration from a hybrid similarity measure proposed by Monge and Elkan (1996). However, unlike +Monge and Elkan, we attempt to match the tokens in each string while incorporating penalties for +tokens that are “missing” in one of the strings. +Suppose we would like to compare a pair of multi-token strings 𝑥 and 𝑦. As a running example, we +consider 𝑥 = “University of California, San Diego” and 𝑦 = “Univ. Calif., San Diego”. Given a +separator character (e.g., a space), we can map each string to a set of tokens. For example, string 𝑥 +from our running example would be mapped to +𝑋 = {“California,”, “Diego”, “of”, “San”, “University”}. +Note that we have used capital 𝑋 to denote the token set7 representation of string 𝑥 – a convention +we adopt throughout this appendix. Also note that 𝑋 is a lossy representation of 𝑥, as it discards +information about the token order. This is desirable for our applications to names and addresses8, +where permutation of the tokens does not significantly change the meaning of the strings. +We propose to measure the distance from 𝑥 to 𝑦 via a generalized edit distance on the token sets 𝑋 +and 𝑌. We consider three elementary edit operations: +• token insertions where a token 𝑏 is appended to the input set; +• token deletions where a token 𝑎 is removed from the input set; and +• token substitutions where a token 𝑎 in the input set is replaced by a token 𝑏 ≠ 𝑎. +Each elementary operation takes an input set 𝑄 to an output set 𝑄′, which we write as 𝑄 → 𝑄′, and +has an associated cost 𝑐(𝑄 → 𝑄′) ≥ 0. We let +𝑐(𝑄 → 𝑄′) = +���� +���� +𝑑𝑖 distinner(𝜆, 𝑏), +if 𝑄 = 𝑄′ \ {𝑏} (insertion), +𝑑𝑑 distinner(𝑎, 𝜆), +if 𝑄 \ {𝑎} = 𝑄′ (deletion), +𝑑𝑠 distinner(𝑎, 𝑏), +if 𝑄 \ {𝑎} = 𝑄′ \ {𝑏} (substitution), +7Technically we consider a multi-set, since we allow tokens to appear multiple times. +8Specifically, the title, venue and authors attributes in cora, and the name and addr attributes in rest. +35 + +where 𝑑𝑖, 𝑑𝑑 and 𝑑𝑠 are non-negative weights; 𝜆 is the null string; and distinner(·, ·) is an inner +distance measure on tokens (strings). We then define the hybrid distance between 𝑥 and 𝑦 as +the minimum average cost of transforming 𝑋 into 𝑌 via a sequence of elementary edit operations +𝑇𝑋,𝑌 = (𝑋 → 𝑄1, 𝑄1 → 𝑄2, . . . , 𝑄𝑙−1 → 𝑌). Symbolically, we write +disthybrid(𝑥, 𝑦) = min +𝑇𝑋,𝑌 +1 +|𝑇𝑋,𝑌 | +∑︁ +(𝑄→𝑄′)∈𝑇𝑋,𝑌 +𝑐(𝑄 → 𝑄′). +We can compute the hybrid distance using an off-the-shelf linear sum assignment problem (LSAP) +solver (Crouse, 2016). In order to do so, we need to add null string tokens to 𝑋 and 𝑌 to account for +all possible insertion and deletion operations. Concretely, we add |𝑌| null tokens to 𝑋 to allow for +insertions and |𝑋| null tokens to 𝑌 to allow deletions. We then construct a pairwise cost matrix +by applying distinner to all pairs of tokens in (the amended) 𝑋 and 𝑌. The resulting matrix is then +passed to the LSAP solver, which returns the optimal set of edit operations and their cost. +Returning to our running example, if we set distinner to the Levenshtein distance, the solution to the +LSAP is +{(“University” ↔ “Univ.”, 5), (“of” ↔ 𝜆, 2), (“California,” ↔ “Calif.,”, 6), +(“San” ↔ “San”, 0), (“Diego” ↔ “Diego”, 0), (𝜆 ↔ 𝜆, 0), (𝜆 ↔ 𝜆, 0), +(𝜆 ↔ 𝜆, 0), (𝜆 ↔ 𝜆, 0)}. +Hence we conclude that disthybrid(𝑥, 𝑦) = 5+2+6+0+0 +5 += 2.6. This distance reflects the semantic +closeness between 𝑥 and 𝑦 better than the Levenshtein distance, which gives a larger value of 14 +when evaluated directly on 𝑥 and 𝑦. +C +Simulation Study +In this appendix, we conduct a simulation study to understand how our model performs in controlled +scenarios. Specifically, we simulate entity resolution data sets where we vary the number of records, +the level of distortion and the level of duplication. Since our model is generative, we could use it to +simulate data, however the resulting data would have negligible specification error for our model, +which is not realistic. We therefore simulate data that is purposefully misspecified for our model by +adding additional dependencies between the entities and entity attributes, and by using a different +process to generate records from entities. We were unable to find an existing data set simulator that +generated such data, so we implemented our own. +C.1 +Data Set Simulator +We provide an overview of our simulator, which generates personal records describing a population +of households. For brevity, we omit low-level details here and refer the reader to the included Python +script. Our simulator operates in two stages: in the first stage it generates a population of households, +then in the second stage it iterates over individuals in all households, generating a random number +of distorted records for each individual. By generating households rather than individuals in the first +36 + +stage, we are able to incorporate additional dependencies between individuals (entities) that are not +present in our ER model (see Section 3.1). +Generating Households. +We now describe how households are generated in the first stage. In +our simplified model, a household may be a couple, a single, a couple or single with children, or a +group of unrelated adults. Individuals within a household are described by the following attributes: +first and last name (first_name and last_name), date of birth (birth_year, birth_month, birth_day), +gender and zipcode. The zipcode is constrained to be the same for all individuals within a household, +and the first name is conditioned on the gender, however the other attributes may vary as described +below. Random values for attributes are generated using the Faker Python library9, which attempts +to mimic real-world frequency distributions. We make the distributions more concentrated for the +name and zipcode attributes to ensure the entities are not too unique (otherwise entity resolution +would be too easy). +We begin by generating the head(s) of the household, which are a male and female couple (for +simplicity) or a single male or female. If a couple is generated, they have a high chance of sharing +the same last name and their birth years are likely not too far apart. Next we randomly decide +whether to generate children. If children are generated, they share the same last name as the head(s) +of the household (the parents) and there is an appropriate gap between their birth year and their +parents’ birth year. If no children are generated, then we randomly decide whether to generate +unrelated adults who live with the head(s) of household. The unrelated adults are constrained to +be of a similar age as the head(s) of the household. When simulating the household composition, +we attempt to follow aggregate statistics from the Current Population Survey (U.S. Census Bureau, +2016). +Generating Records. +In the second stage, records are generated for individuals across all +households. We simulate a single database/file with duplicate records by including an individual +with probability 𝑝inc = 0.9 and sampling the number of records according to a Poisson distribution +with rate parameter 𝜇, truncated to the interval [1, 4]. Each record is obtained by copying the entity +attributes subject to distortion. This is done by iterating over the attributes in a random order, and +deciding whether to activate the distortion process with a probability that varies for each attribute. +The distortion process for birth day, birth month, gender and zipcode involves drawing a replacement +value according to the distribution used in the first stage. The distortion process for birth year +involves adding discrete Gaussian noise to the true birth year. The distortion process for first and +last name may proceed in one of three ways: (1) by making a random typographical error (character +insertion, deletion, substitution or transposition); (2) by replacing the name with a variant drawn +uniformly at random; or (3) by generating a replacement according to the distribution used in the +first stage. Variant names for (2) are sourced from the WeRelate.org Variant Names Project10. +C.2 +Results +We generate 16 data sets using our simulator for each combination of the following variables: +9https://github.com/joke2k/faker +10https://www.werelate.org/wiki/WeRelate:Variant_names_project +37 + +Distortion probability +Attribute +Low distortion +High distortion +first_name +10% +40% +last_name +10% +40% +gender +1% +1% +zipcode +5% +10% +birth_year +1% +10% +birth_month +1% +10% +birth_day +1% +10% +Table S4: Attribute-level distortion probabilities for two levels of distortion: low and high. +Low (μ=0.1) +Medium (μ=1) +High (μ=8) +Very high (μ=100) +1 +2 +3 +4 +1 +2 +3 +4 +1 +2 +3 +4 +1 +2 +3 +4 +0.00 +0.25 +0.50 +0.75 +1.00 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.0 +0.2 +0.4 +0.6 +0.00 +0.25 +0.50 +0.75 +Records per entity (cluster size) +Relative frequency +Figure S5: Distribution of records per entity for each level of duplication: low, medium, high and +very high. The Poisson rate parameter 𝜇 for each level is given in parentheses. +• Number of records. We consider data sets with 1000 and 10000 records in expectation. The +number of records is random and depends on the number of individuals and the Poisson rate +parameter. Since the Poisson rate parameter is fixed (see below), we control the number of +records by varying the number of individuals generated in the first stage. +• Level of distortion. We consider two levels of record distortion which we refer to as “low” and +“high”. These correspond to different choices for the probabilities of activating the distortion +process as detailed in Table S4. +• Level of duplication. We consider four levels of duplication which we refer to as “low”, +“medium”, “high” and “very high”. These levels correspond to Poisson rate parameters of +0.1, 1, 8 and 100, respectively. When the duplication is “low” (𝜇 = 0.1) over 95% of the +entities represented in the data only appear once. Whereas when the duplication is “very high” +(𝜇 = 100) over 95% of the entities represented in the data appear four times. The distribution +of records per entity is plotted for each level in Figure S5. +We perform a comparative evaluation of our model, blink, and Sadinle on the 16 simulated data sets. +The model evaluation procedure is described in Section 5.2 and the blink and Sadinle models are +introduced in Section 5.4. ER evaluation metrics are plotted for each data set in Figure S6. We now +make several observations about the results. +38 + +First, we observe that our model and blink perform similarly when the duplication level is medium, +high or very high. For these duplication levels, blink has a slight advantage in terms of recall when +the distortion level is high. The largest difference is observed for the medium duplication/high +distortion scenario, where the recall for blink is roughly 10 percentage points higher than for our +model. This difference is due to the priors placed on the concentration parameters 𝜌𝑎 in our model, +which favour high concentrations. This corresponds to a prior belief that distortions occur in the +same way, rather than in multiple different ways. However, this is not true for distortions in the +simulated data – e.g., an individual whose first name is “JONATHON” may appear in the data with +four distinct first names: “JOHN”, “JOJN”, “JONATHON” and “ALEX”. If we wanted to exploit +this knowledge, we could increase 𝜏(0) +𝑎 +for our model to favour lower concentrations. +Second, we observe that our model significantly outperforms blink in terms of precision when the +duplication level is low. We believe this is due to the highly informative prior on the linkage structure +used in blink – it uses a coupon prior with 𝑚 fixed to the number of records 𝑁 and 𝜅 → ∞. However +our model under the generalized coupon prior selects a value for 𝑚 of approximately 8 × 𝑁 and 𝜅 of +approximately 100, which allows it to more accurately model a low duplication scenario. +Thirdly, we observe that Sadinle achieves the lowest F1 score when the duplication level is medium, +high or very high. For these duplication levels, the performance gap in F1 score is largest when +the distortion is high – approximately 20 percentage points. The gap is less significant when the +distortion is low – around 5 percentage points. These differences in F1 score are mainly due to lower +recall in most cases – the precision is generally competitive with the other models. +Fourthly, we comment on the effect of the dataset size, measured in terms of the expected number of +records (1000 or 10000). We observe similar trends for all three models: the precision tends to be +larger for the smaller dataset, while the recall tends to be larger for the larger dataset (however there +are exceptions). The difference in performance is most pronounced for the low duplication setting, +where the precision of blink and Sadinle drops considerably for the larger dataset, and the recall of +Sadinle drops also drops considerably for the larger dataset. +D +Study of Linkage Structure Priors and the Distortion Model +In this appendix, we provide additional results for the study of linkage structure priors and distortion +models presented in Section 5.3. Our goal is to study the impact of the modeling contributions +independently, to determine whether each contribution is beneficial in its own right, and/or whether +one contribution is more beneficial than the other. +Recall from Section 5.3 that we considered four parameter regimes for the linkage structure priors – +PY, Ewens, GenCoupon and Coupon – and two distortion models – Ours as proposed in Section 3.1 +and blink as proposed by Steorts (2015). Thus, there are eight model variants to test – one for +each linkage structure prior and distortion model. Figure S7 presents pairwise evaluation metrics +(F1 score, precision and recall) for the eight model variants and four data sets in a single plot. We +interpret the results for each modeling contribution below. +39 + +F1 score +L. Dist. +L. Dup. +F1 score +H. Dist. +L. Dup. +F1 score +L. Dist. +M. Dup. +F1 score +H. Dist. +M. Dup. +F1 score +L. Dist. +H. Dup. +F1 score +H. Dist. +H. Dup. +F1 score +L. Dist. +V. H. Dup. +F1 score +H. Dist. +V. H. Dup. +Recall +L. Dist. +L. Dup. +Recall +H. Dist. +L. Dup. +Recall +L. Dist. +M. Dup. +Recall +H. Dist. +M. Dup. +Recall +L. Dist. +H. Dup. +Recall +H. Dist. +H. Dup. +Recall +L. Dist. +V. H. Dup. +Recall +H. Dist. +V. H. Dup. +Precision +L. Dist. +L. Dup. +Precision +H. Dist. +L. Dup. +Precision +L. Dist. +M. Dup. +Precision +H. Dist. +M. Dup. +Precision +L. Dist. +H. Dup. +Precision +H. Dist. +H. Dup. +Precision +L. Dist. +V. H. Dup. +Precision +H. Dist. +V. H. Dup. +Ours +blink +Sadinle +Ours +blink +Sadinle +Ours +blink +Sadinle +0.25 +0.50 +0.75 +1.00 +0.25 +0.50 +0.75 +0.900 +0.925 +0.950 +0.975 +0.7 +0.8 +0.9 +0.92 +0.94 +0.96 +0.6 +0.7 +0.8 +0.9 +0.90 +0.92 +0.94 +0.96 +0.7 +0.8 +0.9 +0.84 +0.88 +0.92 +0.96 +0.5 +0.6 +0.7 +0.8 +0.84 +0.88 +0.92 +0.96 +0.5 +0.6 +0.7 +0.8 +0.84 +0.87 +0.90 +0.93 +0.5 +0.6 +0.7 +0.8 +0.85 +0.90 +0.95 +0.45 +0.55 +0.65 +0.75 +0.85 +0.25 +0.50 +0.75 +1.00 +0.25 +0.50 +0.75 +1.00 +0.900 +0.925 +0.950 +0.975 +1.000 +0.85 +0.90 +0.95 +1.00 +0.994 +0.996 +0.998 +1.000 +0.97 +0.98 +0.99 +1.00 +0.997 +0.998 +0.999 +1.000 +0.985 +0.990 +0.995 +1.000 +Model +Measure value +Exp. number of records +1000 +10000 +Figure S6: Posterior evaluation metrics for our model, blink, and Sadinle when fitted on simulated +datasets with varying levels of distortion and duplication. Low and high distortion levels are +abbreviated as “L. Dist.” and “H. Dist.” respectively. Low, medium, high and very high duplication +levels are abbreviated as “L. Dup.”, “M. Dup.”, “H. Dup.” and “V. H. Dup.” respectively. The +expected number of records (1000 or 10000) is denoted by the color and shape of the markers. +A point estimate for each evaluation metric is reported based on the median and 95% equi-tailed +credible interval are represented by intervals around the point. +40 + +rest +Precision +rest +Recall +rest +F1 score +cora +Precision +cora +Recall +cora +F1 score +nltcs +Precision +nltcs +Recall +nltcs +F1 score +RLdata +Precision +RLdata +Recall +RLdata +F1 score +PY +Ewens +GenCoupon +Coupon +PY +Ewens +GenCoupon +Coupon +PY +Ewens +GenCoupon +Coupon +0.25 +0.50 +0.75 +1.00 +0.89 +0.90 +0.91 +0.92 +0.93 +0.2 +0.4 +0.6 +0.8 +0.70 +0.75 +0.80 +0.85 +0.95 +0.96 +0.97 +0.98 +0.99 +1.00 +0.90 +0.91 +0.92 +0.93 +0.94 +0.95 +0.2 +0.4 +0.6 +0.8 +0.8 +0.9 +0.25 +0.50 +0.75 +1.00 +0.850 +0.875 +0.900 +0.925 +0.95 +0.96 +0.97 +0.98 +0.99 +0.5 +0.6 +0.7 +0.8 +0.9 +Linkage prior +Measure value +Distortion model +Ours +blink +Figure S7: Evaluation of ER quality as a function of the linkage structure prior (plotted on the +𝑥-axis) and distortion model (indicated by the line color). Three pairwise evaluation measures are +shown (grouped by column) for four data sets (grouped by row). +41 + +100 +1000 +10000 +1e−04 +1e−03 +1e−02 +1e−01 +100 +1000 +10000 +1 +10 +100 +1000 +10000 +PY +Ewens +GenCoupon +RLdata +nltcs +cora +rest +RLdata +nltcs +cora +rest +Data set +RLdata +nltcs +cora +rest +alpha +sigma +alpha +kappa +m +Figure S8: Posterior Ewens-Pitman parameters for three regimes: PY, Ewens and GenCoupon under +our distortion model (Ours). Note that the values of the parameters are presented on log-scales. +Linkage Structure Prior. +While we discuss the effect of the linkage structure prior in Section 5.3, +we only present results for our distortion model (Ours) due to space constraints. In Section 5.3, we +draw two main conclusions from the results in Table 2 which are replicated in Figure S7 (represented +by circular vermilion markers): +1. Our proposal to place vague hyperpriors on the EP parameters (for PY, Ewens and GenCoupon) +improves robustness and yields the highest ER accuracy for three of data sets, as measured +by pairwise F1 score (nltcs is the exception). We observe significantly lower F1 scores +when hyperpriors are not used (see Coupon in Figure S7), particularly for cora and RLdata. +Figure S8 provides further justification for this argument, as it shows vastly different values of +the EP parameters are selected for each data set, facilitated by the vague hyperpriors. +2. Our inferences are relatively insensitive to the EP parameter regime (PY, Ewens or GenCoupon) +despite the fact that each regime is known to exhibit distinct asymptotic behavior (see +Section 2.2). +Figure S7 shows that these conclusions also hold for the blink distortion model (represented by +triangular teal markers). We expect the competitive performance for nltcs under the Coupon linkage +prior may be a coincidence, as the population size under the prior is 3,387, which happens to be +very close to the true value of 3,307 (see Table 1). +Distortion Model. +In Section 5.3, we discuss the effect of the distortion prior under the GenCoupon +linkage structure prior. We now extend the discussion to include results for the three other linkage +structure priors (PY, Ewens and Coupon), as presented in Figure S7. +We find that our distortion model achieves the highest F1 score for all but one of the data sets and +linkage structure priors. The exception is for cora under the Coupon linkage structure prior, where +42 + +the blink distortion model has a slight edge. An explanation for the improved performance of our +model is given in Section 5.3, which we summarize here. The blink distortion model is susceptible +to entering a high distortion mode, particularly for attributes with non-constant distance measures. +This is because it allows a record attribute value to be marked as “distorted” even if it is not actually +distorted. Our model corrects this inconsistency, and in doing so appears to be more robust. In +general, we expect the blink distortion model to result in over-linkage (high recall, low precision), +while our model is expected to be more balanced. Figure S7 supports this argument, with the +difference being most apparent for RLdata, where we see a difference of ∼0.7 in the F1 score. +E +Further Details of Experimental Setup +In this appendix, we provide further details about the experiments presented in Section 5. +Implementation and Hardware. +All experiments were conducted in R version 3.4.4, running on +a local server fitted with two 28-core Intel Xeon Platinum 8180M CPUs and 12 TB of RAM.11 We +developed an open-source R package called exchanger12 which implements variants of our model +(under different linkage structure priors and distortion models) in addition to the blink model (Steorts, +2015). Since an implementation of the model proposed by Sadinle (2014) was not publicly available, +we developed our own which we released as an open-source R package called BDD13. For efficiency +reasons, we implemented inference for all models in C++ using the Rcpp interface (Eddelbuettel and +François, 2011). The data set simulator described in Appendix C.1 was implemented as a Python +script. A Pipfile is provided to specify the dependencies used when running the script. +Hyperparameter Settings. +We followed the recommendations in Section 3.2 when setting +hyperparameters for our model. When setting hyperparameters for the two baseline models, +we attempted to follow the recommendations of the authors. For blink, we set 𝑚 = 𝑁 for the +coupon-collector’s prior and 𝛽(0) +𝑠𝑎 = 𝑁/1000 and 𝛽(1) +𝑠𝑎 = 𝑁/10 for the Beta prior on the distortion +probabilities (here 𝑁 is the total number of records). For Sadinle, we set the agreement levels by +inspecting the distribution of distances for each attribute. We used truncated uniform priors on the +𝑚-probabilities and a uniform prior on the 𝑢-probabilities, as recommended by the author. We set +the lower truncation points for the 𝑚-probabilities to 0.95, based on tuning experiments presented in +Appendix F. +Initialization and MCMC. +For our model and blink, we initialized the linkage structure 𝚲, entity +attributes Y and distortion indicators Z by linking each record to a unique entity and copying the +record attributes into the entity attributes, assuming no distortion. The distortion probabilities 𝚯 +and entity attributes distributions G were initialized by drawing from their conditional distributions. +The Ewens-Pitman parameters and distortion distribution concentration parameters were initialized +using their prior means. +11R scripts are published at github.com/cleanzr/exchanger-experiments. +12Package source code published at github.com/cleanzr/exchanger. +13Package source code published at github.com/cleanzr/BDD. +43 + +A similar initialization was used for the Sadinle model. We assigned each record to a unique entity +(cluster). The 𝑚- and 𝑢-probabilities were initialized by drawing from their conditional distributions. +When fitting each model, we ran Markov chain Monte Carlo (MCMC) for 2 × 105 iterations, +discarding the first 105 iterations as burn-in, and applying thinning with an interval of 10.14 This +produced 104 approximate posterior samples. +F +Tuning Hyperparameters for Sadinle (2014) +Our aim in this appendix is to determine reasonable values for the hyperparameters in the entity +resolution model by Sadinle (2014), which we refer to as Sadinle. +We assume the distance +functions (used to compare attributes) and agreement levels (mappings from real-valued distances +to discrete levels) are fixed, and that flat priors are used, as recommended by Sadinle. Given these +assumptions, the only hyperparameters that remain unspecified are the lower truncation points for +the 𝑚-probabilities. +The 𝑚-probabilities are a set of parameters {𝑚𝑎𝑙} where 𝑚𝑎𝑙 is the probability that a pair of +records referring to the same entity agree at level 𝑙 on attribute 𝑎, given they do not agree at +levels 0, 1, . . . , 𝑙 − 1. Sadinle recommends using truncated flat priors on 𝑚𝑎𝑙, so that the allowed +values lie in the interval [𝜆𝑎𝑙, 1], where 𝜆𝑎𝑙 ∈ [0, 1] is a hyperparameter typically close to 1. More +specifically, he recommends setting 𝜆𝑎𝑙 = 0.95 if attribute 𝑎 is a “nearly-accurate” quasi-identifier +and 𝜆𝑎𝑙 = 0.85 if attribute 𝑎 is an “inaccurate” quasi-identifier. Since it is not clear which setting +for 𝜆𝑎𝑙 is best for our data sets, we run the experiments described in Section 5.4 for four different +values: 𝜆𝑎𝑙 = 0, 0.5, 0.85, 0.95. When setting 𝜆𝑎𝑙, we use the same value for all attributes 𝑎 and +agreement levels 𝑙 for simplicity. The results are reported in Figure S9 and Table S5. +Figure S9 plots the posterior values of the 𝑚-probabilities (on the 𝑥-axis) for each truncation point +𝜆𝑎𝑙 (corresponding to the horizontal panels). We observe that the posterior values of 𝑚𝑎𝑙 tend to be +close to 𝜆𝑎𝑙, especially for agreement level 𝑙 = 0 and 𝜆𝑎𝑙 ≥ 0.5. This suggests that the model favors +small values of 𝑚𝑎𝑙, despite the fact that 𝑚𝑎𝑙 is expected to be close to 1. Consequently, the model +has a tendency to “over-link”—linking records that do not refer to the same entity. Understanding +why the model exhibits this behavior would require further exploration and is beyond the scope of +this paper. However, we speculate that it may be related to the use of flat priors, or known stability +issues with Fellegi-Sunter-type models (Goldstein et al., 2017). As a result, we conclude that the +posterior value of 𝑚𝑎𝑙 is highly sensitive to the choice of 𝜆𝑎𝑙. +Table S5 shows the impact of the truncation point 𝜆𝑎𝑙 on entity resolution performance. It shows that +the performance is relatively stable for cora and rest as a function of 𝜆𝑎𝑠, despite the fact that there +is some variation in the 𝑚-probabilities (as shown in Figure S9). The reason for the stability may be +related to the blocking scheme used for these data sets, which rules out a relatively large number +of potential links, thereby guarding against over-linkage. On the other hand, the performance for +RLdata and nltcs is far less stable. We find that the precision drops considerably as 𝜆𝑎𝑙 is reduced, +14The chain was slower to converge for the cora data set, so we increased the number of iterations to 2.5 × 105 and +the burn-in interval to 1.5 × 105. +44 + +λ = 0 +λ = 0.5 +λ = 0.85 +λ = 0.95 +cora +nltcs +rest +RLdata +0.00 +0.25 +0.50 +0.75 +1.00 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +0.85 +0.90 +0.95 +1.00 +0.95 +0.96 +0.97 +0.98 +0.99 +1.00 +year +venue +title +authors +SEX +REGOFF +DOB_YEAR +DOB_MONTH +DOB_DAY +type +name +city +addr +lname_c1 +fname_c1 +by +bm +bd +m value +Attribute +Agreement level +1 +2 +3 +4 +Figure S9: Posterior estimates of the 𝑚-probabilities (plotted on the 𝑥-axis) in the Sadinle model as +a function of the data set (vertical panels), attribute (𝑦-axis), agreement level (color/marker) and +lower truncation point (horizontal panels). Point estimates are shown based on the median, along +with 95% equi-tailed credible intervals. The model has a tendency to select small values for the +𝑚-probabilities, close to the lower truncation point, especially for agreement level 1. +45 + +Evaluation metric +Data set +Truncation point +Precision +Recall +F1 score +RLdata +0.00 +0.068 (0.067, 0.068) +0.909 (0.906, 0.912) +0.126 (0.125, 0.126) +0.50 +0.069 (0.069, 0.069) +0.921 (0.919, 0.924) +0.128 (0.128, 0.129) +0.85 +0.315 (0.308, 0.321) +0.965 (0.962, 0.967) +0.475 (0.467, 0.481) +0.95 +0.534 (0.524, 0.546) +0.964 (0.962, 0.966) +0.687 (0.679, 0.697) +nltcs +0.00 +0.114 (0.113, 0.114) +0.983 (0.978, 0.987) +0.204 (0.203, 0.205) +0.50 +0.111 (0.110, 0.111) +0.964 (0.958, 0.970) +0.199 (0.197, 0.200) +0.85 +0.162 (0.160, 0.164) +0.972 (0.968, 0.976) +0.278 (0.275, 0.281) +0.95 +0.312 (0.304, 0.319) +0.975 (0.969, 0.979) +0.473 (0.464, 0.480) +cora +0.00 +0.980 (0.979, 0.981) +0.378 (0.377, 0.378) +0.545 (0.544, 0.546) +0.50 +0.981 (0.979, 0.984) +0.390 (0.389, 0.391) +0.558 (0.557, 0.560) +0.85 +0.984 (0.983, 0.984) +0.383 (0.382, 0.384) +0.552 (0.551, 0.553) +0.95 +0.982 (0.981, 0.983) +0.359 (0.357, 0.362) +0.526 (0.524, 0.529) +rest +0.00 +1.000 (0.985, 1.000) +0.607 (0.598, 0.607) +0.756 (0.744, 0.756) +0.50 +0.985 (0.985, 1.000) +0.598 (0.598, 0.607) +0.744 (0.744, 0.756) +0.85 +0.985 (0.985, 1.000) +0.598 (0.598, 0.607) +0.744 (0.744, 0.756) +0.95 +0.993 (0.985, 1.000) +0.603 (0.598, 0.607) +0.750 (0.744, 0.756) +Table S5: Posterior performance of the Sadinle model as a function of the lower truncation point 𝜆𝑎𝑙 +on the 𝑚-probabilities. A point estimate for each evaluation metric is reported based on the median, +along with a 95% equi-tailed credible interval. +while the recall remains relatively stable. This is a sign of over-linkage, which is expected since the +posterior 𝑚-probabilities are significantly smaller. Based on these results, we set 𝜆𝑎𝑙 = 0.95 as the +default value for our other experiments since it seems to achieve balanced performance. +46 + +G +MCMC Diagnostics +G.1 +Study of Linkage Structure Priors +Here we present convergence diagnostics for the models fitted in Section 5.3. We present Geweke +diagnostic plots and trace plots for a selection of model variables for each data set, linkage structure +prior and distortion model. Each pair of plots is preceded by a title of the form “Data set | Linkage +structure prior | Distortion model”. The Geweke diagnostic plot (on the left) depicts a Z-score on the +x-axis for each variable on the y-axis. The Z-score tests for equality of the means of the first 10% +and final 50% of the Markov chain, and is typically expected to be in the range [−2, 2] (Geweke, +1992). The trace plot (on the right) depicts the value of variables (labeled in the right panel) for each +step in the chain (on the x-axis). Note that variable 𝐸 denotes the number of instantiated entities. +We replace integer indices for the attributes by named indices. For instance, 𝜃0,city refers to the +distortion probability in source 0 of the attribute called “city”. +nltcs | PY | Ours +α +ρDOB_DAY +ρDOB_MONTH +ρDOB_YEAR +ρREGOFF +ρSEX +θ0, DOB_DAY +θ0, DOB_MONTH +θ0, DOB_YEAR +θ0, REGOFF +θ0, SEX +d +E +−2 +−1 +0 +1 +2 +Geweke diagnostic +Variable +E +d +θ0, SEX +θ0, REGOFF +θ0, DOB_YEAR +θ0, DOB_MONTH +θ0, DOB_DAY +ρSEX +ρREGOFF +ρDOB_YEAR +ρDOB_MONTH +ρDOB_DAY +α +100000 +125000 +150000 +175000 +200000 +3350 +0.00 +0.02 +0.000 +0.004 +0.000 +0.005 +0.010 +0.000 +0.005 +0.000 +0.003 +0.006 +0.000 +0.004 +0.008 +0e+00 +5e+04 +1e+05 +0e+00 +5e+04 +1e+05 +0e+00 +5e+04 +1e+05 +0e+00 +5e+04 +1e+05 +0 +50000 +3600 +4000 +Iteration +Value +47 + +nltcs | Ewens | Ours +α +ρDOB_DAY +ρDOB_MONTH +ρDOB_YEAR +ρREGOFF +ρSEX +θ0, DOB_DAY +θ0, DOB_MONTH +θ0, DOB_YEAR +θ0, REGOFF +θ0, SEX +E +−2 +−1 +0 +1 +2 +Geweke diagnostic +Variable +E +θ0, SEX +θ0, REGOFF +θ0, DOB_YEAR +θ0, DOB_MONTH +θ0, DOB_DAY +ρSEX +ρREGOFF +ρDOB_YEAR +ρDOB_MONTH +ρDOB_DAY +α +100000 +125000 +150000 +175000 +200000 +3350 +0.000 +0.003 +0.006 +0.000 +0.005 +0.010 +0.015 +0.000 +0.004 +0.000 +0.004 +0.000 +0.005 +0e+00 +5e+04 +1e+05 +0 +50000 +0e+00 +5e+04 +1e+05 +0 +40000 +80000 +0 +50000 +3750 +Iteration +Value +nltcs | GenCoupon | Ours +κ +ρDOB_DAY +ρDOB_MONTH +ρDOB_YEAR +ρREGOFF +ρSEX +θ0, DOB_DAY +θ0, DOB_MONTH +θ0, DOB_YEAR +θ0, REGOFF +θ0, SEX +E +m +−2 +−1 +0 +1 +2 +Geweke diagnostic +Variable +E +m +θ0, SEX +θ0, REGOFF +θ0, DOB_YEAR +θ0, DOB_MONTH +θ0, DOB_DAY +ρSEX +ρREGOFF +ρDOB_YEAR +ρDOB_MONTH +ρDOB_DAY +κ +100000 +125000 +150000 +175000 +200000 +3250 +5000 +0.000 +0.005 +0.010 +0.01 +0.02 +0.00 +0.01 +0.00 +0.01 +0.02 +0.000 +0.005 +0.010 +0 +50000 +0 +30000 +0 +50000 +0e+00 +5e+04 +1e+05 +0 +50000 +400 +Iteration +Value +48 + +nltcs | Coupon | Ours +ρDOB_DAY +ρDOB_MONTH +ρDOB_YEAR +ρREGOFF +ρSEX +θ0, DOB_DAY +θ0, DOB_MONTH +θ0, DOB_YEAR +θ0, REGOFF +θ0, SEX +E +−2 +−1 +0 +1 +2 +Geweke diagnostic +Variable +E +θ0, SEX +θ0, REGOFF +θ0, DOB_YEAR +θ0, DOB_MONTH +θ0, DOB_DAY +ρSEX +ρREGOFF +ρDOB_YEAR +ρDOB_MONTH +ρDOB_DAY +100000 +125000 +150000 +175000 +200000 +3300 +3350 +0.000 +0.004 +0.00 +0.01 +0.000 +0.005 +0.010 +0.000 +0.005 +0.010 +0.000 +0.005 +0.010 +0e+00 +5e+04 +1e+05 +0e+00 +5e+04 +1e+05 +0e+00 +5e+04 +1e+05 +0 +50000 +0 +50000 +Iteration +Value +nltcs | PY | blink +α +θ0, DOB_DAY +θ0, DOB_MONTH +θ0, DOB_YEAR +θ0, REGOFF +θ0, SEX +d +E +−2 +−1 +0 +1 +2 +Geweke diagnostic +Variable +E +d +θ0, SEX +θ0, REGOFF +θ0, DOB_YEAR +θ0, DOB_MONTH +θ0, DOB_DAY +α +100000 +125000 +150000 +175000 +200000 +3300 +3350 +3400 +0.00 +0.01 +0.02 +0.03 +0.00 +0.01 +0.00 +0.01 +0.02 +0.000 +0.005 +0.010 +0.000 +0.005 +0.000 +0.005 +3750 +Iteration +Value +nltcs | Ewens | blink +α +θ0, DOB_DAY +θ0, DOB_MONTH +θ0, DOB_YEAR +θ0, REGOFF +θ0, SEX +E +−2 +−1 +0 +1 +2 +Geweke diagnostic +Variable +E +θ0, SEX +θ0, REGOFF +θ0, DOB_YEAR +θ0, DOB_MONTH +θ0, DOB_DAY +α +100000 +125000 +150000 +175000 +200000 +3350 +3400 +0.00 +0.01 +0.00 +0.01 +0.02 +0.000 +0.005 +0.000 +0.005 +0.010 +0.000 +0.005 +3750 +4250 +Iteration +Value +49 + +nltcs | GenCoupon | blink +κ +θ0, DOB_DAY +θ0, DOB_MONTH +θ0, DOB_YEAR +θ0, REGOFF +θ0, SEX +E +m +−2 +−1 +0 +1 +2 +Geweke diagnostic +Variable +E +m +θ0, SEX +θ0, REGOFF +θ0, DOB_YEAR +θ0, DOB_MONTH +θ0, DOB_DAY +κ +100000 +125000 +150000 +175000 +200000 +3200 +3300 +4800 +5200 +0.00 +0.01 +0.02 +0.00 +0.01 +0.02 +0.03 +0.00 +0.01 +0.02 +0.00 +0.02 +0.00 +0.01 +0.02 +0 +500 +1000 +Iteration +Value +nltcs | Coupon | blink +θ0, DOB_DAY +θ0, DOB_MONTH +θ0, DOB_YEAR +θ0, REGOFF +θ0, SEX +E +−2 +−1 +0 +1 +2 +Geweke diagnostic +Variable +E +θ0, SEX +θ0, REGOFF +θ0, DOB_YEAR +θ0, DOB_MONTH +θ0, DOB_DAY +100000 +125000 +150000 +175000 +200000 +3275 +3325 +0.00 +0.01 +0.02 +0.00 +0.01 +0.02 +0.000 +0.005 +0.010 +0.00 +0.01 +0.02 +0.000 +0.005 +0.010 +Iteration +Value +RLdata | PY | Ours +α +ρbd +ρbm +ρby +ρfname_c1 +ρlname_c1 +θ0, bd +θ0, bm +θ0, by +θ0, fname_c1 +θ0, lname_c1 +d +E +−2 +−1 +0 +1 +2 +Geweke diagnostic +Variable +E +d +θ0, lname_c1 +θ0, fname_c1 +θ0, by +θ0, bm +θ0, bd +ρlname_c1 +ρfname_c1 +ρby +ρbm +ρbd +α +100000 +125000 +150000 +175000 +200000 +8950 +0.5 +0.6 +0.036 +0.044 +0.045 +0.055 +0.065 +0.12 +0.06 +0.09 +0.075 +0.100 +0.125 +0.002 +0.003 +0.004 +0 +50000 +0.00010 +0.00015 +0.00020 +4e−05 +7e−05 +4e−05 +8e−05 +9000 +12000 +15000 +Iteration +Value +50 + +RLdata | Ewens | Ours +α +ρbd +ρbm +ρby +ρfname_c1 +ρlname_c1 +θ0, bd +θ0, bm +θ0, by +θ0, fname_c1 +θ0, lname_c1 +E +−2 +−1 +0 +1 +2 +Geweke diagnostic +Variable +E +θ0, lname_c1 +θ0, fname_c1 +θ0, by +θ0, bm +θ0, bd +ρlname_c1 +ρfname_c1 +ρby +ρbm +ρbd +α +100000 +125000 +150000 +175000 +200000 +8900 +8950 +0.035 +0.040 +0.045 +0.050 +0.045 +0.055 +0.065 +0.12 +0.16 +0.09 +0.10 +0.14 +0.002 +0.003 +0.004 +0.005 +0 +40000 +80000 +120000 +0.000075 +0.000125 +3e−05 +6e−05 +2e−05 +6e−05 +27000 +30000 +Iteration +Value +RLdata | GenCoupon | Ours +κ +ρbd +ρbm +ρby +ρfname_c1 +ρlname_c1 +θ0, bd +θ0, bm +θ0, by +θ0, fname_c1 +θ0, lname_c1 +E +m +−2 +−1 +0 +1 +2 +Geweke diagnostic +Variable +E +m +θ0, lname_c1 +θ0, fname_c1 +θ0, by +θ0, bm +θ0, bd +ρlname_c1 +ρfname_c1 +ρby +ρbm +ρbd +κ +100000 +125000 +150000 +175000 +200000 +8940 +8980 +40000 +45000 +50000 +0.035 +0.040 +0.045 +0.050 +0.055 +0.09 +0.12 +0.06 +0.08 +0.10 +0.09 +0.12 +0.003 +0.004 +0 +40000 +80000 +0.00012 +0.00016 +5e−05 +4e−05 +8e−05 +0 +250 +Iteration +Value +51 + +RLdata | Coupon | Ours +ρbd +ρbm +ρby +ρfname_c1 +ρlname_c1 +θ0, bd +θ0, bm +θ0, by +θ0, fname_c1 +θ0, lname_c1 +E +−2 +0 +2 +Geweke diagnostic +Variable +E +θ0, lname_c1 +θ0, fname_c1 +θ0, by +θ0, bm +θ0, bd +ρlname_c1 +ρfname_c1 +ρby +ρbm +ρbd +100000 +125000 +150000 +175000 +200000 +7700 +7750 +7800 +0.04 +0.05 +0.05 +0.06 +0.50 +0.55 +0.45 +0.55 +0.65 +0.45 +0.50 +0.55 +0.0025 +0e+00 +5e+04 +1e+05 +2e−05 +3e−05 +4e−05 +4.0e−06 +8.0e−06 +1.2e−05 +1.0e−05 +1.5e−05 +Iteration +Value +RLdata | PY | blink +α +θ0, bd +θ0, bm +θ0, by +θ0, fname_c1 +θ0, lname_c1 +d +E +−2 +−1 +0 +1 +2 +Geweke diagnostic +Variable +E +d +θ0, lname_c1 +θ0, fname_c1 +θ0, by +θ0, bm +θ0, bd +α +100000 +125000 +150000 +175000 +200000 +6200 +6300 +6400 +6500 +0.000 +0.025 +0.050 +0.990 +0.995 +1.000 +0.99 +1.00 +0.525 +0.550 +0.44 +0.48 +0.52 +0.475 +0.500 +0.525 +6750 +7250 +Iteration +Value +RLdata | Ewens | blink +α +θ0, bd +θ0, bm +θ0, by +θ0, fname_c1 +θ0, lname_c1 +E +−2 +−1 +0 +1 +2 +Geweke diagnostic +Variable +E +θ0, lname_c1 +θ0, fname_c1 +θ0, by +θ0, bm +θ0, bd +α +100000 +125000 +150000 +175000 +200000 +6300 +0.99 +1.00 +0.99 +1.00 +0.54 +0.475 +0.500 +0.525 +0.51 +0.54 +7000 +Iteration +Value +52 + +RLdata | GenCoupon | blink +κ +θ0, bd +θ0, bm +θ0, by +θ0, fname_c1 +θ0, lname_c1 +E +m +−2 +−1 +0 +1 +2 +Geweke diagnostic +Variable +E +m +θ0, lname_c1 +θ0, fname_c1 +θ0, by +θ0, bm +θ0, bd +κ +100000 +125000 +150000 +175000 +200000 +6200 +6300 +6400 +6500 +10000 +11000 +0.985 +0.990 +0.995 +1.000 +0.98 +0.99 +1.00 +0.52 +0.56 +0.44 +0.46 +0.48 +0.50 +0.48 +0.50 +0.52 +0 +400 +Iteration +Value +RLdata | Coupon | blink +θ0, bd +θ0, bm +θ0, by +θ0, fname_c1 +θ0, lname_c1 +E +−2 +−1 +0 +1 +2 +Geweke diagnostic +Variable +E +θ0, lname_c1 +θ0, fname_c1 +θ0, by +θ0, bm +θ0, bd +100000 +125000 +150000 +175000 +200000 +6250 +6350 +0.98 +0.99 +1.00 +0.985 +0.990 +0.995 +0.500 +0.525 +0.550 +0.44 +0.48 +0.48 +0.50 +0.52 +Iteration +Value +cora | PY | Ours +α +ρauthors +ρtitle +ρvenue +ρyear +θ0, authors +θ0, title +θ0, venue +θ0, year +d +E +−15 +−10 +−5 +0 +5 +Geweke diagnostic +Variable +E +d +θ0, year +θ0, venue +θ0, title +θ0, authors +ρyear +ρvenue +ρtitle +ρauthors +α +150000 +175000 +200000 +225000 +250000 +165 +175 +185 +0.00 +0.05 +0.10 +0.15 +0.025 +0.050 +0.96 +0.98 +0.12 +0.15 +0.18 +0.70 +0.75 +0.80 +0.0025 +1.5 +2.0 +2.5 +0 +50000 +0.6 +1.0 +50 +70 +Iteration +Value +53 + +cora | Ewens | Ours +α +ρauthors +ρtitle +ρvenue +ρyear +θ0, authors +θ0, title +θ0, venue +θ0, year +E +−5.0 +−2.5 +0.0 +2.5 +5.0 +Geweke diagnostic +Variable +E +θ0, year +θ0, venue +θ0, title +θ0, authors +ρyear +ρvenue +ρtitle +ρauthors +α +150000 +175000 +200000 +225000 +250000 +165 +175 +185 +0.025 +0.050 +0.075 +0.96 +0.98 +0.15 +0.70 +0.75 +0.80 +0.001 +0.002 +0.003 +1.5 +2.0 +2.5 +0 +50000 +1 +40 +60 +Iteration +Value +cora | GenCoupon | Ours +κ +ρauthors +ρtitle +ρvenue +ρyear +θ0, authors +θ0, title +θ0, venue +θ0, year +E +m +−2 +0 +2 +Geweke diagnostic +Variable +E +m +θ0, year +θ0, venue +θ0, title +θ0, authors +ρyear +ρvenue +ρtitle +ρauthors +κ +150000 +175000 +200000 +225000 +250000 +170 +180 +200 +300 +400 +500 +0.04 +0.06 +0.08 +0.96 +0.98 +0.15 +0.68 +0.72 +0.76 +0.002 +1.5 +2.0 +2.5 +0 +40000 +80000 +1 +0.5 +1.0 +Iteration +Value +cora | Coupon | Ours +ρauthors +ρtitle +ρvenue +ρyear +θ0, authors +θ0, title +θ0, venue +θ0, year +E +−3 +−2 +−1 +0 +1 +2 +Geweke diagnostic +Variable +E +θ0, year +θ0, venue +θ0, title +θ0, authors +ρyear +ρvenue +ρtitle +ρauthors +150000 +175000 +200000 +225000 +250000 +390 +0.025 +0.050 +0.35 +0.40 +0.08 +0.12 +0.3 +0.4 +0 +50000 +0.5 +1.5 +2.5 +0 +50000 +0.0015 +0.0020 +0.0025 +Iteration +Value +54 + +cora | PY | blink +α +θ0, authors +θ0, title +θ0, venue +θ0, year +d +E +−2 +−1 +0 +1 +2 +Geweke diagnostic +Variable +E +d +θ0, year +θ0, venue +θ0, title +θ0, authors +α +150000 +175000 +200000 +225000 +250000 +175 +180 +185 +0.00 +0.05 +0.10 +0.8 +0.9 +1.0 +0.97 +0.98 +0.99 +1.00 +0.90 +0.95 +1.00 +0.98 +0.99 +1.00 +40 +60 +Iteration +Value +cora | Ewens | blink +α +θ0, authors +θ0, title +θ0, venue +θ0, year +E +−2 +−1 +0 +1 +2 +Geweke diagnostic +Variable +E +θ0, year +θ0, venue +θ0, title +θ0, authors +α +150000 +175000 +200000 +225000 +250000 +175 +180 +185 +190 +0.8 +0.9 +0.98 +0.99 +1.00 +0.92 +0.96 +1.00 +0.98 +0.99 +1.00 +60 +Iteration +Value +cora | GenCoupon | blink +κ +θ0, authors +θ0, title +θ0, venue +θ0, year +E +m +−3 +−2 +−1 +0 +1 +2 +Geweke diagnostic +Variable +E +m +θ0, year +θ0, venue +θ0, title +θ0, authors +κ +150000 +175000 +200000 +225000 +250000 +180 +185 +200 +250 +300 +0.8 +0.9 +1.0 +0.98 +0.99 +1.00 +0.92 +0.96 +1.00 +0.98 +0.99 +1.00 +1 +2 +Iteration +Value +cora | Coupon | blink +θ0, authors +θ0, title +θ0, venue +θ0, year +E +−2 +−1 +0 +1 +2 +Geweke diagnostic +Variable +E +θ0, year +θ0, venue +θ0, title +θ0, authors +150000 +175000 +200000 +225000 +250000 +280 +0.8 +0.9 +1.0 +0.97 +0.98 +0.99 +1.00 +0.92 +0.96 +1.00 +0.98 +0.99 +1.00 +Iteration +Value +55 + +rest | PY | Ours +α +ρaddr +ρcity +ρname +ρtype +θ0, addr +θ0, city +θ0, name +θ0, type +d +E +−2 +−1 +0 +1 +2 +Geweke diagnostic +Variable +E +d +θ0, type +θ0, name +θ0, city +θ0, addr +ρtype +ρname +ρcity +ρaddr +α +100000 +125000 +150000 +175000 +200000 +740 +750 +760 +770 +0.4 +0.7 +0.8 +0.3 +0.4 +0.5 +0.85 +0.90 +0.95 +0.4 +0.5 +0.6 +0.00010 +0.00015 +0.00020 +0.00025 +0e+00 +5e+04 +1e+05 +5e−05 +1e−04 +0e+00 +5e+04 +1e+05 +500 +1500 +Iteration +Value +rest | Ewens | Ours +α +ρaddr +ρcity +ρname +ρtype +θ0, addr +θ0, city +θ0, name +θ0, type +E +−2 +0 +2 +Geweke diagnostic +Variable +E +θ0, type +θ0, name +θ0, city +θ0, addr +ρtype +ρname +ρcity +ρaddr +α +100000 +125000 +150000 +175000 +200000 +740 +750 +760 +0.7 +0.8 +0.3 +0.4 +0.5 +0.85 +0.90 +0.4 +0.5 +0.6 +0.00015 +0.00020 +0.00025 +0e+00 +5e+04 +1e+05 +5e−05 +1e−04 +0 +50000 +1500 +2000 +2500 +Iteration +Value +rest | GenCoupon | Ours +κ +ρaddr +ρcity +ρname +ρtype +θ0, addr +θ0, city +θ0, name +θ0, type +E +m +−2 +−1 +0 +1 +2 +Geweke diagnostic +Variable +E +m +θ0, type +θ0, name +θ0, city +θ0, addr +ρtype +ρname +ρcity +ρaddr +κ +100000 +125000 +150000 +175000 +200000 +740 +750 +760 +2500 +3500 +4500 +0.6 +0.7 +0.8 +0.40 +0.55 +0.85 +0.90 +0.4 +0.5 +0.6 +2e−04 +0e+00 +5e+04 +1e+05 +5e−05 +1e−04 +0e+00 +5e+04 +1e+05 +0 +400 +Iteration +Value +56 + +rest | Coupon | Ours +ρaddr +ρcity +ρname +ρtype +θ0, addr +θ0, city +θ0, name +θ0, type +E +−2 +−1 +0 +1 +2 +Geweke diagnostic +Variable +E +θ0, type +θ0, name +θ0, city +θ0, addr +ρtype +ρname +ρcity +ρaddr +100000 +125000 +150000 +175000 +200000 +710 +720 +0.6 +0.7 +0.3 +0.4 +0.5 +0.6 +0.875 +0.925 +0.5 +0.6 +0.7 +1e−04 +2e−04 +3e−04 +0e+00 +5e+04 +1e+05 +0.00010 +0.00015 +0e+00 +5e+04 +1e+05 +Iteration +Value +rest | PY | blink +α +θ0, addr +θ0, city +θ0, name +θ0, type +d +E +−2 +−1 +0 +1 +2 +Geweke diagnostic +Variable +E +d +θ0, type +θ0, name +θ0, city +θ0, addr +α +100000 +125000 +150000 +175000 +200000 +740 +0.0 +0.4 +0.4 +0.5 +0.6 +0.90 +0.95 +1.00 +0.2 +0.3 +0.94 +0.96 +0.98 +1.00 +1000 +2000 +Iteration +Value +rest | Ewens | blink +α +θ0, addr +θ0, city +θ0, name +θ0, type +E +−2 +−1 +0 +1 +2 +Geweke diagnostic +Variable +E +θ0, type +θ0, name +θ0, city +θ0, addr +α +100000 +125000 +150000 +175000 +200000 +740 +0.4 +0.5 +0.6 +0.7 +0.90 +0.95 +1.00 +0.2 +0.3 +0.950 +0.975 +1.000 +1500 +2000 +2500 +Iteration +Value +rest | GenCoupon | blink +κ +θ0, addr +θ0, city +θ0, name +θ0, type +E +m +−2 +−1 +0 +1 +2 +Geweke diagnostic +Variable +E +m +θ0, type +θ0, name +θ0, city +θ0, addr +κ +100000 +125000 +150000 +175000 +200000 +740 +750 +2000 +3000 +4000 +0.4 +0.5 +0.6 +0.7 +0.90 +0.95 +1.00 +0.15 +0.25 +0.35 +0.950 +0.975 +1.000 +0 +300 +Iteration +Value +57 + +rest | Coupon | blink +θ0, addr +θ0, city +θ0, name +θ0, type +E +−2 +−1 +0 +1 +2 +Geweke diagnostic +Variable +E +θ0, type +θ0, name +θ0, city +θ0, addr +100000 +125000 +150000 +175000 +200000 +700 +710 +720 +0.4 +0.5 +0.6 +0.90 +0.95 +1.00 +0.25 +0.35 +0.950 +0.975 +Iteration +Value +G.2 +Comparison with Baseline Models +Here we present convergence diagnostics for the models fitted in Section 5.4. We present Geweke +diagnostic plots and trace plots for a selection of model variables for each data set and model. Each +pair of plots is preceded by a title of the form “Data set | Model”. The Geweke diagnostic plot (on +the left) depicts a Z-score on the x-axis for each variable on the y-axis. The Z-score tests for equality +of the means of the first 10% and final 50% of the Markov chain, and is typically expected to be +in the range [−2, 2] (Geweke, 1992). The trace plot (on the right) depicts the value of variables +(labeled in the right panel) for each step in the chain (on the x-axis). Note that variable 𝐸 denotes +the number of instantiated entities. We replace integer indices for the attributes by named indices. +For instance, 𝜃0,city refers to the distortion probability in source 0 of the attribute called “city”. +nltcs | blink +θ0, DOB_DAY +θ0, DOB_MONTH +θ0, DOB_YEAR +θ0, REGOFF +θ0, SEX +E +−2 +−1 +0 +1 +2 +Geweke diagnostic +Variable +E +θ0, SEX +θ0, REGOFF +θ0, DOB_YEAR +θ0, DOB_MONTH +θ0, DOB_DAY +100000 +125000 +150000 +175000 +200000 +3300 +0.005 +0.010 +0.015 +0.01 +0.02 +0.004 +0.008 +0.012 +0.01 +0.02 +0.005 +0.010 +Iteration +Value +RLdata | blink +θ0, bd +θ0, bm +θ0, by +θ0, fname_c1 +θ0, lname_c1 +E +−2 +−1 +0 +1 +2 +Geweke diagnostic +Variable +E +θ0, lname_c1 +θ0, fname_c1 +θ0, by +θ0, bm +θ0, bd +100000 +125000 +150000 +175000 +200000 +7350 +7450 +0.34 +0.38 +0.42 +0.30 +0.35 +0.300 +0.325 +0.350 +0.250 +0.275 +0.28 +0.30 +0.32 +Iteration +Value +58 + +cora | blink +θ0, authors +θ0, title +θ0, venue +θ0, year +E +−2 +−1 +0 +1 +2 +Geweke diagnostic +Variable +E +θ0, year +θ0, venue +θ0, title +θ0, authors +100000 +125000 +150000 +175000 +200000 +300 +0.15 +0.25 +0.68 +0.76 +0.4 +0.5 +0.6 +0.75 +0.80 +Iteration +Value +rest | blink +θ0, addr +θ0, city +θ0, name +θ0, type +E +−2 +−1 +0 +1 +2 +Geweke diagnostic +Variable +E +θ0, type +θ0, name +θ0, city +θ0, addr +100000 +125000 +150000 +175000 +200000 +710 +720 +730 +0.35 +0.45 +0.4 +0.5 +0.6 +0.10 +0.15 +0.20 +0.6 +0.7 +Iteration +Value +nltcs | Ours +κ +ρDOB_DAY +ρDOB_MONTH +ρDOB_YEAR +ρREGOFF +ρSEX +θ0, DOB_DAY +θ0, DOB_MONTH +θ0, DOB_YEAR +θ0, REGOFF +θ0, SEX +E +m +−2 +−1 +0 +1 +2 +Geweke diagnostic +Variable +E +m +θ0, SEX +θ0, REGOFF +θ0, DOB_YEAR +θ0, DOB_MONTH +θ0, DOB_DAY +ρSEX +ρREGOFF +ρDOB_YEAR +ρDOB_MONTH +ρDOB_DAY +κ +100000 +125000 +150000 +175000 +200000 +3250 +5000 +0.000 +0.005 +0.010 +0.01 +0.02 +0.00 +0.01 +0.00 +0.01 +0.02 +0.000 +0.005 +0.010 +0 +50000 +0 +30000 +0 +50000 +0e+00 +5e+04 +1e+05 +0 +50000 +400 +Iteration +Value +59 + +RLdata | Ours +κ +ρbd +ρbm +ρby +ρfname_c1 +ρlname_c1 +θ0, bd +θ0, bm +θ0, by +θ0, fname_c1 +θ0, lname_c1 +E +m +−2 +−1 +0 +1 +2 +Geweke diagnostic +Variable +E +m +θ0, lname_c1 +θ0, fname_c1 +θ0, by +θ0, bm +θ0, bd +ρlname_c1 +ρfname_c1 +ρby +ρbm +ρbd +κ +100000 +125000 +150000 +175000 +200000 +8940 +8980 +40000 +45000 +50000 +0.035 +0.040 +0.045 +0.050 +0.055 +0.09 +0.12 +0.06 +0.08 +0.10 +0.09 +0.12 +0.003 +0.004 +0 +40000 +80000 +0.00012 +0.00016 +5e−05 +4e−05 +8e−05 +0 +250 +Iteration +Value +cora | Ours +κ +ρauthors +ρtitle +ρvenue +ρyear +θ0, authors +θ0, title +θ0, venue +θ0, year +E +m +−2 +0 +2 +Geweke diagnostic +Variable +E +m +θ0, year +θ0, venue +θ0, title +θ0, authors +ρyear +ρvenue +ρtitle +ρauthors +κ +150000 +175000 +200000 +225000 +250000 +170 +180 +200 +300 +400 +500 +0.04 +0.06 +0.08 +0.96 +0.98 +0.15 +0.68 +0.72 +0.76 +0.002 +1.5 +2.0 +2.5 +0 +40000 +80000 +1 +0.5 +1.0 +Iteration +Value +60 + +rest | Ours +κ +ρaddr +ρcity +ρname +ρtype +θ0, addr +θ0, city +θ0, name +θ0, type +E +m +−2 +−1 +0 +1 +2 +Geweke diagnostic +Variable +E +m +θ0, type +θ0, name +θ0, city +θ0, addr +ρtype +ρname +ρcity +ρaddr +κ +100000 +125000 +150000 +175000 +200000 +740 +750 +760 +2500 +3500 +4500 +0.6 +0.7 +0.8 +0.40 +0.55 +0.85 +0.90 +0.4 +0.5 +0.6 +2e−04 +0e+00 +5e+04 +1e+05 +5e−05 +1e−04 +0e+00 +5e+04 +1e+05 +0 +400 +Iteration +Value +nltcs | Sadinle +E +mDOB_DAY, 1 +mDOB_MONTH, 1 +mDOB_YEAR, 1 +mREGOFF, 1 +mSEX, 1 +uDOB_DAY, 1 +uDOB_MONTH, 1 +uDOB_YEAR, 1 +uREGOFF, 1 +uSEX, 1 +−2 +−1 +0 +1 +2 +Geweke diagnostic +Variable +E +uSEX, 1 +uREGOFF, 1 +uDOB_YEAR, 1 +uDOB_MONTH, 1 +uDOB_DAY, 1 +mSEX, 1 +mREGOFF, 1 +mDOB_YEAR, 1 +mDOB_MONTH, 1 +mDOB_DAY, 1 +100000 +125000 +150000 +175000 +200000 +2040 +2080 +2120 +0.970 +0.971 +0.900 +0.902 +0.2575 +0.2600 +0.263 +0.266 +0.269 +0.6350 +0.6375 +0.6400 +0.9500 +0.9501 +0.9502 +0.9500 +0.9501 +0.998 +0.999 +1.000 +0.9990 +0.9995 +1.0000 +0.95000 +0.95005 +Iteration +Value +61 + +RLdata | Sadinle +E +mbd, 1 +mbm, 1 +mby, 1 +mfname_c1, 1 +mfname_c1, 2 +mfname_c1, 3 +mfname_c1, 4 +mlname_c1, 1 +mlname_c1, 2 +mlname_c1, 3 +mlname_c1, 4 +ubd, 1 +ubm, 1 +uby, 1 +ufname_c1, 1 +ufname_c1, 2 +ufname_c1, 3 +ufname_c1, 4 +ulname_c1, 1 +ulname_c1, 2 +ulname_c1, 3 +ulname_c1, 4 +−2 +−1 +0 +1 +2 +Geweke diagnostic +Variable +E +ulname_c1, 4 +ulname_c1, 3 +ulname_c1, 2 +ulname_c1, 1 +ufname_c1, 4 +ufname_c1, 3 +ufname_c1, 2 +ufname_c1, 1 +uby, 1 +ubm, 1 +ubd, 1 +mlname_c1, 4 +mlname_c1, 3 +mlname_c1, 2 +mlname_c1, 1 +mfname_c1, 4 +mfname_c1, 3 +mfname_c1, 2 +mfname_c1, 1 +mby, 1 +mbm, 1 +mbd, 1 +100000 +125000 +150000 +175000 +200000 +5625 +5675 +0.126 +0.127 +0.128 +0.0082 +0.0086 +0.0076 +0.0080 +0.01575 +0.01625 +0.0 +0.5 +1.0 +0.99996 +0.99998 +1.00000 +0.39 +0.689 +0.691 +0.01225 +0.081 +0.082 +0.083 +0.0315 +0.0320 +0.0325 +0.0330 +0.950 +0.975 +1.000 +0.950 +0.975 +1.000 +0.99 +1.00 +0.9500 +0.9505 +0.9510 +0.950 +0.975 +1.000 +0.950 +0.975 +1.000 +0.99 +1.00 +0.9500 +0.9504 +0.9508 +0.9512 +0.9500 +0.9502 +0.9504 +0.9500 +0.9504 +0.9500 +0.9505 +Iteration +Value +62 + +cora | Sadinle +E 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+page_content=' Marchant and Benjamin I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Rubinstein School of Computing and Information Systems University of Melbourne Rebecca C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Steorts Departments of Statistical Science and Computer Science Duke University Abstract Entity resolution (record linkage or de-duplication) is the process of identifying and linking duplicate records in databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' In this paper, we propose a Bayesian graphical approach for entity resolution that links records to latent entities, where the prior representation on the linkage structure is exchangeable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' First, we adopt a flexible and tractable set of priors for the linkage structure, which corresponds to a special class of random partition models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Second, we propose a more realistic distortion model for categorical/discrete record attributes, which corrects a logical inconsistency with the standard hit-miss model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Third, we incorporate hyperpriors to improve flexibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Fourth, we employ a partially collapsed Gibbs sampler for inferential speedups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Using a selection of private and non-private data sets, we investigate the impact of our modeling contributions and compare our model with two alternative Bayesian models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' In addition, we conduct a simulation study for household survey data, where we vary distortion, duplication rates and data set size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We find that our model performs more consistently than the alternatives across a variety of scenarios and typically achieves the highest entity resolution accuracy (F1 score).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Open source software is available for our proposed methodology, and we provide a discussion regarding our work and future directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Statement of Significance In survey statistics, entity resolution (record linkage) is used to identify responses submitted by the same individual across multiple surveys, even when unique identifiers such as social ∗This is a pre-copyedited, author-produced version of an article accepted for publication in the Journal of Survey Statistics and Methodology following peer review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' The version of record Marchant, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=', Rubinstein, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=', and Steorts, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' (2023), “Bayesian Graphical Entity Resolution using Exchangeable Random Partition Priors,” Journal of Survey Statistics and Methodology is available online at: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='1093/jssam/smac030.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='02962v1 [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='ME] 8 Jan 2023 security numbers are not recorded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' This paper advances Bayesian methods for entity resolution by: (i) thoroughly evaluating a general class of priors on links between responses and individuals, and (ii) proposing a more realistic model for distortions that appear in individuals’ identifying attributes (name, address, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Both of these contributions are evaluated independently and jointly on a variety of data sets, one of which is a longitudinal medical survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' The results show that the general class of priors – the Ewens-Pitman random partitions – achieve similar accuracy in three previously studied parameter regimes, so long as vague hyperpriors are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' This is an important insight, as it addresses questions in the literature about which parameter regime (if any) is most suitable for entity resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' In addition, the proposed distortion model is found to significantly improve the accuracy of entity resolution, particularly for string-type attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We provide two simulation studies supporting our work and comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' The paper is complemented by an R package that implements the proposed model using an optimized partially collapsed Gibbs sampler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Keywords: entity resolution, record linkage, Bayesian models, exchangeability, random partitions, Ewens-Pitman random partitions 1 Introduction As commonly known in the literature, entity resolution (ER;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' record linkage or de-duplication) is the process of taking large, noisy (dirty) databases and removing duplicate records (often in the absence of a unique identifier) (Doan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=', 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Elmagarmid et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=', 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Naumann and Herschel, 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Getoor and Machanavajjhala, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Christen, 2012a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Christophides et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Ilyas and Chu, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Papadakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Binette and Steorts, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' This problem has become increasingly important in many fields, such as survey methodology, official statistics, computer science, political science, health care, human rights, and others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' In this paper, we are motivated by several applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' For example, we consider a longitudinal health care survey, where information is categorical in nature due to privacy restrictions on the data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' This may be of interest to those in the survey methodology community as they may face similar issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' In addition, we consider categorical and string or textual based data sets (or surveys) such as bibliographic/citation documents, information from restaurants, and a traditional benchmark (synthetic) study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' The goal of analyzing multiple data sets is to make the survey community more aware of data sets that are relevant for entity resolution methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Other overarching goals are to extend recent Bayesian graphical ER methodology for these data sets, provide comparative analyses, simulation studies, and guidance to researchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Moreover, we provide open-source software for the community for our proposed method and two recently proposed methods in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' The idea of entity resolution dates back to Dunn (1946), who envisioned a “book of life” that would piece together information about an individual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Newcombe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' (1959) proposed one of the first methods for performing ER, based on a heuristic statistical test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' This method was later formalized by Fellegi and Sunter (1969), who developed a model based on agreement patterns between pairs of records, and a likelihood ratio test for classifying pairs as linked (referring to the same entity), possibly linked or non-linked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Under some strong assumptions, they showed that their method – now known as the Fellegi-Sunter (FS) method – is statistically optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' The FS method has been advanced over many decades in the ER literature, especially in survey methodology due to 2 its scalability, ease of use and simplicity (Winkler, 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Christen, 2012a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Sadinle and Fienberg, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Enamorado et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' However, it has some inherent limitations: it makes inconsistent (intransitive) predictions, it does not naturally account for uncertainty, it cannot exploit patterns at the entity-level, and it is incompatible with generative modeling approaches (Tancredi and Liseo, 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Some of these limitations can be addressed by adapting the FS method to a Bayesian setting, while also imposing consistency constraints on the links between records (Sadinle, 2014, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' For example, Sadinle (2014) proposed a Bayesian extension of the FS model for performing ER within a single database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' It incorporates consistency (transitivity) constraints by requiring that records are partitioned into groups that are mutually linked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' In addition, the model supports multiple levels of agreement, and incorporates priors on the links and 𝑚/𝑢 probabilities (from the FS model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' In contrast with traditional FS methods, quantification of ER uncertainty is possible by computing the posterior distribution on the links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' However, despite these benefits, the model has not been widely examined in the literature, perhaps in part due to the lack of a publicly-available implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' One of our goals in this paper is to evaluate Sadinle’s model as a representative example of a Bayesian FS model, and compare it to another class of Bayesian entity resolution models, which we now describe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' In parallel to developments in Bayesian FS models, others have proposed a new class of generative models called Bayesian graphical entity resolution models (Tancredi and Liseo, 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Steorts, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Steorts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' In contrast with FS methods, these models do not operate on agreement patterns between pairs of records.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Instead, they model a latent population of entities and the process by which records are generated from entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' They are known as “graphical” models because the fundamental objects in the model form a bipartite graph – the latent entities correspond to one set of vertices, the records correspond to another set of vertices, and the links are edges that connect the vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Tancredi and Liseo (2011) proposed one of the first models of this kind for performing ER across two databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Subsequently, Steorts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' (2016) proposed an extension to multiple databases, while optionally allowing for duplicates within each database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' However, Steorts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' (2016) discovered a limitation of the their model and the model by Tancredi and Liseo (2011) – the uniform prior on the linkage structure is highly informative about the number of entities present in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' They noted that future work ought to consider more appropriate priors on the linkage structure for ER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' The models by Tancredi and Liseo (2011) and Steorts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' (2016) both assume entities are described by a set of latent categorical attributes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=', date of birth, gender) which are distorted in the records.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' However, their model of the distortion process is simple and is unable to capture realistic distortions in string-type attributes, such as names.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Steorts (2015) addressed this problem by proposing a string pseudo-likelihood and an empirically-motivated prior in a model known as blink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' The blink model was later used as a foundation by Marchant et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' (2021) for developing more scalable Bayesian graphical ER techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' They proposed an end-to-end method that jointly performs blocking and ER, where inference can be distributed or parallelized at the block level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Importantly, this enables propagation of blocking uncertainty to the ER task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' They observed a 200× speed-up, which allowed them to scale blink to a data set containing over one million records.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' However, the blink model uses the same uniform prior on the linkage structure as Tancredi and Liseo (2011) and Steorts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 3 (2016), and suffers from the same limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Motivated by the shortcomings of existing Bayesian graphical ER models, we propose and evaluate several modeling refinements in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' First, we propose a flexible and tractable set of priors for the linkage structure that are the Ewens-Pitman (EP) family of random partition models (Pitman, 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' These are the most general family of priors that satisfy exchangeability (an elementary requirement) and they are more flexible than the uniform priors used in previous work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Second, we incorporate hyperpriors on the EP parameters to further increase flexibility and reduce the need for tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' This is motivated by the informativeness of the uniform prior used in previous work (Steorts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Third, we propose a more nuanced distortion model for categorical/discrete attributes, which corrects an inconsistency with the standard hit-miss model used by Tancredi and Liseo (2011);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Steorts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' (2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Steorts (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Fourth, we design a partially collapsed Gibbs sampler to fit our model which incorporates computational optimizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We evaluate our modeling contributions independently and jointly on a selection of private and non-private data sets, and compare our model with the Bayesian graphical ER model by Steorts (2015) and the Bayesian FS model by Sadinle (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We also evaluate our model (and the alternatives) in a controlled simulation study, where we generate synthetic household survey data, with varying numbers of records, levels of distortion, and rates of duplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Overall, we find our model is more robust across the various scenarios tested, and it typically achieves superior ER accuracies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We provide open source software for all of the ER methods under evaluation, and we provide a discussion of our contributions and directions for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' The rest of the paper proceeds as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Section 2 provides background on ER and exchangeable random partitions and outlines notation used throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Section 3 outlines our proposed Bayesian graphical ER model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Section 4 presents a partially collapsed Gibbs sampling algorithm for approximating the posterior distribution and other computational speedups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Section 5 presents an empirical study of our proposed distortion model and linkage structure priors and includes a comparison to two recent Bayesian ER models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='5 summarizes a controlled simulation study that is in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Section 6 summarizes our findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 2 Background and Notation In this section, we provide notation, assumptions, and a review of exchangeable random partitions which are used as a prior in our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Figure 2 includes an index of symbols used throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='1 Notation and Assumptions We review notation and assumptions used throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We assume the data (from one or more sources) is structured, meaning that it has been standardized using schema alignment techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' For the purposes of our paper, the data is represented in a tabular format, where rows correspond to records and columns correspond to attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' This in contrast to unstructured entity resolution, which deals with textual descriptions (paragraphs) or images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' For a full review of these terms, see Papadakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 4 Let 𝑠 ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' , 𝑆} be an index over the data sources and 𝑖 ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' , 𝑁} be an index over the records, which is unique across all sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' The source of the 𝑖-th record is denoted by 𝜍𝑖 ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' , 𝑆} and the record’s attribute values are represented as a tuple x𝑖 = (𝑥𝑖1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' , 𝑥𝑖𝐴) indexed by 𝑎 ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' , 𝐴}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Assume 𝑥𝑖𝑎 ∈ D𝑎 for all 𝑖 and 𝑎, where the domain of the 𝑎-th attribute D𝑎 is a finite set of strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Suppose there exists a (possibly infinite) population of entities indexed by 𝑒 ∈ ℕ, which is represented in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Denote the entity referenced in the 𝑖-th record by 𝜆𝑖 ∈ ℕ and define the linkage structure as 𝚲 = (𝜆1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' , 𝜆𝑁).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We consider the most general case where there are no constraints on 𝚲 – i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=', we permit duplicates within sources and arbitrary links across sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' The linkage structure 𝚲 induces a partition of the records into clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We label the clusters according to their associated entities, allowing for empty clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' The size of cluster 𝑒 is denoted by 𝑁𝑒 = � 𝑖 𝟙[𝜆𝑖 = 𝑒] and the number of non-empty clusters is denoted by 𝐸 = � 𝑒 𝟙[𝑁𝑒 > 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We are interested in the fully unsupervised setting where no information is known about the linkage structure or the entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Our goal is to infer the linkage structure based solely on the observed record attributes {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' , x𝑁} and source identifiers {𝜍1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 𝜍𝑁}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Since we are working in a Bayesian setting, we seek a full posterior (not merely a point estimate) over the linkage structure so that uncertainty can be propagated to post-ER tasks, which may include regression, multiple systems estimation, among other examples (Kaplan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Tancredi and Liseo, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Steorts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Tancredi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Sadinle, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' While the post-ER task is not a goal of this paper, the previous references propose recent approaches of such tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='2 Exchangeable Random Partitions The linkage structure is the primary variable of interest for entity resolution, so we pay special attention to it when designing our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Since we are working in a Bayesian setting, we must specify a prior on the linkage structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We previously noted that the linkage structure can be interpreted as a partition of the records into subsets, where the records in each subset correspond to the same entity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' This interpretation is convenient, as we can drawn on related work on random partitions when considering potential priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' In this section, we review a special class of random partitions called the Ewens-Pitman (EP) family (Pitman, 2006, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 62), which we use as a prior on the linkage structure in our model (see Section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Before defining the EP family of random partitions, we define key concepts and notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Consider a set of 𝑁 records, where [𝑁] = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' , 𝑁} denotes the record identifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' A partition of [𝑁] is a collection of disjoint non-empty subsets of [𝑁].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' For example, {1, 2}, {3} and {1}, {2}, {3} are partitions of [𝑁] for 𝑁 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We can equivalently define a partition in terms of the linkage structure 𝚲 = (𝜆𝑖)𝑖=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='𝑁 where 𝜆𝑖 labels the subset (entity) record 𝑖 is assigned to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='1 Using this notation, the above examples for 𝑁 = 3 could be written as 𝚲 = (1, 1, 2) and 𝚲 = (1, 2, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Let P[𝑁] denote the set of all partitions of [𝑁].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' A random partition of [𝑁] is a random variable whose values lie in P[𝑁].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' The EP family are the most general class of random partitions that satisfy 1One can use any labels to identify the entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' All that matters is that 𝜆𝑖 = 𝜆 𝑗 if records 𝑖 and 𝑗 are assigned to the same entity and 𝜆𝑖 ≠ 𝜆 𝑗 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 5 the following two desirable properties: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Exchangeability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' This means the distribution 𝑃𝑁 over the partitions P𝑁 is invariant under permutations of the record identifiers [𝑁].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Or equivalently, the distribution over 𝚲 is exchangeable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' This is a reasonable requirement if the records have no natural ordering – e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=', it is not known whether one record was generated before or after another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' This is a property of the distribution as 𝑁 varies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' It ensures the distribution is not altered when more records are observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' This is desirable because the model can be learned sequentially in a consistent manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We say that the sequence of distributions 𝑃1, 𝑃2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' over P[1], P[2], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' is consistent if the distribution on P[𝑁] induced by 𝑃𝑀 for 𝑀 > 𝑁 is 𝑃𝑁.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Mathematically, this means 𝑃𝑀({𝜌 : Proj(𝜌, [𝑁]) = 𝜌′}) = 𝑃𝑁 (𝜌′), where Proj(𝜌, [𝑁]) := {𝐴𝑒 ∩ [𝑁] : 𝐴𝑒 ∩ [𝑁] ≠ ∅, 𝐴𝑒 ∈ 𝜌} is the projection of a partition 𝜌 = {𝐴𝑒}𝑒=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='𝐸 of [𝑀] onto [𝑁] and 𝜌′ is a partition of [𝑁].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' In fact, the EP family ensures these properties hold in a limiting sense as 𝑁 → ∞ (Pitman, 2006, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 62).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We can develop an intuitive understanding of the EP family by examining how a random partition is generated sequentially, one record at a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Let (𝜌𝑁)𝑁=1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' be a sequence of EP random partitions where 𝜌𝑁 is a random partition of [𝑁].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We begin at step 1 with 𝜌1 = {1} – i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=', a single record assigned to a single entity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' The random partition 𝜌𝑁 at any later step 𝑁 > 1 is generated conditional on the random partition 𝜌𝑁−1 at step 𝑁 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Let 𝐸 be the number of subsets (occupied entities) in 𝜌𝑁−1 and 𝑁𝑒 be the size of subset (entity) 𝑒 in 𝜌𝑁−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Then 𝜌𝑁 is generated by assigning record 𝑁 to: an existing subset (entity) with probability 𝑁𝑒−𝜎 𝑁+𝛼 , or a “new” subset (entity) with probability 𝛼+𝐸𝜎 𝑁+𝛼 , where 𝜎 and 𝛼 are EP parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' This construction is known as a two-parameter Chinese Restaurant Process and is visualized in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' The allowable values of the EP parameters fall into two regimes depending on the sign of 𝜎: 𝜎 < 0 and 𝛼 = −𝑚𝜎 for some 𝑚 ∈ ℕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We refer to this regime as the generalized coupon partitions, since they are closely related to the coupon-collector’s partition (Pitman, 2006, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 46).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' These partitions are generated by sampling with replacement from a finite population of size 𝑚, where the mixing proportions are drawn from a symmetric Dirichlet distribution with concentration parameter −𝜎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' The coupon-collector’s partition is obtained in the limit 𝜎 → −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 0 ≤ 𝜎 ≤ 1 with 𝛼 > −𝜎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' These are called Pitman-Yor partitions after Pitman and Yor (1997), and are generated by sampling with replacement from an infinite population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' The resulting 6 Entity 1 Entity 2 Entity 𝑘 New entity · · · · 𝑁1 − 𝜎 𝑁 + 𝛼 𝑁2 − 𝜎 𝑁 + 𝛼 𝑁𝐸 − 𝜎 𝑁 + 𝛼 𝛼 + 𝐸𝜎 𝑁 + 𝛼 Figure 1: Illustration of a sequential construction of a Ewens-Pitman random partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' At step 𝑁 a record (black circle) is assigned to one of the occupied entities or a new entity (grey circles) conditioned on the assignments of the previous 𝑁 − 1 records.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' The probabilities assigned to the entities are given inside the grey circles and are dependent on the Ewens-Pitman parameters 𝜎 and 𝛼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' partitions demonstrate preferential attachment behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' The special case 𝜎 = 0 corresponds to the Ewens partition (Kingman, 1978).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' To illustrate the varying behavior of the random partitions as a function of 𝜎, we can examine the asymptotic number of subsets in the partition (entities) 𝐸𝑁 as 𝑁 → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Pitman (2006, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 70) shows 𝐸𝑁 𝑎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' ≍ ���� ���� 𝑚, 𝜎 < 0 and 𝛼 = −𝑚𝜎 for 𝑚 ∈ ℕ, 𝛼 log 𝑁, 𝜎 = 0 and 𝛼 > 0, 𝑆𝜎𝑁𝜎, 0 < 𝜎 < 1 and 𝛼 > −𝜎, (1) where 𝑆𝜎 is a strictly positive random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Thus, by varying 𝜎, we can encode a prior belief that the number of entities 𝐸𝑁 is asymptotically constant, logarithmic, or sub-linear in 𝑁.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 3 Graphical Bayesian ER In this section, we propose a generative model for entity resolution that incorporates a latent population of entities, each with a set of unknown attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Our model employs the Ewens-Pitman class of priors on the linkage structure and a modified record distortion model that deviates from the common “hit-miss” model used by Tancredi and Liseo (2011), Steorts (2015) and Steorts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We provide an index of the model’s variables and an illustration of their dependence structure in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We close the section with a discussion of our model’s hyperparameters, including recommendations about how to set these values when limited prior information is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='1 Model Specification Entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We assume each entity 𝑒 ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='} is associated with a tuple of attribute values y𝑒 = (𝑦𝑒1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' , 𝑦𝑒𝐴), drawn independent and identically distributed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=') from an unknown distribution G with support on D = � 𝑎 D𝑎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' To improve tractability, we assume correlations 7 𝑥𝑖𝑎 𝜆𝑖 π 𝜎 𝛼 𝜁 (1) 𝜁 (0) 𝜒(1) 𝜒(0) 𝑧𝑖𝑎 𝜔𝑖𝑎 𝑦𝑒𝑎 𝜍𝑖 𝐺𝑎 𝐻𝑒𝑎 𝜌𝑎 𝜏(1) 𝑎 𝜏(0) 𝑎 ψ𝑎 𝜐𝑎 𝜙𝑎 𝜃𝑠𝑎 𝛽(1) 𝑠𝑎 𝛽(0) 𝑠𝑎 𝑒 ∈ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 𝐸 𝑖 ∈ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 𝑁 𝑠 ∈ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 𝑆 𝑎 ∈ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 𝐴 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='𝑖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='index over records ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='𝑠 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='index over sources ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='𝑎 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='index over attributes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='𝑒 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='index over entities ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='𝑥𝑖𝑎 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='attribute 𝑎 for record 𝑖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='𝑧𝑖𝑎 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='distortion indicator for attribute 𝑎 of record 𝑖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='𝜔𝑖𝑎 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='distortion propensity for attribute 𝑎 of record 𝑖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='𝜍𝑖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='source of record 𝑖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='𝜆𝑖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='linked entity for record 𝑖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='𝐻𝑒𝑎 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='distortion distribution for attribute 𝑎 of entity 𝑒 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='𝜌𝑎 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='concentration of 𝐻𝑒𝑎 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='ψ𝑎 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='base distribution for 𝐻𝑒𝑎 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='𝜃𝑠𝑎 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='distortion probability for attribute 𝑎 in source 𝑠 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='𝑦𝑒𝑎 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='attribute 𝑎 for entity 𝑒 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='𝐺𝑎 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='distribution over domain for attribute 𝑎 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='π ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='mixing proportions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='𝜎,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 𝛼 Ewens-Pitman parameters D𝑎 domain of attribute 𝑎 dist𝑎 distance measure for attribute 𝑎 Figure 2: Plate diagram and index of symbols for our model under a Pitman-Yor prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 8 between attributes are negligible, and place independent Dirichlet Process (DP) priors on each component of G = (𝐺1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' , 𝐺 𝐴): 𝐺𝑎 ind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' ∼ DP(𝜐𝑎, 𝜙𝑎) , ∀𝑎, 𝑦𝑒𝑎 | 𝐺𝑎 ind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' ∼ Discrete(𝐺𝑎), ∀𝑒, 𝑎, where 𝜐𝑎 > 0 is a concentration parameter and 𝜙𝑎 is a base distribution on domain D𝑎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Each record 𝑖 ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' , 𝑁} is linked to an entity 𝜆𝑖 ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='} which is assumed to be drawn from the population with replacement, according to unknown mixing proportions π = (𝜋1, 𝜋2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' This process induces a partition of the records into clusters according to their linked entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Following the discussion about exchangeability in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='2, we assume the partition is drawn from the Ewens-Pitman (EP) family with parameters (𝜎, 𝛼).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' The corresponding distribution on the mixing proportions π depends on the sign of 𝜎, or equivalently, whether the population of entities is finite or infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' For the finite regime (generalized coupon partitions) we let 𝜎 = −𝜅 < 0 and 𝛼 = 𝑚𝜅 for some 𝑚 ∈ ℕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Our model with hyperpriors on 𝑚 and 𝜅 is as follows: 𝜅 ∼ Gamma(𝜒(0), 𝜒(1)), 𝑚 ∼ NegativeBinomial(𝑟, 𝜈) + 1, π | 𝜅, 𝑚 ∼ Dirichlet(κ), 𝜆𝑖 | π iid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='∼ Categorical(π), ∀𝑖, (2) where 𝜒(0), 𝜒(1), 𝑟 > 0 and 0 < 𝜈 ≤ 1 are hyperparameters, and κ is a vector of length 𝑚 with identical entries 𝜅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' The hyperprior on 𝑚 is a shifted negative binomial distribution with density defined in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' In the infinite regime (Pitman-Yor partitions) the mixing proportions are drawn from a two-parameter Poisson-Dirichlet distribution (Pitman and Yor, 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Our model with hyperpriors on 𝜎 and 𝛼 is as follows: 𝜎 ∼ Beta(𝜁 (0), 𝜁 (1)), 𝛼 ∼ Gamma(𝜒(0), 𝜒(1)), π | 𝜎, 𝛼 ∼ PoissonDirichlet(𝜎, 𝛼), 𝜆𝑖 | π iid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='∼ Categorical(π), ∀𝑖, (3) where 𝜒(0), 𝜒(1), 𝜁 (0), 𝜁 (1) > 0 are hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Here we assume 𝛼 > 0 and 0 < 𝜎 < 1, which is a subset of the admissible parameter space: 0 ≤ 𝜎 ≤ 1 and 𝛼 > −𝜎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We also consider the case where 𝜎 = 0, which corresponds to the Ewens partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' By placing hyperpriors on the EP parameters, we can improve robustness to misspecified hyperparameters which are difficult to set in a non-informative manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Special cases of the above priors have been used in other ER models, albeit with fixed hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Tancredi and Liseo (2011), Steorts (2015) and Steorts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' (2016) used a coupon-collector’s partition with 𝜅 → ∞ and 9 𝑚 fixed, which was shown to be highly informative for the observed population size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Steorts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' (2018) used a Pitman-Yor partition with 𝜎 and 𝛼 fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We assume the data source 𝜍𝑖 ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' , 𝑆} associated with record 𝑖 is drawn i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' from a discrete distribution ξ over the sources {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' , 𝑆}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' There is no need to specify ξ since it is independent of the other model parameters, and the source indicators 𝜍𝑖 are assumed to be fully observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Distortion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We assume the attributes x𝑖 for record 𝑖 are generated by distorting the associated entity attributes y𝜆𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' For simplicity, we assume the distortion process occurs independently for each attribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' To decide whether the 𝑎-th attribute is distorted, a binary indicator 𝑧𝑖𝑎 is drawn which depends on the distortion propensity 𝜔𝑖𝑎 scaled by a source/attribute-level factor 𝜃𝜍𝑖𝑎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We place a Beta prior on 𝜃𝜍𝑖𝑎 and assume the distortion propensity 𝜔𝑖𝑎 is deterministic given the true attribute value 𝑦𝜆𝑖𝑎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Concretely, we have 𝜃𝑠𝑎 ind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' ∼ Beta � 𝛽(0) 𝑠𝑎 , 𝛽(1) 𝑠𝑎 � ∀𝑠, 𝑎 (4) 𝜔𝑖𝑎 | 𝑦𝜆𝑖𝑎 = propensity � min 𝑥∈D𝑎\\{𝑦𝜆𝑖 𝑎} dist𝑎(𝑦𝜆𝑖𝑎, 𝑥), max 𝑥,𝑦∈D𝑎 dist𝑎(𝑦, 𝑥) � ∀𝑖, 𝑎 𝑧𝑖𝑎 | 𝜃𝜍𝑖𝑎, 𝜔𝑖𝑎 ind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' ∼ Bernoulli�𝜃𝜍𝑖𝑎𝜔𝑖𝑎 � ∀𝑖, 𝑎 (5) where 𝛽(0) 𝑠𝑎 , 𝛽(0) 𝑠𝑎 > 0 are hyperparameters and propensity(𝑑min, 𝑑max) := ���� ���� 0, 𝑑min = ∞ and 𝑑max = ∞, 1, 𝑑min = 0 and 𝑑max = 0, e− 𝑑min 2𝑑max , otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' (6) The distortion propensity 𝜔𝑖𝑎 accounts for the fact that some entity attribute values 𝑦𝜆𝑖𝑎 ∈ D𝑎 are more likely to be distorted than others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' It makes use of prior information in the attribute distance measure dist𝑎(𝑦, 𝑥) (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' If 𝑦𝜆𝑖𝑎 is not close to any other values in the domain, it is unlikely to be distorted and 𝜔𝑖𝑎 approaches zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' On the other hand, if 𝑦𝜆𝑖𝑎 is close to at least one other value in the domain, distortion can occur and 𝜔𝑖𝑎 approaches one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' This logic is not included in a similar model by Steorts (2015), which effectively assumes 𝜔𝑖𝑎 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' After drawing the distortion indicator 𝑧𝑖𝑎, record attribute 𝑥𝑖𝑎 is generated by copying the linked entity attribute 𝑦𝜆𝑖𝑎 directly (if 𝑧𝑖𝑎 = 0) or subject to distortion (if 𝑧𝑖𝑎 = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' If 𝑧𝑖𝑎 = 1, the distorted value is drawn from a distortion distribution 𝐻𝜆𝑖𝑎 associated with the linked entity 𝜆𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We assume 𝐻𝜆𝑖𝑎 itself is drawn from a Dirichlet Process: 𝜌𝑎 ∼ Gamma � 𝜏(0) 𝑎 , 𝜏(1) 𝑎 � ∀𝑎, (7) 𝐻𝑒𝑎 | 𝑦𝑒𝑎, 𝜌𝑎 ind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' ∼ DP(𝜌𝑎;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' ψ𝑎(𝑦𝑒𝑎)) ∀𝑒, 𝑎, (8) 10 where 𝜏(0) 𝑎 , 𝜏(1) 𝑎 > 0 are hyperparameters, and ψ𝑎(𝑦𝑒𝑎) is a prior base distribution with support on a subset of D𝑎 \\ {𝑦𝜆𝑖𝑎}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' This differs from models by Tancredi and Liseo (2011), Steorts (2015) and Steorts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' (2016, 2018), which assume 𝐻𝑒𝑎 is deterministic conditional on 𝑦𝑒𝑎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Summarizing this symbolically, we have 𝑥𝑖𝑎 | 𝑧𝑖𝑎, 𝑦𝜆𝑖𝑎, 𝐻𝜆𝑖𝑎 ind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' ∼ � 𝛿(𝑦𝜆𝑖𝑎), if 𝑧𝑖𝑎 = 0, 𝐻𝜆𝑖𝑎, if 𝑧𝑖𝑎 = 1, ∀𝑖, 𝑎 (9) where 𝛿(𝑦) denotes a point mass at 𝑦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' This is reminiscent of a hit-miss model (Copas and Hilton, 1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' However, our construction differs, in that the record and entity attributes are forbidden from matching (𝑥𝑖𝑎 ≠ 𝑦𝜆𝑖𝑎) if the record attribute is distorted (𝑧𝑖𝑎 = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' The hit-miss model of Copas and Hilton (1990) was designed for modeling distortion of continuous attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' For continuous attributes, the probability of drawing the non-distorted value (𝑦𝜆𝑖𝑎) from the miss component 𝐻𝜆𝑖𝑎 is zero, assuming 𝐻𝜆𝑖𝑎 is described by a continuous density function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' This ensures the record value 𝑥𝑖𝑎 is always distorted (𝑥𝑖𝑎 ≠ 𝑦𝜆𝑖𝑎) if 𝑧𝑖𝑎 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Our proposal replicates the same behavior for discrete attributes by ensuring 𝐻𝑒𝑎 has no mass on 𝑦𝑒𝑎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' This is especially important if 𝐻𝑒𝑎 were to place significant mass on 𝑦𝑒𝑎, as the line between distorted and non-distorted values would become blurred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Apart from the modeling advantages, excluding 𝑦𝑒𝑎 from the support of 𝐻𝑒𝑎 also makes inference more tractable as we can collapse 𝐻𝑒𝑎 (see Appendices A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='2 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='2 Choice of Hyperparameters In this section, we provide recommendations for setting the hyperparameters in our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Distance Measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Our proposed distortion model is parameterized by a set of distance measures {dist𝑎}, one for each attribute 𝑎 ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' , 𝐴}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' They encode prior knowledge about the likelihood that a record attribute value 𝑥 appears as a distorted alternative to an entity attribute value 𝑦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' The larger the distance dist𝑎(𝑦, 𝑥), the less likely 𝑥 is a distortion of 𝑦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Since the likelihood of distorting 𝑥 to 𝑦 may not be the same as the likelihood of distorting 𝑦 to 𝑥, we do not require that the distance measures are symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We recommend selecting the distance measures carefully, leveraging prior knowledge about the distortion process where possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' For instance, one might select edit distance to model typographic distortion in a generic string-type attribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' For categorical attributes, one could select a constant distance function dist𝑎(𝑦, 𝑥) ≡ 0, which encodes the prior belief that all values in the domain are equally likely as a distorted alternative to 𝑦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Distortion Base Distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We recommend using the distance measures to set the base distribution ψ𝑎(𝑦𝑒𝑎) in Equation (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Specifically, we recommend a softmax distribution 𝜓𝑎(𝑥 | 𝑦𝑒𝑎) ∝ 𝟙[𝑥 ≠ 𝑦𝑒𝑎] exp(− dist𝑎(𝑦𝑒𝑎, 𝑥)), (10) where the temperature parameter is absorbed in the definition of the distance measure, and the indicator function excludes 𝑦𝑒𝑎 from the support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' This places more weight on values in the domain 11 closer to 𝑦𝑒𝑎 and less weight on values further away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Unlike Steorts (2015), we do not include a factor proportional to the empirical frequency of 𝑥, as distorted values (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=', typographical errors) tend to be infrequent for the applications we consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' For a categorical attribute with dist𝑎(𝑦, 𝑥) ≡ 0, Equation (10) reduces to the uniform distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' In this case, it may be appropriate to incorporate a factor proportional to the empirical frequencies by setting 𝜓𝑎(𝑥 | 𝑦𝑒𝑎) ∝ 𝟙[𝑥 ≠ 𝑦𝑒𝑎] �𝑁 𝑖=1 𝟙[𝑥𝑖𝑎 = 𝑥].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Other Hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' In the absence of prior knowledge, we recommend setting the remaining hyperparameters to yield vague priors – i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=', priors that provide little information relative to the experiment (Tiao and Box, 1973;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Bernardo and Smith, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We note that there are different views in the Bayesian community about how to specify vague and/or uninformative priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' For more on this, we refer to Syversveen (1998) and Irony and Singpurwalla (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Putting aside such debates, our recommendations are as follows: For the shifted negative binomial prior on 𝑚, we set 𝑟 and 𝜈 so that the prior mean is 𝑁 and the prior variance is 𝑁2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' For the gamma prior on 𝛼, we set 𝜒(0) = 1 and 𝜒(1) to be small (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=', 10−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' For the beta prior on 𝜎, we set 𝜁 (0) = 𝜁 (1) = 1 to yield a flat prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' For the Dirichlet prior on the entity attribute distribution, we recommend setting 𝜐𝑎 = 1 and using a uniform base distribution 𝜙𝑎 for all 𝑎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' For the gamma prior on the concentration parameter 𝜌𝑎, we recommend setting 𝜏(0) = 2 and 𝜏(1) small (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=', 10−4) for all 𝑎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' For the beta priors on 𝜃𝑠,𝑎, we encode a weak prior belief of low distortion by setting 𝛽(0) 𝑠𝑎 = 1 and 𝛽(1) 𝑠𝑎 = 4 for all 𝑠, 𝑎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' The hyperparameters can be varied to encourage over-linkage (linking records that do not correspond to the same entity) or under-linkage (failing to link records that correspond to the same entity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Since perfect linkage is not always possible, practitioners may have to decide whether over-linkage or under-linkage is preferred for a given application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We can encourage over-linkage in our model by setting: 𝛽(0) 𝑠𝑎 ≫ 𝛽(1) 𝑠𝑎 (prior belief of high distortion), 𝜐𝑎 ≪ 1 (prior belief of low diversity in attribute 𝑎 among entities), 𝜁 (0) ≪ 𝜁 (1) and 𝜒(0) ≪ 𝜒(1) (prior belief of more links for Pitman-Yor prior), or 𝑟 close to 0 and 𝜈 close to 1 (prior belief of more links for generalized coupon prior).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Similarly, we can encourage under-linkage by reversing the inequalities above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We measure the extent to which our model over- or under-links in our empirical evaluation (Section 5) using precision and recall metrics defined in Equations (12) and (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 12 4 Inference To perform entity resolution using our model, we must find the posterior distribution over the linkage structure conditional on the observed record attributes and their sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Since the posterior is not analytically tractable, we propose an approximate inference scheme based on Markov chain Monte Carlo (MCMC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' MCMC produces approximate samples from the posterior distribution by constructing a Markov chain whose equilibrium distribution matches the posterior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' The samples produced by MCMC are approximate in the sense that they are autocorrelated, and they may only match the equilibrium (posterior) distribution asymptotically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Various algorithms exist within the MCMC framework – we refer the reader to Gamerman and Lopes (2006) or Brooks et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' (2011) for an introduction to the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' In this paper, we use an MCMC algorithm called partially collapsed Gibbs (PCG) sampling (van Dyk and Park, 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' It is a generalization of Gibbs sampling that reduces the extent of conditioning in the variable updates by collapsing (marginalizing out) variables and/or updating variables in groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' This can significantly improve convergence and reduce autocorrelation, so long as prescribed rules are followed to ensure the equilibrium distribution of the Markov chain is preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Ideally, we would like to reduce the extent of conditioning as much as possible, however this must be balanced with computational and mathematical constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' In our proposed sampling scheme, we fully collapse the entity mixing proportions π and the distortion distributions 𝐻𝑒𝑎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We partially-collapse the distortion indicators 𝑧𝑖𝑎 in a joint update for the entity attributes 𝑦𝑒𝑎 and for the distortion distribution concentration 𝜌𝑎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' By collapsing the mixing proportions, we obtain an urn-based scheme for updating the linkage structure similar to those used for nonparametric mixture models (Neal, 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' In the remainder of this section, we highlight some less trivial aspects of inference – full details are provided in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='1 Nonconjugacy While we attempted to maintain conjugacy in our model, we were unable to avoid nonconjugate priors in some cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' This complicates inference, as the posterior conditional distributions used in Gibbs sampling are no longer of a standard form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' There are several well-established methods for dealing with nonconjugacy, including Metropolis-Hastings algorithms (Chib and Greenberg, 1995), rejection sampling (Gilks and Wild, 1992) and auxiliary variable methods (Damlen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=', 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We opt to use auxiliary variable methods owing to their simplicity, as there is no need to design proposals or monitor acceptance rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' There are three sets of parameters in our model for which nonconjugacy is an issue: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' The distortion probabilities 𝜃𝑠𝑎 defined in Equation (4), where the incorporation of the distortion propensities 𝜔𝑖𝑎 breaks the conjugacy of the beta prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We propose an auxiliary variable sampling scheme to update 𝜃𝑠𝑎 in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' The EP parameters: 𝜅 and 𝑚 defined in Equation (2) or 𝛼 and 𝜎 defined in Equation (3), 13 depending on the regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We use an auxiliary variable scheme proposed by Teh (2006), to update 𝛼 and 𝜎 under a gamma and beta prior, as summarized in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We design an auxiliary variable update for 𝜅 and 𝑚 under a gamma and shifted negative binomial prior in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' The distortion distribution concentration 𝜌𝑎 defined in Equation (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We design an auxiliary variable update for 𝜌𝑎 in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='2 Collapsing the Distortion Indicators Marchant et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' (2021) demonstrated the importance of collapsing the distortion indicators {𝑧𝑖𝑎} to improve convergence/mixing for a hit-miss model similar to Equation (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' The posterior factors involving 𝑧𝑖𝑎 factorize over 𝑖 and 𝑎, so that collapsing 𝑧𝑖𝑎 yields: 𝑃(𝑥𝑖𝑎 | 𝜃𝜍𝑖𝑎, 𝜔𝑖𝑎, 𝑦𝜆𝑖𝑎, 𝐻𝜆𝑖𝑎) ∝ 1 ∑︁ 𝑧𝑖𝑎=0 𝑃(𝑥𝑖𝑎 | 𝑧𝑖𝑎, 𝑦𝜆𝑖𝑎, 𝐻𝜆𝑖𝑎)𝑃(𝑧𝑖𝑎 | 𝜃𝜍𝑖𝑎, 𝜔𝑖𝑎) ∝ (1 − 𝜃𝜍𝑖𝑎𝜔𝑖𝑎)𝟙[𝑥𝑖𝑎 = 𝑦𝜆𝑖𝑎] + 𝜃𝜍𝑖𝑎𝜔𝑖𝑎𝐻𝜆𝑖𝑎(𝑥𝑖𝑎).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' (11) We use this result to implement a collapsed update for the entity attributes {𝑦𝑒𝑎}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' While it is possible to implement a collapsed update for the linkage structure {𝜆𝑖}, we opt not to do so, since conditioning on the distortion indicators allows us to reduce computational complexity via indexing (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' This seems to be more efficient empirically (Marchant et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=', 2021), so long as the level of distortion is not too high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='3 Computational Considerations We now discuss ways of improving the computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' The main bottleneck is the update for the linkage structure which scales naïvely as 𝑂(𝑁 · 𝐸) where 𝐸 is the number of instantiated entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='2 The update for the entity attributes may also be problematic for large domains D𝑎 as it scales as 𝑂(𝐸 · |D𝑎|) for the 𝑎-th attribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We are able to reduce the computational complexity of the linkage structure update by exploiting constraints imposed by the distortion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Close inspection of the update for the entity linked to record 𝑖 (see Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='3) reveals that some entities can be immediately excluded from consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Specifically, only those entities whose attributes match the corresponding non- distorted record attributes (𝑥𝑖𝑎 with 𝑧𝑖𝑎 = 0) may be linked to record 𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' In order to efficiently query this set of entities, we maintain inverted indices that map an attribute value 𝑥 ∈ D𝑎 to the set of entities instantiated with that value {𝑒 : 𝑥 = 𝑦𝑒𝑎}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' This approach is considerably more efficient than iterating over all entities sequentially, so long as the level of distortion is relatively low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' However it is important to note that it relies crucially on not collapsing the distortion indicators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' To improve the complexity of the entity attribute update, we can impose a cut-off on the distance 2When stating time complexities in this section, we assume a categorical random variate can be drawn in Θ(𝐶) time where 𝐶 is the number of categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' The algorithm proposed by Vose (1991) satisfies this constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 14 measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Concretely, for attribute 𝑎 we replace the “raw” distance measure dist𝑎 by dist𝑎(𝑦, 𝑥) = � dist𝑎(𝑦, 𝑥), if dist𝑎(𝑦, 𝑥) ≤ 𝑑(cut) 𝑎 , ∞, otherwise, where 𝑑(cut) 𝑎 ∈ (0, ∞) is a configurable cut-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' This approximation eliminates the need to consider unlikely distortions from entity attribute 𝑦 to record attribute 𝑥, for which dist(𝑦, 𝑥) > 𝑑(cut) 𝑎 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' It plays a similar role to blocking in the record linkage literature (Christen, 2012b) and resembles an approach proposed by Marchant et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' To make use of this approximation, we build indices that can efficiently answer range queries – one for each attribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' The index for the 𝑎-th attribute takes a query value 𝑥 ∈ D𝑎 and returns the set of entity attribute values that fall below the cut-off: {𝑦 ∈ D𝑎 : dist𝑎(𝑦, 𝑥) ≤ 𝑑(cut) 𝑎 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 5 Model Comparisons We conduct an empirical study of our ER model using data sets for which the true linkage structure is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='1 describes the data sets used in the study, which are motivated by ER applications in private and non-private settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We explain how our model (and baseline models) are evaluated in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='2, by computing metrics that assess how well the posterior predictions align with the true linkage structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='3 assesses the impact of our modeling contributions by varying the distortion model and the prior on the linkage structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='4 compares our ER model against baselines proposed by Sadinle (2014) and Steorts (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Finally, Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='5 summarizes a controlled simulation study that can be found in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='1 Data Sets We study entity resolution in private and non-private settings, both of which are encountered by practitioners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' The data sets we use in our study are summarized in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Private Setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' In this setting the practitioner has access to de-identified data, where sensitive attributes such as names, addresses, phone numbers, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' are removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' This can make ER quite challenging, as the remaining non-sensitive attributes may carry limited information about the identity of records.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' To study ER in this setting, we use data extracted from the National Long Term Care Survey (Manton, 2010), which we refer to as nltcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Our extract contains de-identified respondent records from the 1982, 1989 and 1994 waves of the survey in the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' state of Alabama.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We use all of the available attributes for ER, which include date of birth (DOB_YEAR, DOB_MONTH, DOB_DAY), registration office (REGOFF) and sex (SEX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Since the data is well-curated, the only distortion that can occur is when a valid attribute value is replaced by another valid attribute value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We, therefore, model the attributes as categorical by employing a constant distance function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Non-private Setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' In this setting, the practitioner has access to data with sensitive attributes, such as names.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We assume unique identifiers such as social security numbers are not available, as 15 Data set Setting Entity type # records (𝑁) # entities nltcs Private People 5,359 3,307 RLdata Non-private People 10,000 9,000 cora Non-private Citations 1,295 125 rest Non-private Restaurants 864 752 Table 1: Summary of data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' ER would otherwise be trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Obtaining real survey data for a non-private setting with ground truth is challenging, so we use three publicly-available data sets from the ER literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Although these data sets cover other domains, they exhibit characteristics one would expect to encounter in real survey data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Namely, we observe the presence of multiple “name-like” attributes, as well as different levels of variation and distortion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Below, we provide a brief description of each data set and the attributes used for ER: RLdata is a synthetic person data set, where 10% of the records are duplicates with random errors (Sariyar and Borg, 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We model the name attributes (fname_c1 and lname_c1) using the normalized Levenshtein distance measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' The attributes related to date of birth – bd, bm and by – are modeled as categorical attributes with a constant distance measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='3 cora is a collection of computer science citation records hosted on the RIDDLE repository (Bilenko, 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' It is the “dirtiest” of all the data sets we consider, as it was extracted from various online sources with different citation styles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' As a pre-processing step, we separate hyphenated words and remove punctuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We also correct several erroneous ground truth labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' The title, venue and authors attributes generally contain multiple words with semantic and character-level variations, and are therefore modeled using a hybrid token/edit distance measure described in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' The year attribute is modeled using normalized Levenshtein distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' rest is a collection of restaurant records from the Fodor and Zagat restaurant guides hosted on the RIDDLE repository (Bilenko, 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' It is not as “dirty” as cora as there are fewer sources and less variation between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We applied the same pre-processing steps as for cora.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' The name and addr attributes generally contain multiple words and are therefore modeled using the same hybrid distance measure as for cora.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' The city and type (cuisine) attributes are modeled as categorical with a constant distance measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='2 Model Evaluation We evaluate an ER model on a data set by comparing the inferred linkage structure ˆΛ to the true linkage structure Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Recall that Λ = (𝜆1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' , 𝜆𝑁) specifies the corresponding entity 𝜆𝑖 for each record 𝑖 in the data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' The agreement between ˆΛ and Λ can be measured using pairwise precision and recall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' The pairwise precision is the proportion of record pairs linked in ˆΛ that are also linked 3This is a benchmark data set that is widely used in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 16 in Λ: Pr( ˆΛ, Λ) = �𝑁 𝑖≠𝑗=1 𝟙[ ˆ𝜆𝑖 = ˆ𝜆 𝑗]𝟙[𝜆𝑖 = 𝜆 𝑗] �𝑁 𝑖≠𝑗=1 𝟙[ ˆ𝜆𝑖 = ˆ𝜆 𝑗] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' (12) It takes on values from 0 to 1, where larger values indicate fewer false positive errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' The pairwise recall is the proportion of record pairs linked in Λ that are also linked in ˆΛ: Re( ˆΛ, Λ) = �𝑁 𝑖≠𝑗=1 𝟙[ ˆ𝜆𝑖 = ˆ𝜆 𝑗]𝟙[𝜆𝑖 = 𝜆 𝑗] �𝑁 𝑖≠𝑗=1 𝟙[𝜆𝑖 = 𝜆 𝑗] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' (13) It takes on values from 0 to 1, where larger values indicate fewer false negative errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' It is rarely possible to achieve high precision and recall – one must usually make a trade-off depending on which types of errors are more costly in a given application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' If precision and recall are equally important, then one can measure the agreement between ˆΛ and Λ using the pairwise F1 score which is the harmonic mean of precision and recall: F1( ˆΛ, Λ) = 2 Pr( ˆΛ, Λ) · Re( ˆΛ, Λ) Pr( ˆΛ, Λ) + Re( ˆΛ, Λ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' (14) Since the models in our study are Bayesian, the inferred (posterior) linkage structure ˆΛ is a random variable and the metrics in Equations (12)–(14) can be regarded as random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We estimate the distribution of the metrics under the posterior using samples generated via MCMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' In doing so, we are able to account for posterior uncertainty in our evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' For each metric, we report a point estimate using the median, along with a 95% equi-tailed credible interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Further details about the MCMC implementation and configuration for each model are provided in Appendix E and MCMC diagnostics are in Appendix G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='3 Study of Linkage Structure Priors and Distortion Model In this section, we study the effect of two modeling contributions proposed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='1 – the Ewens-Pitman (EP) linkage structure priors and the refined distortion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Our objective is to determine the impact of each modeling contribution in isolation, using the blink model as a baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We summarize the results here and refer the reader to Appendix D for comprehensive results covering eight combinations of linkage structure priors and distortion models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Linkage Structure Priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We consider four priors on the linkage structure, which correspond to distinct EP parameter regimes (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='2): 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' PY: Pitman-Yor regime with 𝜎 ∈ (0, 1) and hyperpriors on 𝜎, 𝛼 as detailed in Equation (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Ewens: Ewens regime with 𝜎 = 0 and a hyperprior on 𝛼 as detailed in Equation (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' GenCoupon: generalized coupon regime with 𝜎 = −𝜅 < 0 and hyperpriors on 𝜅, 𝑚 as detailed in Equation (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 17 RLdata nltcs cora rest −10 −5 0 −2 −1 0 1 2 50 100 150 200 −4 −2 0 Coupon GenCoupon Ewens PY Relative error (%) Prior Figure 3: Posterior relative error in the predicted number of entities for all data sets and linkage structure priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Under-linkage is observed for cora, which is likely due to significant noise that is not well-captured by the distortion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Coupon: coupon collector’s partition used by with 𝜅 → ∞ and 𝑚 = 𝑁.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' The first three priors are flexible, in the sense that hyperpriors are placed on the EP parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' The last prior is a particular instance of GenCoupon where the EP parameters are fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' It is used in models by Tancredi and Liseo (2011), Steorts (2015) and Steorts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' (2016), and serves as a baseline here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' ER evaluation metrics are presented in Table 2 for the four linkage structure priors, assuming the rest of the model follows the specification in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Another perspective on ER accuracy is provided in Figure 3, which plots the relative error in the inferred number of entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Both results demonstrate the benefit of placing hyperpriors on the EP parameters, as is done for PY, Ewens, and GenCoupon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' These linkage structure priors achieve superior F1 scores compared to Coupon, where the EP parameters are fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Figure S8 (Appendix D) is consistent with this finding, demonstrating that vastly different values of the EP parameters are inferred for each data set when hyperpriors are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Another interesting observation is the fact that the ER accuracy is relatively similar among PY, Ewens and GenCoupon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' This was unexpected at first, given the three parameter regimes are known to exhibit distinct asymptotic behavior (see equation 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' This suggests all three regimes may be flexible enough to model the linkage structure of the data sets we consider here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' It would be interesting to see if these observations translate to much larger data sets, where the asymptotic behavior of the three regimes would become more apparent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Distortion Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We compare our proposed distortion model (specified in the latter part of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='1) to the distortion model proposed by Steorts (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' For brevity, we refer to our distortion model as Ours and that of Steorts (2015) as blink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Here, we report results for the GenCoupon linkage structure prior – the results for the other linkage structure priors are reported in Appendix D and exhibit similar trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Figure 4 plots the inferred level of distortion for each attribute under both distortion models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' It shows that blink tends to encourage high distortion, particularly for attributes with non-constant distance measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='4 For example, the fname_c1 and lname_c1 attributes for RLdata are predicted 4The attributes modeled with non-constant distance measures are: all attributes for cora, name and addr for rest, and fname_c1 and lname_c1 for RLdata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 18 Evaluation metric Data set EP regime Precision Recall F1 score RLdata PY 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='896 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='879, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='917) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='961 (0.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='893) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='970 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='961, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='978) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='917 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='908, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='931) GenCoupon 0.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='674) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='911 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='893, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='938) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='750 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='722, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='781) Table 2: Posterior evaluation metrics for our model under four linkage structure priors corresponding to distinct Ewens-Pitman (EP) parameter regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' A point estimate for each evaluation metric is reported based on the median, along with a 95% equi-tailed credible interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Similar performance is observed for the three regimes where the EP parameters are permitted to vary (PY, Ewens and GenCoupon).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' A significant drop in performance is observed for the Coupon regime on RLdata and cora.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 19 RLdata nltcs cora rest 0 25 50 75 100 bd bm by fname_c1 lname_c1 dob_day dob_month dob_year regoff sex authors title venue year addr city name type Distortion level (%) Attribute Distortion model Ours blink Figure 4: Comparison of the posterior attribute-level distortion under two distortion models: Ours (red top-most intervals) and blink (teal bottom-most intervals).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' The blink distortion model tends to favor higher levels of distortion – in some cases approaching 100 percent – which is not consistent with expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' to be almost 100% distorted under the blink distortion model, which is inconsistent with expectations for this data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Our distortion model does not appear to suffer from this problem, as it requires disagreement between entity and record attributes in order to classify them as “distorted”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Since high distortion makes reliable linkage more challenging, we expect that our distortion model is likely to perform better in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Indeed, it achieves a better balance between precision and recall in our full results (see Figure S7 in Appendix D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Summary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We return to the original goal of this section and summarize what we have learned in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' First, we have learned that our proposed linkage structure prior is generally more robust due to the use of hyperpriors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' In addition, our inferences are relatively insensitive to the EP parameter regime, which may be due to the fact that the data sets are relatively small in size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' This behavior also holds for the blink distortion model when combined with our proposed linkage structure priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Second, when studying the performance of the distortion models (blink versus our proposed distortion model), we find that ours predicts more reasonable distortion rates and improves the linkage accuracy as measured by F1 score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Thus, based on this study, we would recommend our distortion model and linkage structure priors moving forward for data sets similar to those we have considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' However, we stress that further exploration is needed to provide more general recommendations for other data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 20 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='4 Comparison with Baseline Models In this section, we study how our entity resolution model performs in comparison with models by Steorts (2015) and (Sadinle, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' The blink model by Steorts (2015) is a natural baseline to consider, as it served as inspiration for our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Compared to our model, blink is less Bayesian in its design, as many of the parameters are set empirically or arbitrarily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Both our model and blink, are examples of direct modeling approaches to ER – i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=', they model how the observed records are generated, incorporating the linkage structure as a latent variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' In contrast, the model by Sadinle (2014) (which we refer to as Sadinle) adopts a comparison-based approach to ER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Instead of modeling the data directly, it models attribute-level comparisons between pairs of records, incorporating the presence/absence of a link between the pair as a latent variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Sadinle (2018) compares direct and comparison-based approaches from a methodological perspective, however we are not aware of any empirical comparisons in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Our goal in this section is to provide a comparison for the first time on a variety of data sets, where we make no strong claims that our results generalize to all applications or data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' In order to make the comparison as fair as possible, we use the same distance functions to model the distortion in our model, blink, and Sadinle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' For instance, if we use edit distance to model distortion for a name attribute in our model and blink, then we also use edit distance to compare the same name attribute in Sadinle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We set the distance cut-offs for our model and blink (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='3) to align with the blocking design used for Sadinle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Further information about our experimental setup is provided in Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' ER evaluation metrics are presented in Table 3 for all three models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' For simplicity we only provide results for our model under the GenCoupon prior, which is denoted Ours in the table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Our model achieves the highest (or equal-highest) F1 score for all four data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We expect the poorer performance of blink is due to its use of subjective (inflexible) priors and its distortion model, which tends to favour high distortion and over-linkage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Sadinle achieves the second highest F1 score in the non-private setting (RLdata, cora, rest), and the lowest F1 score in the private setting (nltcs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' The poorer performance for nltcs may be partly related to the blocking scheme, which is less aggressive, leaving the model more susceptible to over-linkage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Another important factor is the sensitivity of Sadinle to the truncation points for the priors on the 𝑚-probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We perform coarse-grained tuning of the truncation points in Appendix F, however fine-grained tuning could result in additional performance gains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Summary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' This study provides evidence that our model achieves a better balance between precision and recall than blink and Sadinle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We stress that our results are based on four data sets – further experimentation is required to determine whether our results generalize to other data sets and applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We speculate that the better performance of our model is mainly due to improved flexibility resulting from the addition of priors and hyperpriors, which can be viewed as performing model selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 21 Evaluation metric Data set Model Precision Recall F1 score RLdata Ours 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='917 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='902, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='932) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='966 (0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='344) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='992 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='988, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='996) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='502 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='492, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='511) Sadinle 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='946) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='751 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='713, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='780) Sadinle 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='993 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='985, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='000) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='603 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='598, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='607) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='750 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='744, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='756) Table 3: Posterior performance of our model against two baselines: blink (Steorts, 2015) and Sadinle (Sadinle, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' A point estimate for each evaluation metric is reported based on the median, along with a 95% equi-tailed credible interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Our model achieves the highest (or equal-highest) F1 score within the credible intervals for all data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='5 Controlled Simulation Study We conduct a simulation study to evaluate our model under controlled conditions, where we vary the size of the data set, the level of distortion, and the level of duplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Due to space constraints, we summarize the study here – full details can be found in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We design a simulator for household survey data sets, where responses are collected for individuals within households.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Since the attributes of individuals within a household are dependent – e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=', the address is the same, family members may share the same last name, the age of individuals may be correlated – the simulated data follows a more complex generative process than our entity resolution model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' This is intentional, as it allows us to evaluate our model in a more realistic setting where it is misspecified for the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' The dataset simulator also incorporates a record generation process which is misspecified for our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Rather than sampling individuals from the population, our data set simulator iterates over all individuals, randomly deciding whether to include the individual, and if so, how many distorted records to create.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We run entity resolution using our model on 16 simulated data sets, using the blink and Sadinle models as baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We summarize the results across three factors below: Duplication level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' The level of duplication has minimal impact on the performance of our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' blink performs well for moderate to high levels of duplication, however it over-links severely when the level of duplication is low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' The performance of Sadinle does not seem to follow a consistent trend as the duplication varies – it achieves a lower F1 score than our model and blink in all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 22 Distortion level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We find the more distorted data sets are more difficult to link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Specifically, we observe a drop in recall of around 10 percentage points for our model and blink when compared to the data sets with low distortion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Larger drops in recall of 15–20 percentage points are observed for Sadinle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Data set size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We find that our model performs similarly for both data set sizes (1000 and 10000 records).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' blink also performs similarly in most scenarios, however, we observe a drop in precision for the larger data set when the level of duplication is low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Sadinle performs significantly worse for the larger data sets in terms of precision, however, the recall is relatively stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' In summary, the simulation study shows that our model achieves the most consistent performance across all scenarios tested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' blink is also competitive, but it is has poor performance in the low duplication scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Sadinle achieves the lowest F1 score when the level of duplication is non-negligible, and is somewhat competitive when the level of distortion is low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 6 Discussion In this section, we summarize our contributions and provide a discussion regarding future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We have proposed a Bayesian model for entity resolution that addresses limitations of previous work (Steorts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Steorts, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Marchant et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Our model can be viewed as performing graphical entity resolution, where observed records are clustered to (unobserved) latent entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' To improve upon the scalability of previous work, we designed a partially collapsed Gibbs sampler with an optimized implementation that can handle data sets of around 10,000 records.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' This allowed us to provide comparisons with models by Steorts (2015) and Sadinle (2014), which was previously only possible for toy-sized data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We provided comparisons to real and synthetic data sets and a controlled simulation study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We observed that our model tends to be less sensitive to changes in the hyperparameters than competing models for the data sets considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Further analysis is required to make more general conclusions beyond the data sets and simulations considered in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' There are many potential avenues for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' First, it would be of interest to explore scaling for our proposed model and the model by Sadinle (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' This could be achieved by designing parallel/distributed inference algorithms, by investigating more efficient MCMC algorithms, or by resorting to blocking techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Another area of interest, is exploring more diverse data sets to understand the strengths and weaknesses of each method in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Although we made recommendations based on four data sets and a simulation study, more comparisons would help to alleviate any selection bias regarding data sets and provide guidance to users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Finally, future work could consider microclustering priors (Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=', 2015) to assess their effectiveness compared to the infinitely-exchangeable linkage structure priors considered here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' This would require modifying the sampling scheme and selecting an appropriate class of microclustering priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Acknowledgements This work was supported by the National Science Foundation [CAREER-1652431], the Alfred Sloan Foundation, the Australian Research Council [DP220102269] and an Australian Government 23 Research Training Program Scholarship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' References Bernardo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' and Smith, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' (2009), Bayesian theory, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 405, John Wiley & Sons.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Bureau of the Census.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 27 A Gibbs Updates In this appendix, we derive updates for the partially collapsed Gibbs sampler used to perform approximate inference for the ER model introduced in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Some of the updates are non-trivial due to non-conjugacy of the proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='1 Update for the Distortion Probabilities In this section, we provide the update for the distortion probability 𝜃𝑠𝑎 (source 𝑠 and attribute 𝑎).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' This update is complicated by the presence of the distortion propensity variables 𝜔𝑖𝑎, which breaks the conjugacy of the beta prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' To overcome this problem, we introduce the following auxiliary variables: 𝑞𝑖𝑎 | 𝜔𝑖𝑎 ∼ Bernoulli(𝜔𝑖𝑎) ∀𝑖, 𝑎 and modify the conditional distribution for the distortion indicators as follows: 𝑧𝑖𝑎 | 𝜃𝜍𝑖𝑎, 𝑞𝑖𝑎 ∼ Bernoulli(𝜃𝜍𝑖𝑎𝑞𝑖𝑎) ∀𝑖, 𝑎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' It is straightforward to show that one recovers the original model in Equation (5) when the auxiliary variables are marginalized out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Observe that the contribution to the posterior involving 𝑞𝑖𝑎 is (𝜃𝜍𝑖𝑎𝑞𝑖𝑎)𝑧𝑖𝑎(1 − 𝜃𝜍𝑖𝑎𝑞𝑖𝑎)1−𝑧𝑖𝑎𝜔𝑞𝑖𝑎 𝑖𝑎 (1 − 𝜔𝑖𝑎)1−𝑞𝑖𝑎 = � 𝜃𝑧𝑖𝑎 𝜍𝑖𝑎(1 − 𝜃𝜍𝑖𝑎)1−𝑧𝑖𝑎𝜔𝑖𝑎 �𝑞𝑖𝑎 [1 − 𝜔𝑖𝑎]1−𝑞𝑖𝑎 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Thus, the distribution of 𝑞𝑖𝑎 conditional on the other variables is: 𝑞𝑖𝑎 | 𝜔𝑖𝑎, 𝜃𝜍𝑖𝑎, 𝑧𝑖𝑎 ∼ Bernoulli � 𝜔𝑖𝑎𝜃𝑧𝑖𝑎 𝜍𝑖𝑎(1 − 𝜃𝜍𝑖𝑎)1−𝑧𝑖𝑎 𝜔𝑖𝑎𝜃𝑧𝑖𝑎 𝜍𝑖𝑎(1 − 𝜃𝜍𝑖𝑎)1−𝑧𝑖𝑎 + 1 − 𝜔𝑖𝑎 � ∀𝑖, 𝑎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' (S15) Next, observe that the contribution to the posterior involving 𝜃𝑠𝑎 is 𝜃𝛽(0) 𝑠𝑎 −1 𝑠𝑎 (1 − 𝜃𝑠𝑎)𝛽(1) 𝑠𝑎 −1 � 𝑖:𝜍𝑖=𝑠 (𝜃𝑠𝑎𝑞𝑖𝑎)𝑧𝑖𝑎(1 − 𝜃𝑠𝑎𝑞𝑖𝑎)1−𝑧𝑖𝑎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Hence, the distribution of 𝜃𝑠𝑎 conditional on the other variables is: 𝜃𝑠𝑎 | Q, Z, S ∼ Beta � 𝛽(0) 𝑠𝑎 + ∑︁ 𝑖:𝜍𝑖=𝑠 𝑧𝑖𝑎, 𝛽(1) 𝑠𝑎 + ∑︁ 𝑖:𝜍𝑖=𝑠 𝑞𝑖𝑎(1 − 𝑧𝑖𝑎) � ∀𝑠, 𝑎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' (S16) It is also straightforward to see that the distribution of 𝑧𝑖𝑎 conditional on the other variables is a point mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' In particular, we have 𝑧𝑖𝑎 | 𝑥𝑖𝑎, 𝜆𝑖, Y = � 1, if 𝑥𝑖𝑎 ≠ 𝑦𝜆𝑖𝑎 0, otherwise (S17) 28 In summary, to update the distortion probabilities, one would first compute the distortion indicators {𝑧𝑖𝑎} using Equation (S17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Then, conditional on the other variables, one would draw auxiliary variables {𝑞𝑖𝑎} using Equation (S15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Finally, one can update the distortion probabilities {𝜃𝑠𝑎} using Equation (S16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' The updates for the other model parameters are unaffected by the introduction of the auxiliary variables {𝑞𝑖𝑎}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='2 Update for the Entity Attributes In this section, we provide the update for the entity attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' When updating entity attribute 𝑦𝑒𝑎, we collapse the base distribution 𝐻𝑒𝑎 and distortion indicators Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' The posterior factors involving 𝑦𝑒𝑎 after collapsing 𝐻𝑒𝑎 are as follows: 𝑃(𝑦𝑒𝑎 | Z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 𝛀,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 𝚯,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 𝐺𝑎,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 𝜌𝑎) ∝ 𝑃(𝑦𝑒𝑎 | 𝐺𝑎) × ∫ � 𝑖:𝜆𝑖=𝑒 𝑃(𝑥𝑖𝑎 | 𝜃𝜍𝑖𝑎,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 𝜔𝑖𝑎,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 𝑦𝑒𝑎,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 𝐻𝑒𝑎)𝑃(𝐻𝑒𝑎 | 𝑦𝑒𝑎,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 𝜌𝑎) d𝐻𝑒𝑎 ∝ 𝑃(𝑦𝑒𝑎 | 𝐺𝑎) � 𝑖:𝜆𝑖=𝑒 𝑥𝑖𝑎=𝑦𝑒𝑎 (1 − 𝜃𝜍𝑖𝑎𝜔𝑖𝑎) � 𝑖:𝜆𝑖=𝑒 𝑥𝑖𝑎≠𝑦𝑒𝑎 (𝜃𝜍𝑖𝑎𝜔𝑖𝑎) × ∫ � 𝑖:𝜆𝑖=𝑒 𝑥𝑖𝑎≠𝑦𝑒𝑎 𝐻𝑒𝑎(𝑥𝑖𝑎) � 𝑣∈D𝑎\\{𝑦𝑒𝑎} 𝐻𝑒𝑎(𝑣)𝜌𝑎𝜓𝑎(𝑣|𝑦𝑒𝑎)−1 B(𝜌𝑎ψ𝑎(𝑦𝑒𝑎)) d𝐻𝑒𝑎 ∝ 𝐺𝑎(𝑦𝑒𝑎) � 𝑖:𝜆𝑖=𝑒 𝑥𝑖𝑎=𝑦𝑒𝑎 (1 − 𝜃𝜍𝑖𝑎𝜔𝑖𝑎) � 𝑖:𝜆𝑖=𝑒 𝑥𝑖𝑎≠𝑦𝑒𝑎 (𝜃𝜍𝑖𝑎𝜔𝑖𝑎) × Γ(𝜌𝑎) Γ( ¯𝑛𝑒𝑎(𝑦𝑒𝑎) + 𝜌𝑎) � 𝑣∈V𝑒𝑎 Γ(𝑛𝑒𝑎(𝑣) + 𝜌𝑎𝜓𝑎(𝑣 | 𝑦𝑒𝑎)) Γ(𝜌𝑎𝜓𝑎(𝑣 | 𝑦𝑒𝑎)) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' where B(·) is the multivariate beta function,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' V𝑒𝑎 = �� 𝑖:𝜆𝑖=𝑒{𝑥𝑖𝑎}� \\ {𝑦𝑒𝑎} are the distorted record values for the 𝑎-th attribute associated with entity 𝑒,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 𝑛𝑒𝑎(𝑣) = � 𝑖:𝜆𝑖=𝑒 𝟙[𝑥𝑖𝑎 = 𝑣] is the number of records linked to entity 𝑒 whose 𝑎-th attribute is equal to 𝑣 and ¯𝑛𝑒𝑎(𝑣) = � 𝑖:𝜆𝑖=𝑒 𝟙[𝑥𝑖𝑎 ≠ 𝑣] is the number of records linked to entity 𝑒 whose 𝑎-th attribute is not equal to 𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We can rewrite the above expression in a more computationally convenient form by repeatedly 29 applying the recurrence relation for the Gamma functions5 to yield: 𝑃(𝑦𝑒𝑎 | Z, 𝛀, 𝚯, S, 𝐺𝑎, 𝜌𝑎) ∝ 𝐺𝑎(𝑦𝑒𝑎) � 𝑣∈V𝑒𝑎 �𝑛𝑒𝑎(𝑣) 𝑗=1 { 𝑗 − 1 + 𝜌𝑎𝜓𝑎(𝑣 | 𝑦𝑒𝑎)} �¯𝑛𝑒𝑎(𝑦𝑒𝑎) 𝑗=1 { 𝑗 − 1 + 𝜌𝑎} × � 𝑖:𝜆𝑖=𝑒 𝑥𝑖𝑎=𝑦𝑒𝑎 (1 − 𝜃𝜍𝑖𝑎𝜔𝑖𝑎) � 𝑖:𝜆𝑖=𝑒 𝑥𝑖𝑎≠𝑦𝑒𝑎 (𝜃𝜍𝑖𝑎𝜔𝑖𝑎).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Observe that the above distribution may only have support on a subset of the full domain D𝑎 when distance thresholds are applied, as discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' In particular, one can show that the support is a subset of � 𝑖:𝜆𝑖=𝑒 {𝑦 ∈ D𝑎 : dist𝑎(𝑦, 𝑥𝑖𝑎) ≤ 𝑑(cut) 𝑎 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' This fact can be used to implement the update more efficiently, since it is not necessary to construct a pmf over the entire domain D𝑎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='3 Update for the Linkage Structure In this section, we provide the update for the linkage structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' When updating the linkage structure, we use an urn-based scheme as described by Neal (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' In doing so, we only need to keep track of entities in the population that are linked to records – any isolated entities not linked to records are ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' This is important, as the population may be infinite in size for some Ewens-Pitman parameter regimes (when 𝜎 ≥ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' To update the linked entity 𝜆𝑖 for record 𝑖, we remove the current link and allow the record to either join one of the remaining instantiated entities (with at least one other record) or instantiate a “new” entity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' The conditional distribution has the following form: 𝑃(𝜆𝑖 = 𝑒 | Z, X, Y , 𝚲−𝑖, {𝜌𝑎}) ∝ ������ ������ 𝐶 |𝑒|−𝜎 𝛼+𝑁−1 � 𝑎 ∫ 𝑃(𝑥𝑖𝑎 | 𝑧𝑖𝑎, 𝑦𝑒𝑎, 𝐻𝑒𝑎)dℍ−𝑖,𝑒𝑎, if 𝑒 is instantiated and |𝑒| > 0, 𝐶 𝛼+𝜎𝐸 𝛼+𝑁−1 � 𝑎 � 𝑦𝑒𝑎∈D𝑎 𝑃(𝑦𝑒𝑎 | 𝐺𝑎) ∫ 𝑃(𝑥𝑖𝑎 | 𝑧𝑖𝑎, 𝑦𝑒𝑎, 𝐻𝑒𝑎)dℍ0,𝑒𝑎, if 𝑒 is “new”, (S18) where 𝐶 is a normalization constant;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 𝚲−𝑖 = (𝜆1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' , 𝜆𝑖−1, 𝜆𝑖+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' , 𝜆𝑁) are the linked entities for all records excluding 𝑖;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 5Repeated application of the recurrence relation for the Gamma function yields Γ(𝑧) = Γ(𝑧 + 𝑛 + 1) 𝑧(𝑧 + 1) · · · (𝑧 + 𝑛) for complex 𝑧 (excluding zero and the negative integers) and non-negative integer 𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 30 |𝑒| = � 𝑖′≠𝑖 𝟙[𝜆𝑖′ = 𝑒] is the number of records (excluding 𝑖) linked to entity 𝑒;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 𝐸 = � 𝑒′≠𝑒 𝟙[|𝑒| > 0] is the number of instantiated entities with at least one linked record;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' ℍ0,𝑒𝑎 is the prior for 𝐻𝑒𝑎;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' and ℍ−𝑖,𝑒𝑎 is the posterior for 𝐻𝑒𝑎 given the observed distorted record attributes 𝑥𝑖′𝑎 for which 𝑖′ ≠ 𝑖 and 𝜆𝑖′ = 𝑒 (also conditioned on 𝑦𝑒𝑎 and 𝜌𝑎).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Recall that the prior ℍ0,𝑒𝑎 for 𝐻𝑒𝑎 conditioned on 𝑦𝑒𝑎 and 𝜌𝑎 is Dirichlet(𝜌𝑎ψ𝑎(𝑦𝑒𝑎)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Since 𝑥𝑖𝑎 is Categorical(𝐻𝑒𝑎) if 𝑧𝑖𝑎 = 1 (and a point mass at 𝑦𝑒𝑎 if 𝑧𝑖𝑎 = 0), the posterior ℍ−𝑖,𝑒𝑎 is also Dirichlet by conjugacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' In particular, one can show that ℍ−𝑖,𝑒𝑎 is Dirichlet(α−𝑖,𝑒𝑎) where 𝛼−𝑖,𝑒𝑎(𝑣) = 𝜌𝑎𝜓𝑎(𝑣 | 𝑦𝑒𝑎) + ∑︁ 𝑖′≠𝑖:𝜆𝑖′=𝑒 𝑧𝑖′𝑎𝟙[𝑥𝑖′𝑎 = 𝑣] for 𝑣 ∈ D𝑎 \\ {𝑦𝑒𝑎}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We can therefore simplify the integral in Equation (S18) with respect to ℍ−𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='𝑒𝑎 as follows: ∫ 𝑃(𝑥𝑖𝑎 | 𝑧𝑖𝑎,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 𝑦𝑒𝑎,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 𝐻𝑒𝑎) 𝑑ℍ−𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='𝑒𝑎 = 𝟙[𝑥𝑖𝑎 = 𝑦𝑒𝑎]1−𝑧𝑖𝑎 ∫ 𝐻𝑒𝑎(𝑥𝑖𝑎)𝑧𝑖𝑎Γ(𝜌𝑎) � 𝑣∈D𝑎\\{𝑦𝑒𝑎} 𝐻𝑒𝑎(𝑣)𝜌𝑎𝜓𝑎(𝑣|𝑦𝑒𝑎)−1 Γ(𝜌𝑎𝜓𝑎(𝑣 | 𝑦𝑒𝑎)) d𝐻𝑒𝑎 = � 𝛼−𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='𝑒𝑎(𝑥𝑖𝑎) � 𝑣 ∈D𝑎\\{𝑦𝑒𝑎 } 𝛼−𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='𝑒𝑎(𝑣),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 𝑧𝑖𝑎 = 1 𝟙[𝑥𝑖𝑎 = 𝑦𝑒𝑎],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 𝑧𝑖𝑎 = 0 (S19) By a similar argument,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' the integral in Equation (S18) with respect to ℍ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='𝑒𝑎 can be simplified to: ∫ 𝑃(𝑥𝑖𝑎 | 𝑧𝑖𝑎,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 𝑦𝑒𝑎,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 𝐻𝑒𝑎) 𝑑ℍ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='𝑒𝑎 = � 𝜓𝑎(𝑥𝑖𝑎 | 𝑦𝑒𝑎),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 𝑧𝑖𝑎 = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 𝟙[𝑥𝑖𝑎 = 𝑦𝑒𝑎],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 𝑧𝑖𝑎 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' (S20) Putting Equations (S19) and (S20) in (S18), then gives: 𝑃(𝜆𝑖 = 𝑒 | Z, X, Y , 𝚲−𝑖, {𝜌𝑎}) ∝ ������� ������� 𝐶 |𝑒|−𝜎 𝛼+𝑁−1 � 𝑎:𝑧𝑖𝑎=1 𝛼−𝑖,𝑒𝑎(𝑥𝑖𝑎) � 𝑣 ∈D𝑎\\{𝑦𝑒𝑎 } 𝛼−𝑖,𝑒𝑎(𝑣), if 𝑒 is instantiated and |𝑒| > 0, 𝐶 𝛼+𝜎𝐸 𝛼+𝑁−1 � 𝑎:𝑧𝑖𝑎=1 � 𝑦∈D𝑎 𝐺𝑎(𝑦)𝜓𝑎(𝑥𝑖𝑎 | 𝑦)𝑧𝑖𝑎𝟙[𝑥𝑖𝑎 = 𝑦]1−𝑧𝑖𝑎, if 𝑒 is “new”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='4 Update for the Ewens-Pitman Parameters In this section, we describe the update for the Ewens-Pitman parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Since the priors on the Ewens-Pitman parameters 𝛼 and 𝜎 are non-conjugate, we cannot perform a direct Gibbs update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Thus, we describe tractable updates which require the introduction of auxiliary variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' The updates (and priors) differ depending on the range of 𝜎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Teh (2006) proposed a scheme for beta/gamma priors when 0 ≤ 𝜎 < 1 and 𝛼 > 0, which is summarized in Section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' In Section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='2 we propose a similar scheme for gamma/shifted negative binomial priors when 𝜎 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 31 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='1 Case 0 ≤ 𝜎 < 1 and 𝛼 > 0 Teh (2006) proposed an auxiliary variable scheme for the regime 0 ≤ 𝜎 < 1 and 𝛼 > 0 such that the priors 𝜎 ∼ Beta � 𝜁 (0), 𝜁 (1)� and 𝛼 ∼ Gamma � 𝜒(0), 𝜒(1)� are conjugate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We provide a summary of the scheme here, but refer the reader to (Teh, 2006) for further details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' The scheme introduces the following sets of auxiliary variables conditional on the two parameters 𝛼 and 𝜎: 𝑤 | 𝑁, 𝛼 ∼ Beta(𝛼 + 1, 𝑁 − 1), 𝑢𝑘 | 𝜎, 𝛼, 𝐸 ∼ Bernoulli � 𝛼 𝛼 + 𝜎𝑘 � , 𝑘 ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' , 𝐸 − 1} 𝑣𝑒 𝑗 | 𝜎, 𝚲 ∼ Bernoulli � 𝑗 − 1 𝑗 − 𝜎 � , ∀𝑒, 𝑗 ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' , 𝑁𝑒 − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' (S21) Here 𝑁𝑒 = |{𝑖 : 𝜆𝑖 = 𝑒}| denotes the number of records linked to entity 𝑒, and 𝐸 = � 𝑒 𝟙[𝑁𝑒 > 1] denotes the number of entities linked to at least one record.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' It follows that the posterior distributions of 𝛼 and 𝜎 conditional on the auxiliary variables and other model parameter are given by: 𝜎 | {𝑢𝑘}, {𝑣𝑒 𝑗}, 𝚲 ∼ Beta�� � 𝜁 (0) + 𝐸−1 ∑︁ 𝑘=1 (1 − 𝑢𝑘), 𝜁 (1) + ∑︁ 𝑒:𝑁𝑒>1 𝑁𝑒−1 ∑︁ 𝑗=1 (1 − 𝑣𝑒 𝑗)�� � , 𝛼 | {𝑢𝑘}, 𝑤, 𝚲 ∼ Gamma � 𝜒(0) + 𝐸−1 ∑︁ 𝑘=1 𝑢𝑘, 𝜒(1) − log 𝑤 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' (S22) Thus, to update 𝛼 and 𝜎, one would first draw auxiliary variables 𝑤, {𝑢𝑘} and {𝑣𝑒 𝑗} conditional on the linkage structure 𝚲 and the old values of 𝛼 and 𝜎 using Equation (S21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Then, conditional on the auxiliary variables and the linkage structure, one would draw new values for 𝛼 and 𝜎 using Equation (S22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='2 Case 𝜎 < 0 and 𝛼 = 𝑚𝜅 for 𝑚 ∈ ℕ We describe an auxiliary variable scheme for updating the Ewens-Pitman parameters in the regime 𝜎 < 0 and 𝛼 = 𝑚𝜅, where 𝑚 ∈ ℕ and 𝜅 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' This scheme is inspired by Teh (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' The likelihood factor associated with the partition of 𝑁 records into 𝐸 entities is as follows (Pitman,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 2006): 𝑃(partition config) = (𝑚)𝐸↓ (𝑚𝜅)𝑁↑ 𝐸 � 𝑒=1 (𝜅)𝑁𝑒↑ = 𝜅𝐸−1(𝑚 − 1)𝐸−1↓ (𝑚𝜅 − 1)𝑁−1↑ 𝐸 � 𝑒=1 (𝜅 − 1)𝑁𝑒−1↑,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' (S23) where “partition config” is a representation of the linkage structure 𝚲 as a partition6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 𝑁𝑒 is the number of records linked to the 𝑒-th entity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' (𝑥)𝑛↑ = �𝑛−1 𝑖=0 (𝑥+𝑖) is the rising factorial,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' and (𝑥)𝑛↓ = �𝑛−1 𝑖=0 (𝑥−𝑖) 6Records 𝑖 and 𝑗 belong to the same subset of the partition if 𝜆𝑖 = 𝜆 𝑗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' and otherwise belong to different subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 32 is the falling factorial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We begin by expressing the denominator in this equation as 1 (𝑚𝜅 − 1)𝑁−1↑ = Γ(𝑚𝜅 + 1) Γ(𝑚𝜅 + 𝑁) = B(𝑚𝜅 + 1, 𝑁 − 1) Γ(𝑁 − 1) = 1 Γ(𝑁 − 1) ∫ 1 0 𝑤𝑚𝜅(1 − 𝑤)𝑁−2 d𝑤, which allows us to introduce the following auxiliary variable: 𝑤 | 𝑚, 𝜅, 𝑁 ∼ Beta(𝑚𝜅 + 1, 𝑁 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' (S24) Expressing each of the latter factors in Equation (S23) as (𝜅 − 1)𝑁𝑒−1↑ = 𝑁𝑒−1 � 𝑗=1 (𝜅 + 𝑗) = 𝑁𝑒−1 � 𝑗=1 ∑︁ 𝑣𝑒 𝑗∈{0,1} 𝜅𝑣𝑒 𝑗 𝑗1−𝑣𝑒 𝑗 permits us to introduce the following additional auxiliary variables: 𝑣𝑒 𝑗 | 𝜅 ∼ Bernoulli � 𝜅 𝜅 + 𝑗 � , ∀𝑒, 𝑗 ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' , 𝑁𝑒 − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' (S25) With this representation, we can place conjugate priors on 𝜅 and 𝑚, namely: 𝜅 ∼ Gamma(𝜒(0), 𝜒(1)) and 𝑚 ∼ NegativeBinomial(𝑟, 𝜈) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' (S26) The distribution on 𝑚 is a shifted negative binomial with support on the positive integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' The parameterization we adopt for the negative binomial is in terms of the number of failures 𝑥 ∈ {0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='} in a sequence of trials before a given number of successes 𝑟 > 0 occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Each trial is an i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' draw from a Bernoulli distribution with success probability 𝜈.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' The density of 𝑥 is given by 𝑃(𝑥 | 𝑟, 𝜈) = (𝑥 + 𝑟 − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' (𝑟 − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='𝑥!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 𝜈𝑟(1 − 𝜈)𝑥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Finally, we combine the priors in Equation (S26) with the likelihood factors to obtain the following posterior distributions for the 𝑚 and 𝜅, conditional on the other model parameters: 𝑚 | 𝑤, 𝜅, 𝚲 ∼ NegBinomial(𝑟 + 𝐸 − 1, 1 − (1 − 𝜈)𝑤𝜅) + 𝐸, 𝜅 | {𝑣𝑒 𝑗}, 𝑤, 𝑚, 𝚲 ∼ Gamma�� � 𝜒(0) + 𝐸 − 1 + 𝐸 ∑︁ 𝑒=1 𝑁𝑒−1 ∑︁ 𝑗=1 𝑣𝑒 𝑗, 𝜒(1) − 𝑚 log 𝑤�� � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' (S27) Thus, to update 𝜅 and 𝑚, one would first draw auxiliary variables 𝑤 and {𝑣𝑒 𝑗} conditional on the linkage structure 𝚲 and the old values of 𝛼 and 𝜎 using Equations (S24) and (S25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Then, conditional on the auxiliary variables and the linkage structure, one would draw new values for 𝜅 and 𝑚 using Equation (S27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 33 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='5 Update for the Distortion Distribution Concentration In this section, we provide the update for the distortion distribution concentration 𝜌𝑎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Since we cannot rely on conjugacy for the update, we propose an auxiliary variable scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' When updating 𝜌𝑎, we condition on the entity attribute values {𝑦𝑒𝑎}𝑒=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='𝐸, the record attribute values {𝑥𝑖𝑎}𝑖=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='𝑁 and the links 𝚲 = {𝜆𝑖}𝑖=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='𝑁.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We collapse the distortion distributions {𝐻𝑒𝑎}𝑒=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' The contribution to the likelihood involving 𝜌𝑎 is: � 𝑒 ∫ � 𝑖:𝜆𝑖=𝑒 𝑃(𝑥𝑖𝑎 | 𝜃𝜍𝑖𝑎, 𝜔𝑖𝑎, 𝑦𝑒𝑎, 𝐻𝑒𝑎)𝑃(𝐻𝑒𝑎 | 𝜌𝑎) d𝐻𝑒𝑎 ∝ � 𝑒 Γ(𝜌𝑎) Γ( ¯𝑛𝑒𝑎(𝑦𝑒𝑎) + 𝜌𝑎) � 𝑣∈V𝑒𝑎 Γ(𝑛𝑒𝑎(𝑣) + 𝜌𝑎𝜓𝑎(𝑣 | 𝑦𝑒𝑎)) Γ(𝜌𝑎𝜓𝑎(𝑣 | 𝑦𝑒𝑎)) = � 𝑒 B(𝜌𝑎, ¯𝑛𝑒𝑎(𝑦𝑒𝑎)) Γ( ¯𝑛𝑒𝑎(𝑦𝑒𝑎)) � 𝑣∈V𝑒𝑎 𝑛𝑒𝑎(𝑣) � 𝑗=1 { 𝑗 − 1 + 𝜌𝑎𝜓𝑎(𝑣 | 𝑦𝑒𝑎)} (S28) where B(·, ·) is the beta function, V𝑒𝑎 := �� 𝑖:𝜆𝑖=𝑒{𝑥𝑖𝑎}� \\ {𝑦𝑒𝑎}, 𝑛𝑒𝑎(𝑣) = � 𝑖:𝜆𝑖=𝑒 𝟙[𝑥𝑖𝑎 = 𝑣] and ¯𝑛𝑒𝑎(𝑣) = � 𝑖:𝜆𝑖=𝑒 𝟙[𝑥𝑖𝑎 ≠ 𝑣].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Expressing the beta function in Equation (S28) as B(𝜌𝑎, ¯𝑛𝑒𝑎(𝑦𝑒𝑎)) = ∫ 1 0 𝑤𝜌𝑎−1 𝑒 (1 − 𝑤𝑒) ¯𝑛𝑒𝑎(𝑦𝑒𝑎)−1 d𝑤𝑒 permits us to introduce the following auxiliary variables: 𝑤𝑒 | 𝜌𝑎, X, Y , 𝚲 ∼ Beta � 𝜌𝑎, ∑︁ 𝑒 ∑︁ 𝑖 𝟙[𝑥𝑖𝑎 ≠ 𝑦𝜆𝑖𝑎] � , for all 𝑒.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We can also express each of the latter factors in Equation (S28) as 𝑗 − 1 + 𝜌𝑎𝜓𝑎(𝑣 | 𝑦𝑒𝑎) = 1 ∑︁ 𝑢𝑒𝑣 𝑗=0 𝜌𝑎𝜓𝑎(𝑣 | 𝑦𝑒𝑎)𝑢𝑒𝑣 𝑗 ( 𝑗 − 1)1−𝑢𝑒𝑣 𝑗 which permits us to introduce the following auxiliary variables: 𝑢𝑒𝑣 𝑗 | 𝜌𝑎, X, Y , 𝚲 ∼ Bernoulli � 𝜌𝑎𝜓𝑎(𝑣 | 𝑦𝑒𝑎) 𝑗 − 1 + 𝜌𝑎𝜓𝑎(𝑣 | 𝑦𝑒𝑎) � (S29) for all 𝑒, 𝑣 ∈ V𝑒𝑎 and 𝑗 ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' , 𝑛𝑒𝑎(𝑣)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Now since the prior on 𝜌𝑎 is Gamma(𝜏(0) 𝑎 , 𝜏(1) 𝑎 ), we obtain the following posterior distribution for 𝜌𝑎 conditional on the other parameters: 𝜌𝑎 | {𝑤𝑒}, {𝑢𝑒𝑣 𝑗}, X, Y , 𝚲 ∼ Gamma�� � 𝜏(0) 𝑎 + ∑︁ 𝑒 ∑︁ 𝑣∈V𝑒𝑎 𝑛𝑒𝑎(𝑣) ∑︁ 𝑗=1 𝑢𝑒𝑣 𝑗, 𝜏(1) 𝑎 − ∑︁ 𝑒 log 𝑤𝑒�� � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' (S30) 34 Thus to update 𝜌𝑎, one would first draw auxiliary variables {𝑤𝑒} and {𝑢𝑒𝑣 𝑗} conditional on the record attributes X, entity attributes Y , linkage structure 𝚲, and the previous value of 𝜌𝑎 using Equations (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='5) and (S29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Then, conditional on the auxiliary variables and X, Y , 𝚲, one would draw a new value for 𝜌𝑎 using Equation (S30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' B Hybrid Distance Measure In this appendix, we describe a hybrid distance measure that is useful for comparing text strings containing multiple tokens (words), where individual tokens may be subject to distortion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We use the measure in this paper for comparing name and address attributes in the cora and rest data sets (see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='1), however it may have wider applications beyond this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Our measure draws inspiration from a hybrid similarity measure proposed by Monge and Elkan (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' However, unlike Monge and Elkan, we attempt to match the tokens in each string while incorporating penalties for tokens that are “missing” in one of the strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Suppose we would like to compare a pair of multi-token strings 𝑥 and 𝑦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' As a running example, we consider 𝑥 = “University of California, San Diego” and 𝑦 = “Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Calif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=', San Diego”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Given a separator character (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=', a space), we can map each string to a set of tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' For example, string 𝑥 from our running example would be mapped to 𝑋 = {“California,”, “Diego”, “of”, “San”, “University”}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Note that we have used capital 𝑋 to denote the token set7 representation of string 𝑥 – a convention we adopt throughout this appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Also note that 𝑋 is a lossy representation of 𝑥, as it discards information about the token order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' This is desirable for our applications to names and addresses8, where permutation of the tokens does not significantly change the meaning of the strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We propose to measure the distance from 𝑥 to 𝑦 via a generalized edit distance on the token sets 𝑋 and 𝑌.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We consider three elementary edit operations: token insertions where a token 𝑏 is appended to the input set;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' token deletions where a token 𝑎 is removed from the input set;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' and token substitutions where a token 𝑎 in the input set is replaced by a token 𝑏 ≠ 𝑎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Each elementary operation takes an input set 𝑄 to an output set 𝑄′, which we write as 𝑄 → 𝑄′, and has an associated cost 𝑐(𝑄 → 𝑄′) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We let 𝑐(𝑄 → 𝑄′) = ���� ���� 𝑑𝑖 distinner(𝜆, 𝑏), if 𝑄 = 𝑄′ \\ {𝑏} (insertion), 𝑑𝑑 distinner(𝑎, 𝜆), if 𝑄 \\ {𝑎} = 𝑄′ (deletion), 𝑑𝑠 distinner(𝑎, 𝑏), if 𝑄 \\ {𝑎} = 𝑄′ \\ {𝑏} (substitution), 7Technically we consider a multi-set, since we allow tokens to appear multiple times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 8Specifically, the title, venue and authors attributes in cora, and the name and addr attributes in rest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 35 where 𝑑𝑖, 𝑑𝑑 and 𝑑𝑠 are non-negative weights;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 𝜆 is the null string;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' and distinner(·, ·) is an inner distance measure on tokens (strings).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We then define the hybrid distance between 𝑥 and 𝑦 as the minimum average cost of transforming 𝑋 into 𝑌 via a sequence of elementary edit operations 𝑇𝑋,𝑌 = (𝑋 → 𝑄1, 𝑄1 → 𝑄2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' , 𝑄𝑙−1 → 𝑌).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Symbolically, we write disthybrid(𝑥, 𝑦) = min 𝑇𝑋,𝑌 1 |𝑇𝑋,𝑌 | ∑︁ (𝑄→𝑄′)∈𝑇𝑋,𝑌 𝑐(𝑄 → 𝑄′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We can compute the hybrid distance using an off-the-shelf linear sum assignment problem (LSAP) solver (Crouse, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' In order to do so, we need to add null string tokens to 𝑋 and 𝑌 to account for all possible insertion and deletion operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Concretely, we add |𝑌| null tokens to 𝑋 to allow for insertions and |𝑋| null tokens to 𝑌 to allow deletions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We then construct a pairwise cost matrix by applying distinner to all pairs of tokens in (the amended) 𝑋 and 𝑌.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' The resulting matrix is then passed to the LSAP solver, which returns the optimal set of edit operations and their cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Returning to our running example, if we set distinner to the Levenshtein distance, the solution to the LSAP is {(“University” ↔ “Univ.”, 5), (“of” ↔ 𝜆, 2), (“California,” ↔ “Calif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=',”, 6), (“San” ↔ “San”, 0), (“Diego” ↔ “Diego”, 0), (𝜆 ↔ 𝜆, 0), (𝜆 ↔ 𝜆, 0), (𝜆 ↔ 𝜆, 0), (𝜆 ↔ 𝜆, 0)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Hence we conclude that disthybrid(𝑥, 𝑦) = 5+2+6+0+0 5 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' This distance reflects the semantic closeness between 𝑥 and 𝑦 better than the Levenshtein distance, which gives a larger value of 14 when evaluated directly on 𝑥 and 𝑦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' C Simulation Study In this appendix, we conduct a simulation study to understand how our model performs in controlled scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Specifically, we simulate entity resolution data sets where we vary the number of records, the level of distortion and the level of duplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Since our model is generative, we could use it to simulate data, however the resulting data would have negligible specification error for our model, which is not realistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We therefore simulate data that is purposefully misspecified for our model by adding additional dependencies between the entities and entity attributes, and by using a different process to generate records from entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We were unable to find an existing data set simulator that generated such data, so we implemented our own.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='1 Data Set Simulator We provide an overview of our simulator, which generates personal records describing a population of households.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' For brevity, we omit low-level details here and refer the reader to the included Python script.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Our simulator operates in two stages: in the first stage it generates a population of households, then in the second stage it iterates over individuals in all households, generating a random number of distorted records for each individual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' By generating households rather than individuals in the first 36 stage, we are able to incorporate additional dependencies between individuals (entities) that are not present in our ER model (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Generating Households.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We now describe how households are generated in the first stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' In our simplified model, a household may be a couple, a single, a couple or single with children, or a group of unrelated adults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Individuals within a household are described by the following attributes: first and last name (first_name and last_name), date of birth (birth_year, birth_month, birth_day), gender and zipcode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' The zipcode is constrained to be the same for all individuals within a household, and the first name is conditioned on the gender, however the other attributes may vary as described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Random values for attributes are generated using the Faker Python library9, which attempts to mimic real-world frequency distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We make the distributions more concentrated for the name and zipcode attributes to ensure the entities are not too unique (otherwise entity resolution would be too easy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We begin by generating the head(s) of the household, which are a male and female couple (for simplicity) or a single male or female.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' If a couple is generated, they have a high chance of sharing the same last name and their birth years are likely not too far apart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Next we randomly decide whether to generate children.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' If children are generated, they share the same last name as the head(s) of the household (the parents) and there is an appropriate gap between their birth year and their parents’ birth year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' If no children are generated, then we randomly decide whether to generate unrelated adults who live with the head(s) of household.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' The unrelated adults are constrained to be of a similar age as the head(s) of the household.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' When simulating the household composition, we attempt to follow aggregate statistics from the Current Population Survey (U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Census Bureau, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Generating Records.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' In the second stage, records are generated for individuals across all households.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We simulate a single database/file with duplicate records by including an individual with probability 𝑝inc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='9 and sampling the number of records according to a Poisson distribution with rate parameter 𝜇, truncated to the interval [1, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Each record is obtained by copying the entity attributes subject to distortion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' This is done by iterating over the attributes in a random order, and deciding whether to activate the distortion process with a probability that varies for each attribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' The distortion process for birth day, birth month, gender and zipcode involves drawing a replacement value according to the distribution used in the first stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' The distortion process for birth year involves adding discrete Gaussian noise to the true birth year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' The distortion process for first and last name may proceed in one of three ways: (1) by making a random typographical error (character insertion, deletion, substitution or transposition);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' (2) by replacing the name with a variant drawn uniformly at random;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' or (3) by generating a replacement according to the distribution used in the first stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Variant names for (2) are sourced from the WeRelate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='org Variant Names Project10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='2 Results We generate 16 data sets using our simulator for each combination of the following variables: 9https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='com/joke2k/faker 10https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='werelate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='org/wiki/WeRelate:Variant_names_project 37 Distortion probability Attribute Low distortion High distortion first_name 10% 40% last_name 10% 40% gender 1% 1% zipcode 5% 10% birth_year 1% 10% birth_month 1% 10% birth_day 1% 10% Table S4: Attribute-level distortion probabilities for two levels of distortion: low and high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Low (μ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='1) Medium (μ=1) High (μ=8) Very high (μ=100) 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='75 Records per entity (cluster size) Relative frequency Figure S5: Distribution of records per entity for each level of duplication: low, medium, high and very high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' The Poisson rate parameter 𝜇 for each level is given in parentheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Number of records.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We consider data sets with 1000 and 10000 records in expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' The number of records is random and depends on the number of individuals and the Poisson rate parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Since the Poisson rate parameter is fixed (see below), we control the number of records by varying the number of individuals generated in the first stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Level of distortion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We consider two levels of record distortion which we refer to as “low” and “high”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' These correspond to different choices for the probabilities of activating the distortion process as detailed in Table S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Level of duplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We consider four levels of duplication which we refer to as “low”, “medium”, “high” and “very high”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' These levels correspond to Poisson rate parameters of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='1, 1, 8 and 100, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' When the duplication is “low” (𝜇 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='1) over 95% of the entities represented in the data only appear once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Whereas when the duplication is “very high” (𝜇 = 100) over 95% of the entities represented in the data appear four times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' The distribution of records per entity is plotted for each level in Figure S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We perform a comparative evaluation of our model, blink, and Sadinle on the 16 simulated data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' The model evaluation procedure is described in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='2 and the blink and Sadinle models are introduced in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' ER evaluation metrics are plotted for each data set in Figure S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We now make several observations about the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 38 First, we observe that our model and blink perform similarly when the duplication level is medium, high or very high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' For these duplication levels, blink has a slight advantage in terms of recall when the distortion level is high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' The largest difference is observed for the medium duplication/high distortion scenario, where the recall for blink is roughly 10 percentage points higher than for our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' This difference is due to the priors placed on the concentration parameters 𝜌𝑎 in our model, which favour high concentrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' This corresponds to a prior belief that distortions occur in the same way, rather than in multiple different ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' However, this is not true for distortions in the simulated data – e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=', an individual whose first name is “JONATHON” may appear in the data with four distinct first names: “JOHN”, “JOJN”, “JONATHON” and “ALEX”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' If we wanted to exploit this knowledge, we could increase 𝜏(0) 𝑎 for our model to favour lower concentrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Second, we observe that our model significantly outperforms blink in terms of precision when the duplication level is low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We believe this is due to the highly informative prior on the linkage structure used in blink – it uses a coupon prior with 𝑚 fixed to the number of records 𝑁 and 𝜅 → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' However our model under the generalized coupon prior selects a value for 𝑚 of approximately 8 × 𝑁 and 𝜅 of approximately 100, which allows it to more accurately model a low duplication scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Thirdly, we observe that Sadinle achieves the lowest F1 score when the duplication level is medium, high or very high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' For these duplication levels, the performance gap in F1 score is largest when the distortion is high – approximately 20 percentage points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' The gap is less significant when the distortion is low – around 5 percentage points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' These differences in F1 score are mainly due to lower recall in most cases – the precision is generally competitive with the other models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Fourthly, we comment on the effect of the dataset size, measured in terms of the expected number of records (1000 or 10000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We observe similar trends for all three models: the precision tends to be larger for the smaller dataset, while the recall tends to be larger for the larger dataset (however there are exceptions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' The difference in performance is most pronounced for the low duplication setting, where the precision of blink and Sadinle drops considerably for the larger dataset, and the recall of Sadinle drops also drops considerably for the larger dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' D Study of Linkage Structure Priors and the Distortion Model In this appendix, we provide additional results for the study of linkage structure priors and distortion models presented in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Our goal is to study the impact of the modeling contributions independently, to determine whether each contribution is beneficial in its own right, and/or whether one contribution is more beneficial than the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Recall from Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='3 that we considered four parameter regimes for the linkage structure priors – PY, Ewens, GenCoupon and Coupon – and two distortion models – Ours as proposed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='1 and blink as proposed by Steorts (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Thus, there are eight model variants to test – one for each linkage structure prior and distortion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Figure S7 presents pairwise evaluation metrics (F1 score, precision and recall) for the eight model variants and four data sets in a single plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We interpret the results for each modeling contribution below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 39 F1 score L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Dup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' F1 score H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Dup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' F1 score L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Dup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' F1 score H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Dup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' F1 score L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Dup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' F1 score H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Dup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' F1 score L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Dup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' F1 score H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Dup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Recall L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Dup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Recall H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Dup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Recall L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Dup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Recall H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Dup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Recall L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Dup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Recall H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Dup.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='985 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='990 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='995 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='000 Model Measure value Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' number of records 1000 10000 Figure S6: Posterior evaluation metrics for our model, blink, and Sadinle when fitted on simulated datasets with varying levels of distortion and duplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Low and high distortion levels are abbreviated as “L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Dist.” and “H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Dist.” respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Low, medium, high and very high duplication levels are abbreviated as “L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Dup.”, “M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Dup.”, “H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Dup.” and “V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Dup.” respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' The expected number of records (1000 or 10000) is denoted by the color and shape of the markers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' A point estimate for each evaluation metric is reported based on the median and 95% equi-tailed credible interval are represented by intervals around the point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 40 rest Precision rest Recall rest F1 score cora Precision cora Recall cora F1 score nltcs Precision nltcs Recall nltcs F1 score RLdata Precision RLdata Recall RLdata F1 score PY Ewens GenCoupon Coupon PY Ewens GenCoupon Coupon PY Ewens GenCoupon Coupon 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='00 0.' metadata={'source': 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Figure S7: Evaluation of ER quality as a function of the linkage structure prior (plotted on the 𝑥-axis) and distortion model (indicated by the line color).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Three pairwise evaluation measures are shown (grouped by column) for four data sets (grouped by row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 41 100 1000 10000 1e−04 1e−03 1e−02 1e−01 100 1000 10000 1 10 100 1000 10000 PY Ewens GenCoupon RLdata nltcs cora rest RLdata nltcs cora rest Data set RLdata nltcs cora rest alpha sigma alpha kappa m Figure S8: Posterior Ewens-Pitman parameters for three regimes: PY, Ewens and GenCoupon under our distortion model (Ours).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Note that the values of the parameters are presented on log-scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Linkage Structure Prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' While we discuss the effect of the linkage structure prior in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='3, we only present results for our distortion model (Ours) due to space constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' In Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='3, we draw two main conclusions from the results in Table 2 which are replicated in Figure S7 (represented by circular vermilion markers): 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Our proposal to place vague hyperpriors on the EP parameters (for PY, Ewens and GenCoupon) improves robustness and yields the highest ER accuracy for three of data sets, as measured by pairwise F1 score (nltcs is the exception).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We observe significantly lower F1 scores when hyperpriors are not used (see Coupon in Figure S7), particularly for cora and RLdata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Figure S8 provides further justification for this argument, as it shows vastly different values of the EP parameters are selected for each data set, facilitated by the vague hyperpriors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Our inferences are relatively insensitive to the EP parameter regime (PY, Ewens or GenCoupon) despite the fact that each regime is known to exhibit distinct asymptotic behavior (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Figure S7 shows that these conclusions also hold for the blink distortion model (represented by triangular teal markers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We expect the competitive performance for nltcs under the Coupon linkage prior may be a coincidence, as the population size under the prior is 3,387, which happens to be very close to the true value of 3,307 (see Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Distortion Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' In Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='3, we discuss the effect of the distortion prior under the GenCoupon linkage structure prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We now extend the discussion to include results for the three other linkage structure priors (PY, Ewens and Coupon), as presented in Figure S7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We find that our distortion model achieves the highest F1 score for all but one of the data sets and linkage structure priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' The exception is for cora under the Coupon linkage structure prior, where 42 the blink distortion model has a slight edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' An explanation for the improved performance of our model is given in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='3, which we summarize here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' The blink distortion model is susceptible to entering a high distortion mode, particularly for attributes with non-constant distance measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' This is because it allows a record attribute value to be marked as “distorted” even if it is not actually distorted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Our model corrects this inconsistency, and in doing so appears to be more robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' In general, we expect the blink distortion model to result in over-linkage (high recall, low precision), while our model is expected to be more balanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Figure S7 supports this argument, with the difference being most apparent for RLdata, where we see a difference of ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='7 in the F1 score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' E Further Details of Experimental Setup In this appendix, we provide further details about the experiments presented in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Implementation and Hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' All experiments were conducted in R version 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='4, running on a local server fitted with two 28-core Intel Xeon Platinum 8180M CPUs and 12 TB of RAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='11 We developed an open-source R package called exchanger12 which implements variants of our model (under different linkage structure priors and distortion models) in addition to the blink model (Steorts, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Since an implementation of the model proposed by Sadinle (2014) was not publicly available, we developed our own which we released as an open-source R package called BDD13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' For efficiency reasons, we implemented inference for all models in C++ using the Rcpp interface (Eddelbuettel and François, 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' The data set simulator described in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='1 was implemented as a Python script.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' A Pipfile is provided to specify the dependencies used when running the script.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Hyperparameter Settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We followed the recommendations in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='2 when setting hyperparameters for our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' When setting hyperparameters for the two baseline models, we attempted to follow the recommendations of the authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' For blink, we set 𝑚 = 𝑁 for the coupon-collector’s prior and 𝛽(0) 𝑠𝑎 = 𝑁/1000 and 𝛽(1) 𝑠𝑎 = 𝑁/10 for the Beta prior on the distortion probabilities (here 𝑁 is the total number of records).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' For Sadinle, we set the agreement levels by inspecting the distribution of distances for each attribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We used truncated uniform priors on the 𝑚-probabilities and a uniform prior on the 𝑢-probabilities, as recommended by the author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We set the lower truncation points for the 𝑚-probabilities to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='95, based on tuning experiments presented in Appendix F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Initialization and MCMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' For our model and blink, we initialized the linkage structure 𝚲, entity attributes Y and distortion indicators Z by linking each record to a unique entity and copying the record attributes into the entity attributes, assuming no distortion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' The distortion probabilities 𝚯 and entity attributes distributions G were initialized by drawing from their conditional distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' The Ewens-Pitman parameters and distortion distribution concentration parameters were initialized using their prior means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 11R scripts are published at github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='com/cleanzr/exchanger-experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 12Package source code published at github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='com/cleanzr/exchanger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 13Package source code published at github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='com/cleanzr/BDD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 43 A similar initialization was used for the Sadinle model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We assigned each record to a unique entity (cluster).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' The 𝑚- and 𝑢-probabilities were initialized by drawing from their conditional distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' When fitting each model, we ran Markov chain Monte Carlo (MCMC) for 2 × 105 iterations, discarding the first 105 iterations as burn-in, and applying thinning with an interval of 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='14 This produced 104 approximate posterior samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' F Tuning Hyperparameters for Sadinle (2014) Our aim in this appendix is to determine reasonable values for the hyperparameters in the entity resolution model by Sadinle (2014), which we refer to as Sadinle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We assume the distance functions (used to compare attributes) and agreement levels (mappings from real-valued distances to discrete levels) are fixed, and that flat priors are used, as recommended by Sadinle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Given these assumptions, the only hyperparameters that remain unspecified are the lower truncation points for the 𝑚-probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' The 𝑚-probabilities are a set of parameters {𝑚𝑎𝑙} where 𝑚𝑎𝑙 is the probability that a pair of records referring to the same entity agree at level 𝑙 on attribute 𝑎, given they do not agree at levels 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' , 𝑙 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Sadinle recommends using truncated flat priors on 𝑚𝑎𝑙, so that the allowed values lie in the interval [𝜆𝑎𝑙, 1], where 𝜆𝑎𝑙 ∈ [0, 1] is a hyperparameter typically close to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' More specifically, he recommends setting 𝜆𝑎𝑙 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='95 if attribute 𝑎 is a “nearly-accurate” quasi-identifier and 𝜆𝑎𝑙 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='85 if attribute 𝑎 is an “inaccurate” quasi-identifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Since it is not clear which setting for 𝜆𝑎𝑙 is best for our data sets, we run the experiments described in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='4 for four different values: 𝜆𝑎𝑙 = 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='85, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' When setting 𝜆𝑎𝑙, we use the same value for all attributes 𝑎 and agreement levels 𝑙 for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' The results are reported in Figure S9 and Table S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Figure S9 plots the posterior values of the 𝑚-probabilities (on the 𝑥-axis) for each truncation point 𝜆𝑎𝑙 (corresponding to the horizontal panels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We observe that the posterior values of 𝑚𝑎𝑙 tend to be close to 𝜆𝑎𝑙, especially for agreement level 𝑙 = 0 and 𝜆𝑎𝑙 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' This suggests that the model favors small values of 𝑚𝑎𝑙, despite the fact that 𝑚𝑎𝑙 is expected to be close to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Consequently, the model has a tendency to “over-link”—linking records that do not refer to the same entity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Understanding why the model exhibits this behavior would require further exploration and is beyond the scope of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' However, we speculate that it may be related to the use of flat priors, or known stability issues with Fellegi-Sunter-type models (Goldstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' As a result, we conclude that the posterior value of 𝑚𝑎𝑙 is highly sensitive to the choice of 𝜆𝑎𝑙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Table S5 shows the impact of the truncation point 𝜆𝑎𝑙 on entity resolution performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' It shows that the performance is relatively stable for cora and rest as a function of 𝜆𝑎𝑠, despite the fact that there is some variation in the 𝑚-probabilities (as shown in Figure S9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' The reason for the stability may be related to the blocking scheme used for these data sets, which rules out a relatively large number of potential links, thereby guarding against over-linkage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' On the other hand, the performance for RLdata and nltcs is far less stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We find that the precision drops considerably as 𝜆𝑎𝑙 is reduced, 14The chain was slower to converge for the cora data set, so we increased the number of iterations to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='5 × 105 and the burn-in interval to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='5 × 105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 44 λ = 0 λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='5 λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='85 λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='95 cora nltcs rest RLdata 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='9 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='99 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='00 year venue title authors SEX REGOFF DOB_YEAR DOB_MONTH DOB_DAY type name city addr lname_c1 fname_c1 by bm bd m value Attribute Agreement level 1 2 3 4 Figure S9: Posterior estimates of the 𝑚-probabilities (plotted on the 𝑥-axis) in the Sadinle model as a function of the data set (vertical panels), attribute (𝑦-axis), agreement level (color/marker) and lower truncation point (horizontal panels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Point estimates are shown based on the median, along with 95% equi-tailed credible intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' The model has a tendency to select small values for the 𝑚-probabilities, close to the lower truncation point, especially for agreement level 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 45 Evaluation metric Data set Truncation point Precision Recall F1 score RLdata 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='068 (0.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='756) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='993 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='985, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='000) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='603 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='598, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='607) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='750 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='744, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='756) Table S5: Posterior performance of the Sadinle model as a function of the lower truncation point 𝜆𝑎𝑙 on the 𝑚-probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' A point estimate for each evaluation metric is reported based on the median, along with a 95% equi-tailed credible interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' while the recall remains relatively stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' This is a sign of over-linkage, which is expected since the posterior 𝑚-probabilities are significantly smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Based on these results, we set 𝜆𝑎𝑙 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='95 as the default value for our other experiments since it seems to achieve balanced performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 46 G MCMC Diagnostics G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='1 Study of Linkage Structure Priors Here we present convergence diagnostics for the models fitted in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We present Geweke diagnostic plots and trace plots for a selection of model variables for each data set, linkage structure prior and distortion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Each pair of plots is preceded by a title of the form “Data set | Linkage structure prior | Distortion model”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' The Geweke diagnostic plot (on the left) depicts a Z-score on the x-axis for each variable on the y-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' The Z-score tests for equality of the means of the first 10% and final 50% of the Markov chain, and is typically expected to be in the range [−2, 2] (Geweke, 1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' The trace plot (on the right) depicts the value of variables (labeled in the right panel) for each step in the chain (on the x-axis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Note that variable 𝐸 denotes the number of instantiated entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We replace integer indices for the attributes by named indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' For instance, 𝜃0,city refers to the distortion probability in source 0 of the attribute called “city”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' nltcs | PY | Ours α ρDOB_DAY ρDOB_MONTH ρDOB_YEAR ρREGOFF ρSEX θ0, DOB_DAY θ0, DOB_MONTH θ0, DOB_YEAR θ0, REGOFF θ0, SEX d E −2 −1 0 1 2 Geweke diagnostic Variable E d θ0, SEX θ0, REGOFF θ0, DOB_YEAR θ0, DOB_MONTH θ0, DOB_DAY ρSEX ρREGOFF ρDOB_YEAR ρDOB_MONTH ρDOB_DAY α 100000 125000 150000 175000 200000 3350 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='008 0e+00 5e+04 1e+05 0e+00 5e+04 1e+05 0e+00 5e+04 1e+05 0e+00 5e+04 1e+05 0 50000 3600 4000 Iteration Value 47 nltcs | Ewens | Ours α ρDOB_DAY ρDOB_MONTH ρDOB_YEAR ρREGOFF ρSEX θ0, DOB_DAY θ0, DOB_MONTH θ0, DOB_YEAR θ0, REGOFF θ0, SEX E −2 −1 0 1 2 Geweke diagnostic Variable E θ0, SEX θ0, REGOFF θ0, DOB_YEAR θ0, DOB_MONTH θ0, DOB_DAY ρSEX ρREGOFF ρDOB_YEAR ρDOB_MONTH ρDOB_DAY α 100000 125000 150000 175000 200000 3350 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='005 0e+00 5e+04 1e+05 0 50000 0e+00 5e+04 1e+05 0 40000 80000 0 50000 3750 Iteration Value nltcs | GenCoupon | Ours κ ρDOB_DAY ρDOB_MONTH ρDOB_YEAR ρREGOFF ρSEX θ0, DOB_DAY θ0, DOB_MONTH θ0, DOB_YEAR θ0, REGOFF θ0, SEX E m −2 −1 0 1 2 Geweke diagnostic Variable E m θ0, SEX θ0, REGOFF θ0, DOB_YEAR θ0, DOB_MONTH θ0, DOB_DAY ρSEX ρREGOFF ρDOB_YEAR ρDOB_MONTH ρDOB_DAY κ 100000 125000 150000 175000 200000 3250 5000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='010 0 50000 0 30000 0 50000 0e+00 5e+04 1e+05 0 50000 400 Iteration Value 48 nltcs | Coupon | Ours ρDOB_DAY ρDOB_MONTH ρDOB_YEAR ρREGOFF ρSEX θ0, DOB_DAY θ0, DOB_MONTH θ0, DOB_YEAR θ0, REGOFF θ0, SEX E −2 −1 0 1 2 Geweke diagnostic Variable E θ0, SEX θ0, REGOFF θ0, DOB_YEAR θ0, DOB_MONTH θ0, DOB_DAY ρSEX ρREGOFF ρDOB_YEAR ρDOB_MONTH ρDOB_DAY 100000 125000 150000 175000 200000 3300 3350 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='010 0e+00 5e+04 1e+05 0e+00 5e+04 1e+05 0e+00 5e+04 1e+05 0 50000 0 50000 Iteration Value nltcs | PY | blink α θ0, DOB_DAY θ0, DOB_MONTH θ0, DOB_YEAR θ0, REGOFF θ0, SEX d E −2 −1 0 1 2 Geweke diagnostic Variable E d θ0, SEX θ0, REGOFF θ0, DOB_YEAR θ0, DOB_MONTH θ0, DOB_DAY α 100000 125000 150000 175000 200000 3300 3350 3400 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='005 3750 Iteration Value nltcs | Ewens | blink α θ0, DOB_DAY θ0, DOB_MONTH θ0, DOB_YEAR θ0, REGOFF θ0, SEX E −2 −1 0 1 2 Geweke diagnostic Variable E θ0, SEX θ0, REGOFF θ0, DOB_YEAR θ0, DOB_MONTH θ0, DOB_DAY α 100000 125000 150000 175000 200000 3350 3400 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='005 3750 4250 Iteration Value 49 nltcs | GenCoupon | blink κ θ0, DOB_DAY θ0, DOB_MONTH θ0, DOB_YEAR θ0, REGOFF θ0, SEX E m −2 −1 0 1 2 Geweke diagnostic Variable E m θ0, SEX θ0, REGOFF θ0, DOB_YEAR θ0, DOB_MONTH θ0, DOB_DAY κ 100000 125000 150000 175000 200000 3200 3300 4800 5200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='02 0 500 1000 Iteration Value nltcs | Coupon | blink θ0, DOB_DAY θ0, DOB_MONTH θ0, DOB_YEAR θ0, REGOFF θ0, SEX E −2 −1 0 1 2 Geweke diagnostic Variable E θ0, SEX θ0, REGOFF θ0, DOB_YEAR θ0, DOB_MONTH θ0, DOB_DAY 100000 125000 150000 175000 200000 3275 3325 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='010 Iteration Value RLdata | PY | Ours α ρbd ρbm ρby ρfname_c1 ρlname_c1 θ0, bd θ0, bm θ0, by θ0, fname_c1 θ0, lname_c1 d E −2 −1 0 1 2 Geweke diagnostic Variable E d θ0, lname_c1 θ0, fname_c1 θ0, by θ0, bm θ0, bd ρlname_c1 ρfname_c1 ρby ρbm ρbd α 100000 125000 150000 175000 200000 8950 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='036 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='044 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='045 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='055 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='065 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='004 0 50000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='00010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='00015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='00020 4e−05 7e−05 4e−05 8e−05 9000 12000 15000 Iteration Value 50 RLdata | Ewens | Ours α ρbd ρbm ρby ρfname_c1 ρlname_c1 θ0, bd θ0, bm θ0, by θ0, fname_c1 θ0, lname_c1 E −2 −1 0 1 2 Geweke diagnostic Variable E θ0, lname_c1 θ0, fname_c1 θ0, by θ0, bm θ0, bd ρlname_c1 ρfname_c1 ρby ρbm ρbd α 100000 125000 150000 175000 200000 8900 8950 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='040 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='045 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='045 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='055 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='065 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='005 0 40000 80000 120000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='000075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='000125 3e−05 6e−05 2e−05 6e−05 27000 30000 Iteration Value RLdata | GenCoupon | Ours κ ρbd ρbm ρby ρfname_c1 ρlname_c1 θ0, bd θ0, bm θ0, by θ0, fname_c1 θ0, lname_c1 E m −2 −1 0 1 2 Geweke diagnostic Variable E m θ0, lname_c1 θ0, fname_c1 θ0, by θ0, bm θ0, bd ρlname_c1 ρfname_c1 ρby ρbm ρbd κ 100000 125000 150000 175000 200000 8940 8980 40000 45000 50000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='040 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='045 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='055 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='004 0 40000 80000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='00012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='00016 5e−05 4e−05 8e−05 0 250 Iteration Value 51 RLdata | Coupon | Ours ρbd ρbm ρby ρfname_c1 ρlname_c1 θ0, bd θ0, bm θ0, by θ0, fname_c1 θ0, lname_c1 E −2 0 2 Geweke diagnostic Variable E θ0, lname_c1 θ0, fname_c1 θ0, by θ0, bm θ0, bd ρlname_c1 ρfname_c1 ρby ρbm ρbd 100000 125000 150000 175000 200000 7700 7750 7800 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='0025 0e+00 5e+04 1e+05 2e−05 3e−05 4e−05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='0e−06 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='0e−06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='2e−05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='0e−05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='5e−05 Iteration Value RLdata | PY | blink α θ0, bd θ0, bm θ0, by θ0, fname_c1 θ0, lname_c1 d E −2 −1 0 1 2 Geweke diagnostic Variable E d θ0, lname_c1 θ0, fname_c1 θ0, by θ0, bm θ0, bd α 100000 125000 150000 175000 200000 6200 6300 6400 6500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='990 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='995 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='99 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='525 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='550 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='475 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='525 6750 7250 Iteration Value RLdata | Ewens | blink α θ0, bd θ0, bm θ0, by θ0, fname_c1 θ0, lname_c1 E −2 −1 0 1 2 Geweke diagnostic Variable E θ0, lname_c1 θ0, fname_c1 θ0, by θ0, bm θ0, bd α 100000 125000 150000 175000 200000 6300 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='99 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='99 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='475 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='525 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='51 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='54 7000 Iteration Value 52 RLdata | GenCoupon | blink κ θ0, bd θ0, bm θ0, by θ0, fname_c1 θ0, lname_c1 E m −2 −1 0 1 2 Geweke diagnostic Variable E m θ0, lname_c1 θ0, fname_c1 θ0, by θ0, bm θ0, bd κ 100000 125000 150000 175000 200000 6200 6300 6400 6500 10000 11000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='985 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='990 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='995 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='99 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='52 0 400 Iteration Value RLdata | Coupon | blink θ0, bd θ0, bm θ0, by θ0, fname_c1 θ0, lname_c1 E −2 −1 0 1 2 Geweke diagnostic Variable E θ0, lname_c1 θ0, fname_c1 θ0, by θ0, bm θ0, bd 100000 125000 150000 175000 200000 6250 6350 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='99 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='985 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='990 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='995 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='525 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='550 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='52 Iteration Value cora | PY | Ours α ρauthors ρtitle ρvenue ρyear θ0, authors θ0, title θ0, venue θ0, year d E −15 −10 −5 0 5 Geweke diagnostic Variable E d θ0, year θ0, venue θ0, title θ0, authors ρyear ρvenue ρtitle ρauthors α 150000 175000 200000 225000 250000 165 175 185 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='0025 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='5 0 50000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='0 50 70 Iteration Value 53 cora | Ewens | Ours α ρauthors ρtitle ρvenue ρyear θ0, authors θ0, title θ0, venue θ0, year E −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='0 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='0 Geweke diagnostic Variable E θ0, year θ0, venue θ0, title θ0, authors ρyear ρvenue ρtitle ρauthors α 150000 175000 200000 225000 250000 165 175 185 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='003 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='5 0 50000 1 40 60 Iteration Value cora | GenCoupon | Ours κ ρauthors ρtitle ρvenue ρyear θ0, authors θ0, title θ0, venue θ0, year E m −2 0 2 Geweke diagnostic Variable E m θ0, year θ0, venue θ0, title θ0, authors ρyear ρvenue ρtitle ρauthors κ 150000 175000 200000 225000 250000 170 180 200 300 400 500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='002 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='5 0 40000 80000 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='0 Iteration Value cora | Coupon | Ours ρauthors ρtitle ρvenue ρyear θ0, authors θ0, title θ0, venue θ0, year E −3 −2 −1 0 1 2 Geweke diagnostic Variable E θ0, year θ0, venue θ0, title θ0, authors ρyear ρvenue ρtitle ρauthors 150000 175000 200000 225000 250000 390 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='4 0 50000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='5 0 50000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='0015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='0020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='0025 Iteration Value 54 cora | PY | blink α θ0, authors θ0, title θ0, venue θ0, year d E −2 −1 0 1 2 Geweke diagnostic Variable E d θ0, year θ0, venue θ0, title θ0, authors α 150000 175000 200000 225000 250000 175 180 185 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='99 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='99 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='00 40 60 Iteration Value cora | Ewens | blink α θ0, authors θ0, title θ0, venue θ0, year E −2 −1 0 1 2 Geweke diagnostic Variable E θ0, year θ0, venue θ0, title θ0, authors α 150000 175000 200000 225000 250000 175 180 185 190 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='99 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='92 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='96 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='99 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='00 60 Iteration Value cora | GenCoupon | blink κ θ0, authors θ0, title θ0, venue θ0, year E m −3 −2 −1 0 1 2 Geweke diagnostic Variable E m θ0, year θ0, venue θ0, title θ0, authors κ 150000 175000 200000 225000 250000 180 185 200 250 300 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='99 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='92 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='96 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='99 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='00 1 2 Iteration Value cora | Coupon | blink θ0, authors θ0, title θ0, venue θ0, year E −2 −1 0 1 2 Geweke diagnostic Variable E θ0, year θ0, venue θ0, title θ0, authors 150000 175000 200000 225000 250000 280 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='99 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='92 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='96 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='99 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='00 Iteration Value 55 rest | PY | Ours α ρaddr ρcity ρname ρtype θ0, addr θ0, city θ0, name θ0, type d E −2 −1 0 1 2 Geweke diagnostic Variable E d θ0, type θ0, name θ0, city θ0, addr ρtype ρname ρcity ρaddr α 100000 125000 150000 175000 200000 740 750 760 770 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='00010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='00015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='00020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='00025 0e+00 5e+04 1e+05 5e−05 1e−04 0e+00 5e+04 1e+05 500 1500 Iteration Value rest | Ewens | Ours α ρaddr ρcity ρname ρtype θ0, addr θ0, city θ0, name θ0, type E −2 0 2 Geweke diagnostic Variable E θ0, type θ0, name θ0, city θ0, addr ρtype ρname ρcity ρaddr α 100000 125000 150000 175000 200000 740 750 760 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='00015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='00020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='00025 0e+00 5e+04 1e+05 5e−05 1e−04 0 50000 1500 2000 2500 Iteration Value rest | GenCoupon | Ours κ ρaddr ρcity ρname ρtype θ0, addr θ0, city θ0, name θ0, type E m −2 −1 0 1 2 Geweke diagnostic Variable E m θ0, type θ0, name θ0, city θ0, addr ρtype ρname ρcity ρaddr κ 100000 125000 150000 175000 200000 740 750 760 2500 3500 4500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='6 2e−04 0e+00 5e+04 1e+05 5e−05 1e−04 0e+00 5e+04 1e+05 0 400 Iteration Value 56 rest | Coupon | Ours ρaddr ρcity ρname ρtype θ0, addr θ0, city θ0, name θ0, type E −2 −1 0 1 2 Geweke diagnostic Variable E θ0, type θ0, name θ0, city θ0, addr ρtype ρname ρcity ρaddr 100000 125000 150000 175000 200000 710 720 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='875 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='925 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='7 1e−04 2e−04 3e−04 0e+00 5e+04 1e+05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='00010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='00015 0e+00 5e+04 1e+05 Iteration Value rest | PY | blink α θ0, addr θ0, city θ0, name θ0, type d E −2 −1 0 1 2 Geweke diagnostic Variable E d θ0, type θ0, name θ0, city θ0, addr α 100000 125000 150000 175000 200000 740 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='98 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='00 1000 2000 Iteration Value rest | Ewens | blink α θ0, addr θ0, city θ0, name θ0, type E −2 −1 0 1 2 Geweke diagnostic Variable E θ0, type θ0, name θ0, city θ0, addr α 100000 125000 150000 175000 200000 740 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='950 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='975 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='000 1500 2000 2500 Iteration Value rest | GenCoupon | blink κ θ0, addr θ0, city θ0, name θ0, type E m −2 −1 0 1 2 Geweke diagnostic Variable E m θ0, type θ0, name θ0, city θ0, addr κ 100000 125000 150000 175000 200000 740 750 2000 3000 4000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='950 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='975 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='000 0 300 Iteration Value 57 rest | Coupon | blink θ0, addr θ0, city θ0, name θ0, type E −2 −1 0 1 2 Geweke diagnostic Variable E θ0, type θ0, name θ0, city θ0, addr 100000 125000 150000 175000 200000 700 710 720 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='950 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='975 Iteration Value G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='2 Comparison with Baseline Models Here we present convergence diagnostics for the models fitted in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We present Geweke diagnostic plots and trace plots for a selection of model variables for each data set and model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Each pair of plots is preceded by a title of the form “Data set | Model”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' The Geweke diagnostic plot (on the left) depicts a Z-score on the x-axis for each variable on the y-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' The Z-score tests for equality of the means of the first 10% and final 50% of the Markov chain, and is typically expected to be in the range [−2, 2] (Geweke, 1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' The trace plot (on the right) depicts the value of variables (labeled in the right panel) for each step in the chain (on the x-axis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' Note that variable 𝐸 denotes the number of instantiated entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' We replace integer indices for the attributes by named indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' For instance, 𝜃0,city refers to the distortion probability in source 0 of the attribute called “city”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' nltcs | blink θ0, DOB_DAY θ0, DOB_MONTH θ0, DOB_YEAR θ0, REGOFF θ0, SEX E −2 −1 0 1 2 Geweke diagnostic Variable E θ0, SEX θ0, REGOFF θ0, DOB_YEAR θ0, DOB_MONTH θ0, DOB_DAY 100000 125000 150000 175000 200000 3300 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='010 Iteration Value RLdata | blink θ0, bd θ0, bm θ0, by θ0, fname_c1 θ0, lname_c1 E −2 −1 0 1 2 Geweke diagnostic Variable E θ0, lname_c1 θ0, fname_c1 θ0, by θ0, bm θ0, bd 100000 125000 150000 175000 200000 7350 7450 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='300 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='325 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='350 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='250 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='275 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='32 Iteration Value 58 cora | blink θ0, authors θ0, title θ0, venue θ0, year E −2 −1 0 1 2 Geweke diagnostic Variable E θ0, year θ0, venue θ0, title θ0, authors 100000 125000 150000 175000 200000 300 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='80 Iteration Value rest | blink θ0, addr θ0, city θ0, name θ0, type E −2 −1 0 1 2 Geweke diagnostic Variable E θ0, type θ0, name θ0, city θ0, addr 100000 125000 150000 175000 200000 710 720 730 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='7 Iteration Value nltcs | Ours κ ρDOB_DAY ρDOB_MONTH ρDOB_YEAR ρREGOFF ρSEX θ0, DOB_DAY θ0, DOB_MONTH θ0, DOB_YEAR θ0, REGOFF θ0, SEX E m −2 −1 0 1 2 Geweke diagnostic Variable E m θ0, SEX θ0, REGOFF θ0, DOB_YEAR θ0, DOB_MONTH θ0, DOB_DAY ρSEX ρREGOFF ρDOB_YEAR ρDOB_MONTH ρDOB_DAY κ 100000 125000 150000 175000 200000 3250 5000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='010 0 50000 0 30000 0 50000 0e+00 5e+04 1e+05 0 50000 400 Iteration Value 59 RLdata | Ours κ ρbd ρbm ρby ρfname_c1 ρlname_c1 θ0, bd θ0, bm θ0, by θ0, fname_c1 θ0, lname_c1 E m −2 −1 0 1 2 Geweke diagnostic Variable E m θ0, lname_c1 θ0, fname_c1 θ0, by θ0, bm θ0, bd ρlname_c1 ρfname_c1 ρby ρbm ρbd κ 100000 125000 150000 175000 200000 8940 8980 40000 45000 50000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='040 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='045 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='055 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='004 0 40000 80000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='00012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='00016 5e−05 4e−05 8e−05 0 250 Iteration Value cora | Ours κ ρauthors ρtitle ρvenue ρyear θ0, authors θ0, title θ0, venue θ0, year E m −2 0 2 Geweke diagnostic Variable E m θ0, year θ0, venue θ0, title θ0, authors ρyear ρvenue ρtitle ρauthors κ 150000 175000 200000 225000 250000 170 180 200 300 400 500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='002 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='5 0 40000 80000 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='0 Iteration Value 60 rest | Ours κ ρaddr ρcity ρname ρtype θ0, addr θ0, city θ0, name θ0, type E m −2 −1 0 1 2 Geweke diagnostic Variable E m θ0, type θ0, name θ0, city θ0, addr ρtype ρname ρcity ρaddr κ 100000 125000 150000 175000 200000 740 750 760 2500 3500 4500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='6 2e−04 0e+00 5e+04 1e+05 5e−05 1e−04 0e+00 5e+04 1e+05 0 400 Iteration Value nltcs | Sadinle E mDOB_DAY, 1 mDOB_MONTH, 1 mDOB_YEAR, 1 mREGOFF, 1 mSEX, 1 uDOB_DAY, 1 uDOB_MONTH, 1 uDOB_YEAR, 1 uREGOFF, 1 uSEX, 1 −2 −1 0 1 2 Geweke diagnostic Variable E uSEX, 1 uREGOFF, 1 uDOB_YEAR, 1 uDOB_MONTH, 1 uDOB_DAY, 1 mSEX, 1 mREGOFF, 1 mDOB_YEAR, 1 mDOB_MONTH, 1 mDOB_DAY, 1 100000 125000 150000 175000 200000 2040 2080 2120 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='970 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='971 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='900 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='902 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='2575 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='2600 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='263 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='266 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='269 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='6350 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='6375 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='6400 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='9500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='9501 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='9502 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='9500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='9501 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='998 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='999 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='9990 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='9995 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='0000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='95000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content='95005 Iteration Value 61 RLdata | Sadinle E mbd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 1 mbm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 1 mby,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 1 mfname_c1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 1 mfname_c1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 2 mfname_c1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 3 mfname_c1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 4 mlname_c1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 1 mlname_c1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 2 mlname_c1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 3 mlname_c1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 4 ubd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 1 ubm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 1 uby,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 1 ufname_c1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 1 ufname_c1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 2 ufname_c1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 3 ufname_c1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 4 ulname_c1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 1 ulname_c1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 2 ulname_c1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 3 ulname_c1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 4 −2 −1 0 1 2 Geweke diagnostic Variable E ulname_c1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 4 ulname_c1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 3 ulname_c1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 2 ulname_c1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 1 ufname_c1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 4 ufname_c1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 3 ufname_c1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 2 ufname_c1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 1 uby,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 1 ubm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfKQOD/content/2301.02962v1.pdf'} +page_content=' 1 ubd,' 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b/adE5T4oBgHgl3EQfeg9M/content/tmp_files/2301.05619v1.pdf.txt @@ -0,0 +1,945 @@ +Understanding and improving social factors in education: a +computational social science approach1 +Nabeel Gillani1 (ORCID), Rebecca Eynon2 (ORCID) +1College of Arts, Media and Design and D’Amore-McKim School of Business, +Northeastern University, Boston, MA, USA +2Oxford Internet Institute and Department of Education, University of Oxford, +Oxford, UK +Abstract +Over the past decade, an explosion in the availability of education-related datasets has enabled new +computational research in education. Much of this work has investigated digital traces of online +learners in order to better understand and optimize their cognitive learning processes. Yet cognitive +learning on digital platforms does not equal education. Instead, education is an inherently social, +cultural, economic, and political process manifesting in physical spaces, and educational outcomes +are influenced by many factors that precede and shape the cognitive learning process. Many of +these are social factors like children’s connections to schools (including teachers, counselors, and role +models), parents and families, and the broader neighborhoods in which they live. In this article, we +briefly discuss recent studies of learning through large-scale digital platforms, but largely focus on +those exploring sociological aspects of education. We believe computational social scientists can +creatively advance this emerging research frontier—and in doing so, help facilitate more equitable +educational and life outcomes. +Keywords: Education, Learning Analytics, Neighborhood Effects, Educational Data Science, +Social Data Science +Introduction +The journalist Thomas Friedman’s famous declaration of 2012 as “The Year of the MOOC” +(Friedman, 2013) heralded the beginning of a new era of education. MOOCs—or massive, +open, online courses—drew enthusiasm and optimism from a wide audience as potential +enablers of more equitable global access to quality education. In parallel, they sparked a +new wave of computational research in education. Emerging platforms and the large +datasets they created inspired researchers to analyze how students engage with lectures and +quizzes online (Kizilcec et al, 2013; Breslow et al., 2013); try and predict who is most likely to +drop out of courses (Kloft et al., 2014); experiment with new methods for sequencing +learning content (Zhao et al., 2018); and even deploy interventions designed to improve +course completion rates (Kizilcec et al., 2017; Kizilcec et al., 2020). We, too, were among +these researchers, analyzing patterns of engagement in MOOC discussion forums to better +understand the nature of communication and social engagement in these spaces (Gillani & +Eynon, 2014; Gillani et al., 2014; Eynon et al., 2016). Yet almost as quickly as they rose to +1 This is a draft of the chapter. The final version will be available in Handbook of Computational Social Science +edited by Taha Yasseri, forthcoming 2023, Edward Elgar Publishing Ltd. The material cannot be used for any +other purpose without further permission of the publisher and is for private use only. Please cite as: N. Gillani +and R. Eynon (2023). Understanding and improving social factors in education: a computational social science +approach. In: T. Yasseri (Ed.), Handbook of Computational Social Science. Edward Elgar Publishing Ltd. +1 + +prominence, it became clear that MOOCs would not be a silver bullet for addressing +disparities in educational access and outcomes, or ‘disrupt’ higher education as many +believed: with the exception of several notable cases (Cadwalladr, 2012), most early MOOC +participants were well-educated adults, hailing mostly from developed countries (Emanuel, +2013). +MOOCs, and digital learning environments more broadly, have helped shed light on learner +behaviors and patterns that may have previously been difficult—if not impossible—to +measure. For example, after initial school closures due to the COVID-19 pandemic, data +from digital learning platforms helped reveal how students in the US from lower-income +neighborhoods were engaging much less with academic content than those in more affluent +areas (Chetty et al., 2020). Yet understanding learning processes through data from digital +platforms hardly tells us everything we need in order to improve educational access and +outcomes. Across the world, there are still tremendous global achievement gaps (Graetz et +al., 2020)—gaps that persist even within specific developed countries like the US, stemming +from a myriad of factors like continued racial and income segregation in schools (Reardon & +Owens); racial and gender biases among some teachers and other education leaders (Starck, +2020); and, broadly speaking, the crippling effects of poverty on nutrition (Walker, 2011), +attention and cognition (Mani et al., 2013; McCloyd, 1998), self-confidence (Browman et al., +2019), and other out-of-school factors that impact the extent to which children are able to +learn and grow. +In this light, it is clear that digital learning platforms, no matter how advanced, will always be +limited in the extent to which they can improve educational and life outcomes for all +students—especially those experiencing various structural disadvantages like poverty and +racism. So, too, will be the potential impact of computational research as a whole—even if +such research becomes more solutions-oriented, as some have called for (Watts, 2017). Yet +the collection and discovery of new education-related datasets, combined with advances in +computational methods spanning exploratory data analyses, machine learning, social +network analysis, and other approaches offer promise in equipping researchers across +disciplines with new tools to ask questions that can surface knowledge about educational +processes and systems in ways that were previously difficult to imagine. +This promise has motivated several research efforts in the past one to two decades, parallel +to the interest in digital learning platforms, to explore education-related datasets using a +myriad of computational approaches. Journals have hosted special issues on “Educational +Data Science” (McFarland et al., 2021) and “Educational Research in a New Data +Environment” (Reardon & Stuart, 2019), featuring research hailing from both social scientists +who are increasingly leveraging computational methods in their work, and computer/data +scientists with an interest in the social sciences. As these and other related articles +highlight, researchers are using advances in natural language processing to identify gender +biases in textbooks (Lucy et al., 2020); social network analysis to design effective +anti-bullying interventions (Paluck et al., 2016); and quasi-experimental methods to infer the +effectiveness of teachers (Chetty et al., 2014) and guidance counselors (Mulhern, 2020), to +name a few. Instead of confining themselves to digital learning platforms, these and other +studies represent a growing body of work that seeks to use computational methods to +explore issues germain to education systems and institutions as they are experienced every +day, “in real life”, by students and families. +2 + +In this chapter, we focus on research that explores one dimension of such education +systems: the social factors that shape educational access and outcomes for children aged +birth through (approximately) 18 years of age. In particular, we discuss recent +computational work exploring how schools, families, and neighborhoods shape children’s +educational and life outcomes from an early age. Many researchers and practitioners agree +that schools, neighborhoods, and families all operate on and affect children’s educational +trajectories in meaningful ways (Purpose Built Communities, 2019), but often debate the +relative influence of each. After briefly reviewing several studies, we discuss several +directions of opportunity for future work, and how computational social scientists may +creatively apply their unique disciplinary and methodological backgrounds to pursue them. +Before proceeding, we make three notes. One: while we discuss digital learning platforms +and the aforementioned social factors as two separate categories of research and practice, +our purpose in doing so is not to create a false dichotomy. Society and technology are +interwoven (Selwyn, 2019), and education is no different. Instead, we make this distinction +largely to highlight the emerging body of work in the latter, and encourage computational +social scientists with an interest in applications to education to consider investigating these +social factors even when the datasets may not be as readily available or easy to capture +compared to data generated from digital learning ecosystems. Indeed, There is significantly +more work to do to better conceptualise the relationships between education, digital +technologies and society to facilitate meaningful social computational science in education. +Two: most of the examples we use are drawn from studies conducted in the US. While many +of the themes we discuss vis-a-vis the US are relevant in other countries, we also +acknowledge the importance of more research specifically focused on, and conducted +within, other international contexts—especially developing contexts, given that much of +what works in the developed world cannot be force-fitted into developing countries (Irani et +al., 2010). Three: while some of the studies we highlight leverage large datasets and recent +advances in machine learning and other data science techniques, several others use more +traditional quantitative methods (like linear regression analyses used for program evaluation +/ causal inference) as their main methodological tools. We include these different types of +studies to contrast what is meant by ‘computational’, inviting readers to conceptualize a +broad methodological landscape for conducting computational social science research in +education. +Computational approaches for understanding and improving +learning: a learning science and analytics view +As noted earlier, the proliferation of digital learning environments is generating large-scale +“digital trace” data describing how learners engage with online lectures, assignments, and +other materials. Much of the academic research exploring these questions has focused on +understanding and optimizing “cognitive learning” processes—i.e., the processes through +which students acquire knowledge or skills pertaining to specific academic topics (Mayer, +2012). Furthermore, many of these studies have been conducted by researchers with +backgrounds in computer science, data science, and learning or cognitive science. While a +full review of this vast literature is out of scope for this Chapter, below, we briefly highlight +several studies across two broad categories—intelligent tutoring and learning analytics—to +demonstrate different ways in which researchers are applying computational techniques to +3 + +make sense of, and even shape, data in digital learning settings. Throughout these sections, +we refer to “AI” (artificial intelligence) and machine learning; we refer those who may be +unfamiliar with these terms or their broader applications and risks in education to Gillani et +al., 2023. We note that many of the approaches described below may straddle the line +between computational social science and computational cognitive science—especially +when the focus is largely on optimizing the individual’s learning process. +Intelligent tutoring +Intelligent tutoring systems (ITS) are tools that seek to adapt to a student’s learning style and +state in order to help them learn content and build skills in a way that is uniquely suited to +their needs. Given the ease-of-assessment for simple mathematics problems, many of these +ITS have focused on helping students learn math, though there are also examples in other +disciplines, e.g. language learning . +2 +The “I” in ITS often has different definitions for different tools. For example, some ITS are +machine learning-based systems that seek to infer a student’s knowledge state based on +which problems they answer correctly or incorrectly (Ritter, 2007). These systems then +provide students with problems that are most likely to be at their “learning edge”—i.e. the +problems they haven’t yet answered that they are most likely to get correct, given their prior +history of answers. Other ITS, like (Kelly et al., 2013), use simple rules or heuristics to +determine if and when a student has mastered some concept (e.g. if they answer three or +more of a particular type of question correctly in a row). Experimental evidence has largely +shown ITS to be effective in increasing students’ grades and test scores (Shank, 2019). Of +course, grades and test scores offer only one (limited) view into student learning, and +methodological challenges in evaluating the efficacy of ITS—e.g., “site selection bias” +(Allcott, 2015)—may limit our ability to fully understand their impact on educational +outcomes. +More recently, some researchers have argued that the real value of ITS may not lie in their +problem recommendations, but instead, in what they can reveal about the granular +misconceptions students harbor vis-a-vis course material in order to better inform and +support how human educators teach (Baker, 2016). For example, in a recent paper, an +intelligent tutoring algorithm that used deep neural networks to model students’ knowledge +states also produced a granular map of how different types of concepts and questions relate +to one another. This map was a byproduct of which questions students answered correctly +and incorrectly (Piech et al., 2015). Such interpretations shift computational learner +modeling away from a cognition-optimization process to one that aims to scaffold +teacher-student interactions through “learning analytics”. +Learning analytics +In addition to bootstrapping new ITS, the proliferation of data from digital learning +environments has also inspired the development and use of “learning analytics” to improve +teaching and learning practices (Gašević et al., 2015). +A large body of research over the past 10 years has illustrated the myriad of ways that +methods from artificial intelligence (particularly machine learning) can be applied to extract +2 Duolingo.com and Busuu.com, for example. +4 + +insights from learner data. For example, one of the first studies on digital trace data +generated in MOOCs used unsupervised machine learning to infer a typology of participants +based on which types of course activities they engage with, and for how long (Kizilcec et al., +2013). Another study applied linear regression to system log files from a learning +management system (LMS)—which captured data on usage frequency and system access +patterns—to illustrate how more “regular” learning (proxied by how regularly a user logs +into the LMS) positively predicts performance on a final test (Jo et al., 2014). To illustrate +the value of data generated by learners as a part of using intelligent tutoring system usage, +(Xing & Goggins, 2015) build a machine learning model to detect when students are “going +off task” based on platform usage. +These are but a few of the numerous learning analytics studies that currently exist. There is, +undoubtedly, great potential in using machine learning techniques to make sense of the vast +amounts of data being generated in learning contexts; however, learning analytics as a +discipline is still too nascent to make conclusive claims about how mining and analyzing +learners’ digital traces can enhance teaching and learning practices. Some researchers have +explicitly called this out, highlighting that there is still very little evidence on how learning +analytics supports learning and teaching—and of the reported evidence, how very little of it +shows negative effects, perhaps suggesting a skew in the research community towards +reporting positive results (Ferguson & Clow, 2017). Other researchers have cautioned +against reducing learning analytics to “counting clicks”, calling instead for an approach to +analysis that is grounded in existing theories of learning—and hence, more likely to enhance +learning outcomes (Gašević et al., 2015). Finally, in many cases, there is still a large gap +between applications of computational methods to learner trace data and to what extent +these applications end up being useful to teachers and learners—inspiring researchers to +define new roles like “educational data scientists” (Agasisti & Bowers, 2017) and “learning +engineers” (Thille, 2016) to try and bridge these gaps in order to make the technical +contributions better serve humans. +Computational approaches for understanding and improving +learning: a social factors view +While computer scientists have driven a large portion of the work behind the +above-described computational approaches to analyzing data from learning platforms, much +of the computational social science research on the role of schools, families, and +neighborhoods in shaping children’s educational and life outcomes has been generated by +applied micro economists and sociologists. We review several of these studies below. +Schools +The local neighborhood has had an important role in the planning and development of many +school systems across the world. For example, in the US, locality-specific movements were +central to enabling free primary and secondary schooling (Goldin & Katz, 2008). +Unfortunately, one enduring legacy of this primarily place-based movement to expand +access to education is a continued relationship between neighborhood characteristics, +especially racial demographics and household income, and academic achievement. These +relationships have resulted in achievement gaps where low-income children of color are +significantly less likely to perform well in school when compared to their higher-income, +White counterparts (Reardon et al., 2018). +5 + +To appreciate why these gaps matter, it is worth reflecting on why schools matter. Namely, +what is the purpose of schooling? Economists are often interested in how teachers and +schools impact intergenerational outcomes, like future earnings (Chetty et al., 2011), though +some have also explored how education might also lead to greater happiness in adulthood +(Oreopoulos & Salvanes, 2011). Through these lenses, the purpose of education is to help +equip a child with the knowledge, skills, and attitudes needed to achieve these outcomes. +However, the philosopher Biesta argues that the purpose of education is defined by our +values, and thus, a lack of explicit articulation of these values in recent conversations about +education and educational measurement makes such conversations incomplete. Without +providing “an answer”, Biesta offers a framework to help structure debates about the +purpose of education: education is about “qualification”, or equipping young people with +skills; “socialization”, or helping young people to become a part of a collective social, +cultural, political order; and “subjectification”, or helping individuals become independent +and autonomous citizens (Biesta, 2009; Eynon, 2022). +Differing notions of the purpose of education (and, implicitly, the values that shape those +perceived purposes) have fueled different measures of what constitutes quality schooling. +One measure that has been shown to correlate with long-term outcomes is +“effectiveness”—or how much a child learns and grows, over time, at their school (as +opposed to snapshot measures like test scores alone, which are only weakly correlated with +growth/effectiveness measures). Under these measures of effectiveness, “learning and +growing” are usually defined in terms of changes in performance on standardized tests. +While this is still an inherently limited measure—test scores do not capture the full breadth +of a child’s educational journey or outcomes—there has been evidence that students who +are exposed to more effective educational settings are also more likely to attend college, +earn more as adults, and less likely to have teen pregnancies (Chetty et al., 2014). +What makes schools effective? Ensuring adequate funding and resources for students, +regardless of socioeconomic or demographic background, is of critical importance. +However, a recent study analyzing data from several charter schools in New York City +showed that even schools with higher per-pupil expenditures do not always improve +learning outcomes (Fryer & Dobbie, 2013). This suggests there are other—perhaps more +difficult-to-measure factors—that also matter. One answer is that effective schools have +effective adults in them: teachers, counselors, and other staff who are well-suited to help +students achieve their potential. There is a large literature investigating the development of +“teacher value-added” models—i.e., data-driven, often quasi-experimental methods for +computing the causal effect that teachers have on children’s learning outcomes (Koretz, +2008). These models have helped identify the impact teachers have on both shorter-term +measures (like test scores), but also, longer-term outcomes like those highlighted above. +Teachers, of course, are one of many adults that students may be exposed to in school: +guidance counselors, too, can have a pivotal impact on the life trajectories of children. A +recent paper exploited the fact that many high schools assign students to guidance +counselors based on the starting letter of the students’ last names—an effectively random +assignment—to identify the impact of counselors on students’ academic achievement and +college-going behaviors (Mulhern, 2020). The study found that effective counselors (based +on their impact on students’ performance in other years) had roughly the same impact on +students’ academic outcomes as effective teachers—a surprising insight given that +6 + +counselors in US public schools often serve an order of magnitude more students (hundreds) +than teachers do. +Beyond their effectiveness in delivering instruction or offering guidance on +education-related matters, teachers and counselors often play an important role in the lives +of students by serving as role models and mentors. A recent study leveraged a range of +regression specifications to identify positive relationships between a child having at least +one in-school mentor and their future academic achievement (Kraft et al., 2021). A prior +study exploiting random assignment of students to classrooms found that Black students +matched to same-race teachers were more likely to graduate high school and enroll in +college (Gershenson et al., 2018)—perhaps due to students’ abilities to see themselves and +their life experiences reflected in these adult stakeholders. The counselors' study discussed +above found similar relationships. Even when such opportunities for students to recognize +their own unique background and circumstances in adult role-model and mentor-like figures +are short-lived, they can potentially have important effects. For example (Riley, 2019) +showed that screening a movie depicting a strong, relevant female role model could lead +students in Uganda—particularly lower-performing females—to achieve higher scores on +their final exams in Math. Another study demonstrated how a motivational speech by +former US First Lady Michelle Obama, delivered at a lower-performing school for girls in the +UK, raised end-of-year performance on standardized tests (Burgess, 2016). +Schools, therefore, can serve as conduits to connecting children to effective teachers, +counselors, role models, and mentors. Still, the relationships between place, income, race, +and education have limited the extent to which children and their families can access +schools that afford such opportunities for students. To combat this, cities around the world +are increasingly offering “school choice”: opportunities for families to select schools for their +children beyond their immediate neighborhood contexts. Increased choice, however, does +not always translate into increased quality of education. The resources, knowledge, and +other assets families can tap into are likely to shape their ability to take advantage of such +choices in schooling, with a reproduction of advantage often the outcome of such policy +strategies (Ball et al., 2014). +Parents and families +Parents and families influence education in multiple ways; here, we focus on one of those +ways: school choice. Family units, and parents in particular, are often tasked with selecting +schools for children. Parents, however, are influenced by their own prior beliefs and other +priorities for what constitutes a “good school” for their child. For example, many parents' +notions of school quality, and subsequent choices, are influenced by what they hear through +their social networks vs. more formal indicators of quality (Ball & Vincent, 1998)—which may +help explain why some parents' school preferences are more closely correlated to measures +of “peer quality” at the school (measured by standardized test score achievement) than +measures of school effectiveness (Abdulkadiroglu et al., 2019). Furthermore, some parents +are also influenced by school characteristics that do not necessarily directly indicate the +quality of education the school offers—for example, the school's proximity to their home or +other geographic characteristics (Bell, 2009). These preferences may be a reflection of what +parents value, or a symptom of various barriers they face, like limitations in their ability to +transport students to/from certain schools; scheduling conflicts that prevent them from +visiting schools to actively evaluate how well they are likely to serve their children; or other +7 + +challenges that families living in poverty or facing other hardships may be disproportionately +more likely to experience (Robertson et al., 2021). +Whether parents have an explicit choice in which schools they select for their children, or +this choice is implicit by way of where families end up living, in recent years, parents have +turned to school reviews platforms like GreatSchools.org for insights into the quality of +potential schools their children may attend (Lovenheim & Walsh, 2017). A recent study, +however, used quasi-experimental methods (treating the expansion of GreatSchools’ ratings +availability across the US as effectively random) to argue that more availability of school +ratings information has actually exacerbated racial and income segregation (Hasan & Kumar, +2019). While perhaps counterintuitive, this is not surprising: school ratings websites often +highly-weight snapshot-in-time test scores—which are notoriously correlated with the racial +and income demographics of the schools (Barnum & LeMarr LeMee, 2019)—and hence +indirectly drive parents to make decisions based on these factors instead of measures that +more accurately capture school effectiveness. Our own research has underscored this +possibility, leveraging recent advances in natural language processing to analyze how +parents talk about schools in the reviews they post on GreatSchools and finding that such +reviews reflect racial and socioeconomic differences in schools instead of their actual quality +(i.e., effectiveness ) (Gillani et al., 2021). This raises questions about what kinds of +3 +information these sites contain and how this information might be repositioned, or changed +altogether, to minimize unintended and potentially-harmful consequences. +While more information can sometimes exacerbate inequalities in education, in other cases, +it can reduce “information frictions” and knowledge gaps parents might face when seeking +to support their children’s education. Here, too, computational methods—combined with +behavioral science-inspired interventions—have played an important role in reducing such +frictions (Bergman, 2019). Many of these studies fall beyond the realm of school choice and +seek to engage parents as active sources of support in their children’s educational +experiences. For example, (Bergman & Chen, 2019) designed an SMS-based system to +inform parents about their children’s class absences and performance, finding that doing so +significantly reduced the rate of course failures. Focusing on lower-income parents of +prekindergarten children, (York and Loeb, 2014) ran an 8-month-long text messaging +campaign in which they shared information highlighting the importance of literacy as well as +suggestions for small-scale literacy-related activities that could be done around the home. +They found a significant increase in parental engagement and literacy levels after children +started school. In addition to their results, what’s particularly notable about these studies is +that they move from observational analyses to intervention design and evaluation. As we +discuss later in the chapter, computational social scientists may be particularly well-suited to +help bridge existing gaps between data analysis and the design/evaluation of interventions +that seek to improve educational outcomes. +Neighborhoods +There is an active debate in the social sciences about how much neighborhoods—versus +families or schools—influence a child's future outcomes. Some scholars argue that a child's +family has the greatest impact on their development, especially at an early age (Heckman, +3 Though, in late 2020, GreatSchools overhauled its ratings methodology in an effort to address these issues, +downweighting test scores and upweighting measures of student growth and how well schools serve students +from different backgrounds: https://www.greatschools.org/gk/articles/why-we-changed-our-ratings/. +8 + +2006), in part, by facilitating the development of character and other metacognitive traits +like motivation (Heckman & Kautz, 2013). Referencing the Perry Preschool Project, a famous +early childhood intervention administered in the 1960s that included home visits to help +strengthen parents' abilities to support their children's learning (Schweinhart et al., 1993), +these scholars note how some of the intervention's greatest long-term effects (e.g. higher +earnings, lower crime rates) accrued to those who lived in the worst neighborhoods. They +use this to argue for “the relative unimportance of ZIP codes in explaining the observed +intergenerational program effects on the children of Perry participants” (Heckman & +Karapakula, 2019). +Others argue that schools, more than neighborhoods, shape a child's future trajectory. E.g., +Fryer & Katz (2013) use quasi-experimental evidence documenting changes in children's +neighborhood and school quality—comparing findings from the randomized experimental +voucher-based “Moving to Opportunity” (MTO) study (Katz et al., 2001) to academic gains of +those who received lottery admissions to a high-quality charter school in New York City +(Dobbie & Fryer, 2009)—to suggest high-quality schools affect a child's long-term academic +achievement and earnings more than better neighborhoods. +Yet recent experimental and quasi-experimental analyses have revealed the large impact a +child's neighborhood can have on their academics and future earnings. These studies +critically note that how much a neighborhood shapes a child's future is, on average, a +function of how much time the child actually spends growing up there. For example, a few +years after Fryer & Katz, 2013 was published, several researchers revisited the MTO study +with two new lenses: i) longer-term earnings data for children (via tax returns) who +participated in the study, and ii) an analysis of how future earnings for these children +differed depending on how old they were at the time of the move. The researchers found +that when children moved to a lower-poverty neighborhood before the age of 13, their +college attendance rates and future earnings increased. On the other hand, older children +experienced no such effects (Chetty et al., 2016). +Members of this research team have conducted several additional studies leveraging +administrative (intergenerational tax and Census) records to quantify the impact a child's +geography has on their future outcomes, like earnings. For example, they analyze data +describing millions of families who moved across commuting zones in the US to find that +exposure to better neighborhoods leads children's future earnings to converge at a rate of +4% per year that they spend in that neighborhood to the future earnings of children who +were already living there (Chetty & Hendren, 2015). The authors leverage several +econometric tools to control for potential confounders (like family effects) and identify this +effect as the “causal effect of place” on a child's future outcomes. While identifying the +underlying mechanisms driving these effects will require more research, the authors find a +variety of neighborhood characteristics that positively correlate with upward mobility, +including: i) lower poverty rates, ii) more social capital, iii) higher-quality schools (as +measured by higher test scores), and iv) more college graduates (Chetty et al., 2018a). +In a separate paper, researchers disaggregated the effects of neighborhoods on future +outcomes by children's race and gender to find that black boys were much less likely to +achieve upward mobility than white boys, even when they grew up in similar +neighborhoods. Interestingly, one of the features of neighborhoods with smaller gaps +9 + +between black and white boys' future mobility rates was the presence of more black fathers +per capita (not necessarily a child's own father within the household)—pointing to the +possible impact, again, of role model figures on facilitating better outcomes (Chetty et al., +2018b). Finally, a recent collection of follow-on studies leveraged an unprecedented scale of +Facebook data (70 billion friendships across 20 million individuals) to shed new empirical +light on a long-standing topic: the belief that social capital, defined as friendships across +socioeconomic divides, has a causal effect on intergenerational upward mobility (Chetty et +al., 2022a; Chetty et al., 2022b). These new studies illustrate how digital trace data and +administrative records, together, can help illuminate potential pathways to improved life +outcomes for youth. +Thus, much like schools, neighborhoods may affect children through how they expose (or fail +to expose) children to role models and mentors. This possibility is underscored by another +recent study, which highlighted how children who grow up in geographies with more +inventors are more likely to become inventors themselves in the same technology classes +that are prevalent in those geographies (Bell et al., 2018). Thus, children growing up in the +Bay Area near Silicon Valley are more likely to invent new software/computer technologies; +children growing up in the Midwestern US are more likely to invent medical devices; etc. +The authors argue that such differences point to a causal impact of exposure to innovation: +children who are exposed to innovation are more likely to become inventors themselves +when they grow up. Unfortunately, girls, children of color, and those living in low-income +environments are often less likely to be exposed to inventors who share their backgrounds, +reflecting longstanding structural inequity in society. +These recent findings suggest that boundaries between schools, families, and +neighborhoods may actually be more fluid and interleaving than some researchers have +previously suggested. If we think of schools as a part of many children's neighborhood +experiences, and generalize the influence a family has to the impact their social network +(which includes family, but also neighbors, peers, role models and others) has—it is clear +that neighborhoods constitute a “meta-mechanism” that influence children's futures +through a range of possible channels that include schools and family-like figures. +Looking ahead: exploring social factors in education with +computational social science +As many of the citations so far suggest, much of the existing computational work exploring +social factors in education has been conducted by applied microeconomists. However, there +is a need and opportunity to draw more sociologists and philosophers of education into +these debates—particularly to offer new perspectives on theories of social justice and the +values that influence how we think about the role and purpose of education in an +ever-evolving social context. Some have argued that rapidly-expanding datasets and new +computational methods will bring about “the end of theory” (Anderson, 2008), yet the +creativity and nuance of many of the studies we’ve discussed in this chapter should make it +clear that asking meaningful questions will require much more than data mining alone. +Computational social scientists can partner with sociologists and philosophers to incorporate +their rich theories to inform new empirical questions, while also illuminating how empirical +inquiry can spark the development of novel theories and hypotheses (Blades, 2021). Playing +this intermediary role is sorely needed to more holistically investigate how social factors +10 + +shape educational and life outcomes. Below, we outline some of the ways in which +computational social scientists can help bridge these worlds. +Measuring a broader set of inputs and outcomes +One of the inherent limitations in exploring social factors that shape children’s educational +and life outcomes is capturing the “right” data. One simple way to think of such data is in +terms of inputs and outcomes. For example, throughout this Chapter, we’ve shared studies +that explore relationships between inputs like students or teachers, and outcomes like +academic achievements (as measured by performance on standardized tests). Despite the +important role of social factors in shaping educational and life outcomes, few studies +develop robust ways of measuring the strength of social ties (e.g. by capturing social +network data between participants ), or other related measures that have been theoretically +4 +linked to improved outcomes for youth and families, like social capital (Chetty et al., 2018a; +Putnam, 2000). Thus, looking ahead, can we develop more robust ways of capturing and +incorporating social network information into existing models? For example, perhaps by +using data from existing social networking platforms (a la recent efforts by Chetty et al., +2022a and Chetty, et al., 2022b)? Or by creating new methods for capturing the (evolving) +social networks of students and critical peer and adult figures in their lives? Doing so may +help offer more insights into the nature and kinds of social ties that are most associated with +achievement. More ambitiously, future work could make even stronger connections with +the long standing literature that makes far more visible the process and practices of +schools—what is sometimes called the 'new sociology of education' (e.g. Young, 1972)—and +with those focused on critical pedagogy (e.g. Friere, 1970). +Furthermore, in terms of outcomes, how might we broaden outcome measures beyond +performance on tests to capture more about the impact social factors have on children’s +conceptions of self, feelings about their place and purpose in the world and in relation to +others, or other aspects of living a quality of life that may be correlated with—but still +distinct from—easier-to-measure, inherently quantitative factors like test scores (or even +longer-term measures like future income)? These are difficult questions that will benefit +from fresh, creative thinking from computational social scientists interested in developing +new input and outcome measures from existing and new datasets—but that can only be +achieved through meaningful engagement with other academic communities that have +different philosophical, conceptual and often methodological approaches. +Starting with the questions, not methods +One of the benefits of established disciplines is that they often have built up, over decades +or even centuries, established approaches and methods to different types of research +questions. For example, linear regression models and their many variants are often the +go-to choices for many applied microeconomists. While such methodological focus can +produce efficiencies in data analysis and sharing, it can also—over time—constrain the types +of questions that even seem plausible to ask. For example, linear regression is limited as a +tool for natural language processing, where the words and phrases may have complex, +non-linear dependencies in sentences; where the feature space may be much larger than +the number of observations in the dataset itself (and hence, prone to overfitting); etc. In +4 There are a few notable exceptions, e.g. (Paluck et al., 2016). +11 + +recognition of this, social scientists are developing a richer cadre of “Text as Data” methods +(Gentzkow et al., 2019). +By starting with the questions and turning to methods as needed to explore and answer +such questions as deeply and richly as possible, computational social scientists can help +draw attention to a wider range of topics on the education research agenda—including +those that may not directly identify causal relationships (which is of great interest to many +applied microeconomists) but still help yield valuable insights that inform downstream +causal analyses, design-based research, etc. (Singer, 2019). A byproduct of such an approach +may be a greater degree of methodological diversity applied to different research problems, +and over time, an expansion of the questions researchers and practitioners even deem +possible to ask. +Unpacking the underlying mechanisms driving certain observed outcomes +Much of the applied microeconomics literature on neighborhood effects or teacher +effectiveness outline how much neighborhoods or teachers can increase children’s shorter +and longer-term outcomes, but few unpack at a granular level what it is about +neighborhoods, or teachers, or other social factors that make them more or less effective. +Indeed, looking back through the history of the sociology of education, we see a similar +criticism of early work from economists and quantitative researchers in the 1960s and 70s +exploring questions of equity and schooling, many of whom treated schools and associated +institutions as a black box (Weiss, 2016). It is naive to think data and computational +methods alone will enable researchers to answer these questions more deeply in the coming +years; a rich body of qualitative research will continue to be indispensable. This will +encompass a continuation of research that is broadly considered more qualitative or +quantitative in the traditional sense, but there may also be opportunities for novel +computational work that spans across these boundaries. Nevertheless, there is still a critical +role of computational methods in this pursuit. For example, might we use techniques from +computer vision applied to historical corpora of neighborhood-level Google Street View +images (a la Naik et al., 2017) to better understand environmental predictors, and +eventually, causal factors responsible for different rates of upward mobility across +neighborhoods? Can we access and merge datasets spanning notes and audio/video +recordings describing teacher classroom observations (a la Kelly et al., 2018), students’ +written feedback, and other inputs to identify behaviors and practices that lead to larger +gains in life outcomes for students? These and other questions may benefit from a fresh +look from computational social scientists hailing from different disciplinary backgrounds. +They will also require thoughtful considerations of data privacy and ethics, as new streams +of data will bring with them new questions about how they should or should not be +processed—and what their unintended consequences might be (Hakimi et al., 2021). +Unpacking mechanisms is not just about providing richer research about particular +phenomena, it is also about meaningfully engaging with researchers that have very different +ontological and epistemological worldviews to further refine and develop the conceptual +frames often seen in education data science (Eynon, 2023). Furthermore, unpacking +mechanisms requires challenging our own notions of access and “success” in education, and +not treating education as a straightforwardly positive and neutral ‘thing’. What we have +seen from the discussion above is that both the learning analytics/sciences and social factors +perspectives offer a strong focus on distributional issues related to education. In other +12 + +words, there is a tendency to focus primarily on questions related to the resources required +to ensure everyone has access to a good quality education (typically measured in quite +narrow ways related to achievement on standardized tests) (Keddie, 2012). This may be, for +example, through the use of more technology in the classroom, the use of grants or +vouchers, using technology to provide more or better information to help with school +choice, or trying to ensure that where a person can afford to live does not impact the quality +of the school systems a young person can access. +These are all important issues. However, questions of equity and social justice—which are +intrinsically linked to issues of quality education—have multiple dimensions (North, 2006). +Beyond questions of distribution, we must also explore the cultural and political dimensions +of injustice (Fraser, 2008). Specifically, questions of misrecognition (such as cultural +domination and disrespect) that marginalised communities may experience at school, +through, for example, the ways that young people are treated, and what is / what is not +included in the curriculum (Power and Frandiji, 2010:388). And questions of +misrepresentation may arise due to potential deficits of educational governance across +school systems which make participation for some groups significantly challenging (Sayed, et +al., 2020). +Relatedly, we can enhance awareness of how some interventions designed to make +schooling more equal can inadvertently reinforce the status quo or cause harm to different +groups. Research has shown, for example, how the common focus on distributive +approaches to social justice in schools tends only to actually further stigmatise “the poor” +rather than change economic structures to make society fairer (e.g. North, 2006; Leibowitz & +Bozalek, 2016). Achieving equity and justice in schools (and indeed in society more broadly) +requires attention to the complex interplay between economic, cultural and political factors +(Fraser, 2008). +Computational social scientists may partner with sociologists, educationalists, and +philosophers of education or familiarize themselves with related theories and methods, to +bring a more nuanced understanding of the complex role that education plays in society +(beyond delivering learning and qualification) and ways of theorising about social justice and +education that would add to the existing research in this domain. +Bridging analysis and design +As computational social scientists begin to more deeply unpack the underlying mechanisms +responsible for observed effects, they will be uniquely suited to also help inform, and +perhaps even lead, the design and development of interventions that seek to improve +educational and life outcomes for youth. Currently, a wide range of researchers participate +in the design and evaluation of education-related interventions. Often, these interventions +come in the form of applied microeconomists running randomized control trials like those +highlighted earlier or in more international settings (Gibson & Sautmann, 2021). They also +come, however, in the form of computing and design researchers using both +human-centered and participatory design methods to develop frameworks and systems that +seek to serve different segments of the population (e.g. Costanza-Chock, 2020). Not all +computational social scientists may wish to participate in design-related activities that pick +up where analyses of secondary data (or even preliminary causal analyses) leave off, but +their methodological and domain insights may lend them to fill an important “in-between” +13 + +space that connects theory and methods from research domains to the practical challenges +and considerations of intervention in field settings. Engaging in these in-between spaces +may help create new fruitful directions for research, and also, meaningful social +change—with due deference and humility in light of how different cultures and groups of +people may desire (or not desire) such interventions (Irani et al., 2010). +Expanding the audience of research and evaluating/improving “how it lands” +Learning analysts and engineers often see digital platform developers or certain groups of +users (e.g. teachers and school administrators) as important audiences for their findings. +Applied micro economists and sociologists may largely target policymakers—often at state +and national levels—as their primary audiences. As computational social scientists select +questions to work on, they also have an opportunity to think creatively about which +audience(s) they wish to engage directly in dialogue with their work. Much like our point +about methods above, letting the questions drive the thinking around which audiences are +appropriate for the work—instead of assuming a priori that the audience must be a +government policymaker, or platform designer, or some other known stakeholder—may help +surface new ideas for people and systems that may have interest in learning about and +building upon the research findings that computational social scientists help produce. +Simply picking an audience, however, does not guarantee positive social change; there are +significant barriers to communicating findings from research and science. The COVID-19 +pandemic has made this particularly clear (if it wasn’t already before): for example, +school-based mask mandates were a hotly contentious issue across school districts in the US +(Cottle, 2021), and more generally, vaccine hesitancy levels remain high (Sreedhar & Gopal, +2021). The ways in which individuals, organizations, policymakers, and governments as a +whole process and act upon scientific findings is a complex function of individual +motivations and preferences; sociopolitical forces; notions of fairness and morality; and +several other factors (Haidt, 2012). There are also more tactical factors at play, like time +scarcity (Rogers & Lasky-Fink, 2020) and the ability of decision-makers to identify the +relevance of existing research to their own contexts (Nakajima, 2021). There is, therefore, a +tremendous need for research that explores not only which policies and practices help +improve educational outcomes, but also, how those practices can most effectively be +communicated and positioned in order to increase the chances of their eventual +implementation. These communications-based research questions are certainly pertinent to +education, but also other domains, and could benefit from the creative thinking and new +questions that computational social scientists are well-positioned to put forth and test. +Conclusion +In this Chapter, we have presented two perspectives on how computational social scientists +might engage with research pertaining to education. We have focused more on the social +factors that shape educational outcomes—namely, schools, families, and +neighborhoods—given the impact such factors have on influencing educational access, +experiences, and outcomes. We believe it is an exciting time for computational social +scientists to apply their interstitial interests and skill sets to questions in education—and to +creatively come up with new questions that we can’t even imagine yet. +14 + +Further readings +Interested readers may look to (Fischer et al., 2020) for a more in-depth review of the ways +in which computational methods are being used to better understand data from digital +learning environments. Readers who wish to learn more about using behavioral science to +foster greater parental engagement in schools may look to (Bergman, 2019). Finally, for a +broader view on trends in the use of data science and other computational methods in +education, we refer readers to (McFarland et al., 2021) and (reardon & Stuart, 2019). +References +Abdulkadiroglu, A., Pathak, P. A., Schellenberg, J., Walters, C. R. (2019). Do parents value +school effectiveness? NBER Working Paper No. 23912. +Agasisti, T. and Bowers, A.J. 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Proceedings of +ACM CHI Conference on Human Factors in Computing Systems. +23 + diff --git a/adE5T4oBgHgl3EQfeg9M/content/tmp_files/load_file.txt b/adE5T4oBgHgl3EQfeg9M/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..95af7421cbd9c2bbb664ad04fd194c35e72a7421 --- /dev/null +++ b/adE5T4oBgHgl3EQfeg9M/content/tmp_files/load_file.txt @@ -0,0 +1,1153 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf,len=1152 +page_content='Understanding and improving social factors in education: a computational social science approach1 Nabeel Gillani1 (ORCID),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Rebecca Eynon2 (ORCID) 1College of Arts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Media and Design and D’Amore-McKim School of Business,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Northeastern University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Boston,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' MA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' USA 2Oxford Internet Institute and Department of Education,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' University of Oxford,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Oxford,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' UK Abstract Over the past decade,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' an explosion in the availability of education-related datasets has enabled new computational research in education.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Much of this work has investigated digital traces of online learners in order to better understand and optimize their cognitive learning processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Yet cognitive learning on digital platforms does not equal education.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Instead, education is an inherently social, cultural, economic, and political process manifesting in physical spaces, and educational outcomes are influenced by many factors that precede and shape the cognitive learning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Many of these are social factors like children’s connections to schools (including teachers, counselors, and role models), parents and families, and the broader neighborhoods in which they live.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' In this article, we briefly discuss recent studies of learning through large-scale digital platforms, but largely focus on those exploring sociological aspects of education.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' We believe computational social scientists can creatively advance this emerging research frontier—and in doing so, help facilitate more equitable educational and life outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Keywords: Education, Learning Analytics, Neighborhood Effects, Educational Data Science, Social Data Science Introduction The journalist Thomas Friedman’s famous declaration of 2012 as “The Year of the MOOC” (Friedman, 2013) heralded the beginning of a new era of education.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' MOOCs—or massive, open, online courses—drew enthusiasm and optimism from a wide audience as potential enablers of more equitable global access to quality education.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' In parallel, they sparked a new wave of computational research in education.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Emerging platforms and the large datasets they created inspired researchers to analyze how students engage with lectures and quizzes online (Kizilcec et al, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Breslow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=', 2013);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' try and predict who is most likely to drop out of courses (Kloft et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=', 2014);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' experiment with new methods for sequencing learning content (Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=', 2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' and even deploy interventions designed to improve course completion rates (Kizilcec et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Kizilcec et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' We, too, were among these researchers, analyzing patterns of engagement in MOOC discussion forums to better understand the nature of communication and social engagement in these spaces (Gillani & Eynon, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Gillani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Eynon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Yet almost as quickly as they rose to 1 This is a draft of the chapter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' The final version will be available in Handbook of Computational Social Science edited by Taha Yasseri, forthcoming 2023, Edward Elgar Publishing Ltd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' The material cannot be used for any other purpose without further permission of the publisher and is for private use only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Please cite as: N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Gillani and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Eynon (2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Understanding and improving social factors in education: a computational social science approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' In: T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Yasseri (Ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' ), Handbook of Computational Social Science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Edward Elgar Publishing Ltd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' 1 prominence, it became clear that MOOCs would not be a silver bullet for addressing disparities in educational access and outcomes, or ‘disrupt’ higher education as many believed: with the exception of several notable cases (Cadwalladr, 2012), most early MOOC participants were well-educated adults, hailing mostly from developed countries (Emanuel, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' MOOCs, and digital learning environments more broadly, have helped shed light on learner behaviors and patterns that may have previously been difficult—if not impossible—to measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' For example, after initial school closures due to the COVID-19 pandemic, data from digital learning platforms helped reveal how students in the US from lower-income neighborhoods were engaging much less with academic content than those in more affluent areas (Chetty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Yet understanding learning processes through data from digital platforms hardly tells us everything we need in order to improve educational access and outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Across the world, there are still tremendous global achievement gaps (Graetz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=', 2020)—gaps that persist even within specific developed countries like the US, stemming from a myriad of factors like continued racial and income segregation in schools (Reardon & Owens);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' racial and gender biases among some teachers and other education leaders (Starck, 2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' and, broadly speaking, the crippling effects of poverty on nutrition (Walker, 2011), attention and cognition (Mani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' McCloyd, 1998), self-confidence (Browman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=', 2019), and other out-of-school factors that impact the extent to which children are able to learn and grow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' In this light, it is clear that digital learning platforms, no matter how advanced, will always be limited in the extent to which they can improve educational and life outcomes for all students—especially those experiencing various structural disadvantages like poverty and racism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' So, too, will be the potential impact of computational research as a whole—even if such research becomes more solutions-oriented, as some have called for (Watts, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Yet the collection and discovery of new education-related datasets, combined with advances in computational methods spanning exploratory data analyses, machine learning, social network analysis, and other approaches offer promise in equipping researchers across disciplines with new tools to ask questions that can surface knowledge about educational processes and systems in ways that were previously difficult to imagine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' This promise has motivated several research efforts in the past one to two decades, parallel to the interest in digital learning platforms, to explore education-related datasets using a myriad of computational approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Journals have hosted special issues on “Educational Data Science” (McFarland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=', 2021) and “Educational Research in a New Data Environment” (Reardon & Stuart, 2019), featuring research hailing from both social scientists who are increasingly leveraging computational methods in their work, and computer/data scientists with an interest in the social sciences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' As these and other related articles highlight, researchers are using advances in natural language processing to identify gender biases in textbooks (Lucy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=', 2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' social network analysis to design effective anti-bullying interventions (Paluck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=', 2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' and quasi-experimental methods to infer the effectiveness of teachers (Chetty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=', 2014) and guidance counselors (Mulhern, 2020), to name a few.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Instead of confining themselves to digital learning platforms, these and other studies represent a growing body of work that seeks to use computational methods to explore issues germain to education systems and institutions as they are experienced every day, “in real life”, by students and families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' 2 In this chapter, we focus on research that explores one dimension of such education systems: the social factors that shape educational access and outcomes for children aged birth through (approximately) 18 years of age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' In particular, we discuss recent computational work exploring how schools, families, and neighborhoods shape children’s educational and life outcomes from an early age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Many researchers and practitioners agree that schools, neighborhoods, and families all operate on and affect children’s educational trajectories in meaningful ways (Purpose Built Communities, 2019), but often debate the relative influence of each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' After briefly reviewing several studies, we discuss several directions of opportunity for future work, and how computational social scientists may creatively apply their unique disciplinary and methodological backgrounds to pursue them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Before proceeding, we make three notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' One: while we discuss digital learning platforms and the aforementioned social factors as two separate categories of research and practice, our purpose in doing so is not to create a false dichotomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Society and technology are interwoven (Selwyn, 2019), and education is no different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Instead, we make this distinction largely to highlight the emerging body of work in the latter, and encourage computational social scientists with an interest in applications to education to consider investigating these social factors even when the datasets may not be as readily available or easy to capture compared to data generated from digital learning ecosystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Indeed, There is significantly more work to do to better conceptualise the relationships between education, digital technologies and society to facilitate meaningful social computational science in education.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Two: most of the examples we use are drawn from studies conducted in the US.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' While many of the themes we discuss vis-a-vis the US are relevant in other countries, we also acknowledge the importance of more research specifically focused on, and conducted within, other international contexts—especially developing contexts, given that much of what works in the developed world cannot be force-fitted into developing countries (Irani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=', 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Three: while some of the studies we highlight leverage large datasets and recent advances in machine learning and other data science techniques, several others use more traditional quantitative methods (like linear regression analyses used for program evaluation / causal inference) as their main methodological tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' We include these different types of studies to contrast what is meant by ‘computational’, inviting readers to conceptualize a broad methodological landscape for conducting computational social science research in education.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Computational approaches for understanding and improving learning: a learning science and analytics view As noted earlier, the proliferation of digital learning environments is generating large-scale “digital trace” data describing how learners engage with online lectures, assignments, and other materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Much of the academic research exploring these questions has focused on understanding and optimizing “cognitive learning” processes—i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=', the processes through which students acquire knowledge or skills pertaining to specific academic topics (Mayer, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Furthermore, many of these studies have been conducted by researchers with backgrounds in computer science, data science, and learning or cognitive science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' While a full review of this vast literature is out of scope for this Chapter, below, we briefly highlight several studies across two broad categories—intelligent tutoring and learning analytics—to demonstrate different ways in which researchers are applying computational techniques to 3 make sense of, and even shape, data in digital learning settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Throughout these sections, we refer to “AI” (artificial intelligence) and machine learning;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' we refer those who may be unfamiliar with these terms or their broader applications and risks in education to Gillani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=', 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' We note that many of the approaches described below may straddle the line between computational social science and computational cognitive science—especially when the focus is largely on optimizing the individual’s learning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Intelligent tutoring Intelligent tutoring systems (ITS) are tools that seek to adapt to a student’s learning style and state in order to help them learn content and build skills in a way that is uniquely suited to their needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Given the ease-of-assessment for simple mathematics problems, many of these ITS have focused on helping students learn math, though there are also examples in other disciplines, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' language learning .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' 2 The “I” in ITS often has different definitions for different tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' For example, some ITS are machine learning-based systems that seek to infer a student’s knowledge state based on which problems they answer correctly or incorrectly (Ritter, 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' These systems then provide students with problems that are most likely to be at their “learning edge”—i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' the problems they haven’t yet answered that they are most likely to get correct, given their prior history of answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Other ITS, like (Kelly et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=', 2013), use simple rules or heuristics to determine if and when a student has mastered some concept (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' if they answer three or more of a particular type of question correctly in a row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Experimental evidence has largely shown ITS to be effective in increasing students’ grades and test scores (Shank, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Of course, grades and test scores offer only one (limited) view into student learning, and methodological challenges in evaluating the efficacy of ITS—e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=', “site selection bias” (Allcott, 2015)—may limit our ability to fully understand their impact on educational outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' More recently, some researchers have argued that the real value of ITS may not lie in their problem recommendations, but instead, in what they can reveal about the granular misconceptions students harbor vis-a-vis course material in order to better inform and support how human educators teach (Baker, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' For example, in a recent paper, an intelligent tutoring algorithm that used deep neural networks to model students’ knowledge states also produced a granular map of how different types of concepts and questions relate to one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' This map was a byproduct of which questions students answered correctly and incorrectly (Piech et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Such interpretations shift computational learner modeling away from a cognition-optimization process to one that aims to scaffold teacher-student interactions through “learning analytics”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Learning analytics In addition to bootstrapping new ITS, the proliferation of data from digital learning environments has also inspired the development and use of “learning analytics” to improve teaching and learning practices (Gašević et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' A large body of research over the past 10 years has illustrated the myriad of ways that methods from artificial intelligence (particularly machine learning) can be applied to extract 2 Duolingo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content='com and Busuu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content='com, for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' 4 insights from learner data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' For example, one of the first studies on digital trace data generated in MOOCs used unsupervised machine learning to infer a typology of participants based on which types of course activities they engage with, and for how long (Kizilcec et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Another study applied linear regression to system log files from a learning management system (LMS)—which captured data on usage frequency and system access patterns—to illustrate how more “regular” learning (proxied by how regularly a user logs into the LMS) positively predicts performance on a final test (Jo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' To illustrate the value of data generated by learners as a part of using intelligent tutoring system usage, (Xing & Goggins, 2015) build a machine learning model to detect when students are “going off task” based on platform usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' These are but a few of the numerous learning analytics studies that currently exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' There is, undoubtedly, great potential in using machine learning techniques to make sense of the vast amounts of data being generated in learning contexts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' however, learning analytics as a discipline is still too nascent to make conclusive claims about how mining and analyzing learners’ digital traces can enhance teaching and learning practices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Some researchers have explicitly called this out, highlighting that there is still very little evidence on how learning analytics supports learning and teaching—and of the reported evidence, how very little of it shows negative effects, perhaps suggesting a skew in the research community towards reporting positive results (Ferguson & Clow, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Other researchers have cautioned against reducing learning analytics to “counting clicks”, calling instead for an approach to analysis that is grounded in existing theories of learning—and hence, more likely to enhance learning outcomes (Gašević et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Finally, in many cases, there is still a large gap between applications of computational methods to learner trace data and to what extent these applications end up being useful to teachers and learners—inspiring researchers to define new roles like “educational data scientists” (Agasisti & Bowers, 2017) and “learning engineers” (Thille, 2016) to try and bridge these gaps in order to make the technical contributions better serve humans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Computational approaches for understanding and improving learning: a social factors view While computer scientists have driven a large portion of the work behind the above-described computational approaches to analyzing data from learning platforms, much of the computational social science research on the role of schools, families, and neighborhoods in shaping children’s educational and life outcomes has been generated by applied micro economists and sociologists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' We review several of these studies below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Schools The local neighborhood has had an important role in the planning and development of many school systems across the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' For example, in the US, locality-specific movements were central to enabling free primary and secondary schooling (Goldin & Katz, 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Unfortunately, one enduring legacy of this primarily place-based movement to expand access to education is a continued relationship between neighborhood characteristics, especially racial demographics and household income, and academic achievement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' These relationships have resulted in achievement gaps where low-income children of color are significantly less likely to perform well in school when compared to their higher-income, White counterparts (Reardon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' 5 To appreciate why these gaps matter, it is worth reflecting on why schools matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Namely, what is the purpose of schooling?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Economists are often interested in how teachers and schools impact intergenerational outcomes, like future earnings (Chetty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=', 2011), though some have also explored how education might also lead to greater happiness in adulthood (Oreopoulos & Salvanes, 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Through these lenses, the purpose of education is to help equip a child with the knowledge, skills, and attitudes needed to achieve these outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' However, the philosopher Biesta argues that the purpose of education is defined by our values, and thus, a lack of explicit articulation of these values in recent conversations about education and educational measurement makes such conversations incomplete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Without providing “an answer”, Biesta offers a framework to help structure debates about the purpose of education: education is about “qualification”, or equipping young people with skills;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' “socialization”, or helping young people to become a part of a collective social, cultural, political order;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' and “subjectification”, or helping individuals become independent and autonomous citizens (Biesta, 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Eynon, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Differing notions of the purpose of education (and, implicitly, the values that shape those perceived purposes) have fueled different measures of what constitutes quality schooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' One measure that has been shown to correlate with long-term outcomes is “effectiveness”—or how much a child learns and grows, over time, at their school (as opposed to snapshot measures like test scores alone, which are only weakly correlated with growth/effectiveness measures).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Under these measures of effectiveness, “learning and growing” are usually defined in terms of changes in performance on standardized tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' While this is still an inherently limited measure—test scores do not capture the full breadth of a child’s educational journey or outcomes—there has been evidence that students who are exposed to more effective educational settings are also more likely to attend college, earn more as adults, and less likely to have teen pregnancies (Chetty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' What makes schools effective?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Ensuring adequate funding and resources for students, regardless of socioeconomic or demographic background, is of critical importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' However, a recent study analyzing data from several charter schools in New York City showed that even schools with higher per-pupil expenditures do not always improve learning outcomes (Fryer & Dobbie, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' This suggests there are other—perhaps more difficult-to-measure factors—that also matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' One answer is that effective schools have effective adults in them: teachers, counselors, and other staff who are well-suited to help students achieve their potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' There is a large literature investigating the development of “teacher value-added” models—i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=', data-driven, often quasi-experimental methods for computing the causal effect that teachers have on children’s learning outcomes (Koretz, 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' These models have helped identify the impact teachers have on both shorter-term measures (like test scores), but also, longer-term outcomes like those highlighted above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Teachers, of course, are one of many adults that students may be exposed to in school: guidance counselors, too, can have a pivotal impact on the life trajectories of children.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' A recent paper exploited the fact that many high schools assign students to guidance counselors based on the starting letter of the students’ last names—an effectively random assignment—to identify the impact of counselors on students’ academic achievement and college-going behaviors (Mulhern, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' The study found that effective counselors (based on their impact on students’ performance in other years) had roughly the same impact on students’ academic outcomes as effective teachers—a surprising insight given that 6 counselors in US public schools often serve an order of magnitude more students (hundreds) than teachers do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Beyond their effectiveness in delivering instruction or offering guidance on education-related matters, teachers and counselors often play an important role in the lives of students by serving as role models and mentors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' A recent study leveraged a range of regression specifications to identify positive relationships between a child having at least one in-school mentor and their future academic achievement (Kraft et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' A prior study exploiting random assignment of students to classrooms found that Black students matched to same-race teachers were more likely to graduate high school and enroll in college (Gershenson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=', 2018)—perhaps due to students’ abilities to see themselves and their life experiences reflected in these adult stakeholders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=" The counselors' study discussed above found similar relationships." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Even when such opportunities for students to recognize their own unique background and circumstances in adult role-model and mentor-like figures are short-lived, they can potentially have important effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' For example (Riley, 2019) showed that screening a movie depicting a strong, relevant female role model could lead students in Uganda—particularly lower-performing females—to achieve higher scores on their final exams in Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Another study demonstrated how a motivational speech by former US First Lady Michelle Obama, delivered at a lower-performing school for girls in the UK, raised end-of-year performance on standardized tests (Burgess, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Schools, therefore, can serve as conduits to connecting children to effective teachers, counselors, role models, and mentors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Still, the relationships between place, income, race, and education have limited the extent to which children and their families can access schools that afford such opportunities for students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' To combat this, cities around the world are increasingly offering “school choice”: opportunities for families to select schools for their children beyond their immediate neighborhood contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Increased choice, however, does not always translate into increased quality of education.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' The resources, knowledge, and other assets families can tap into are likely to shape their ability to take advantage of such choices in schooling, with a reproduction of advantage often the outcome of such policy strategies (Ball et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Parents and families Parents and families influence education in multiple ways;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' here, we focus on one of those ways: school choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Family units, and parents in particular, are often tasked with selecting schools for children.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Parents, however, are influenced by their own prior beliefs and other priorities for what constitutes a “good school” for their child.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=" For example, many parents' notions of school quality, and subsequent choices, are influenced by what they hear through their social networks vs." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=" more formal indicators of quality (Ball & Vincent, 1998)—which may help explain why some parents' school preferences are more closely correlated to measures of “peer quality” at the school (measured by standardized test score achievement) than measures of school effectiveness (Abdulkadiroglu et al." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=" Furthermore, some parents are also influenced by school characteristics that do not necessarily directly indicate the quality of education the school offers—for example, the school's proximity to their home or other geographic characteristics (Bell, 2009)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' These preferences may be a reflection of what parents value, or a symptom of various barriers they face, like limitations in their ability to transport students to/from certain schools;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' scheduling conflicts that prevent them from visiting schools to actively evaluate how well they are likely to serve their children;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' or other 7 challenges that families living in poverty or facing other hardships may be disproportionately more likely to experience (Robertson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Whether parents have an explicit choice in which schools they select for their children, or this choice is implicit by way of where families end up living, in recent years, parents have turned to school reviews platforms like GreatSchools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content='org for insights into the quality of potential schools their children may attend (Lovenheim & Walsh, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' A recent study, however, used quasi-experimental methods (treating the expansion of GreatSchools’ ratings availability across the US as effectively random) to argue that more availability of school ratings information has actually exacerbated racial and income segregation (Hasan & Kumar, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' While perhaps counterintuitive, this is not surprising: school ratings websites often highly-weight snapshot-in-time test scores—which are notoriously correlated with the racial and income demographics of the schools (Barnum & LeMarr LeMee, 2019)—and hence indirectly drive parents to make decisions based on these factors instead of measures that more accurately capture school effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Our own research has underscored this possibility, leveraging recent advances in natural language processing to analyze how parents talk about schools in the reviews they post on GreatSchools and finding that such reviews reflect racial and socioeconomic differences in schools instead of their actual quality (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=', effectiveness ) (Gillani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' This raises questions about what kinds of 3 information these sites contain and how this information might be repositioned, or changed altogether, to minimize unintended and potentially-harmful consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' While more information can sometimes exacerbate inequalities in education, in other cases, it can reduce “information frictions” and knowledge gaps parents might face when seeking to support their children’s education.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Here, too, computational methods—combined with behavioral science-inspired interventions—have played an important role in reducing such frictions (Bergman, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Many of these studies fall beyond the realm of school choice and seek to engage parents as active sources of support in their children’s educational experiences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' For example, (Bergman & Chen, 2019) designed an SMS-based system to inform parents about their children’s class absences and performance, finding that doing so significantly reduced the rate of course failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Focusing on lower-income parents of prekindergarten children, (York and Loeb, 2014) ran an 8-month-long text messaging campaign in which they shared information highlighting the importance of literacy as well as suggestions for small-scale literacy-related activities that could be done around the home.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' They found a significant increase in parental engagement and literacy levels after children started school.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' In addition to their results, what’s particularly notable about these studies is that they move from observational analyses to intervention design and evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' As we discuss later in the chapter, computational social scientists may be particularly well-suited to help bridge existing gaps between data analysis and the design/evaluation of interventions that seek to improve educational outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=" Neighborhoods There is an active debate in the social sciences about how much neighborhoods—versus families or schools—influence a child's future outcomes." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=" Some scholars argue that a child's family has the greatest impact on their development, especially at an early age (Heckman, 3 Though, in late 2020, GreatSchools overhauled its ratings methodology in an effort to address these issues, downweighting test scores and upweighting measures of student growth and how well schools serve students from different backgrounds: https://www." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content='greatschools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content='org/gk/articles/why-we-changed-our-ratings/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' 8 2006), in part, by facilitating the development of character and other metacognitive traits like motivation (Heckman & Kautz, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=" Referencing the Perry Preschool Project, a famous early childhood intervention administered in the 1960s that included home visits to help strengthen parents' abilities to support their children's learning (Schweinhart et al." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=", 1993), these scholars note how some of the intervention's greatest long-term effects (e." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' higher earnings, lower crime rates) accrued to those who lived in the worst neighborhoods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' They use this to argue for “the relative unimportance of ZIP codes in explaining the observed intergenerational program effects on the children of Perry participants” (Heckman & Karapakula, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=" Others argue that schools, more than neighborhoods, shape a child's future trajectory." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=", Fryer & Katz (2013) use quasi-experimental evidence documenting changes in children's neighborhood and school quality—comparing findings from the randomized experimental voucher-based “Moving to Opportunity” (MTO) study (Katz et al." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=", 2001) to academic gains of those who received lottery admissions to a high-quality charter school in New York City (Dobbie & Fryer, 2009)—to suggest high-quality schools affect a child's long-term academic achievement and earnings more than better neighborhoods." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=" Yet recent experimental and quasi-experimental analyses have revealed the large impact a child's neighborhood can have on their academics and future earnings." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=" These studies critically note that how much a neighborhood shapes a child's future is, on average, a function of how much time the child actually spends growing up there." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' For example, a few years after Fryer & Katz, 2013 was published, several researchers revisited the MTO study with two new lenses: i) longer-term earnings data for children (via tax returns) who participated in the study, and ii) an analysis of how future earnings for these children differed depending on how old they were at the time of the move.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' The researchers found that when children moved to a lower-poverty neighborhood before the age of 13, their college attendance rates and future earnings increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' On the other hand, older children experienced no such effects (Chetty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=" Members of this research team have conducted several additional studies leveraging administrative (intergenerational tax and Census) records to quantify the impact a child's geography has on their future outcomes, like earnings." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=" For example, they analyze data describing millions of families who moved across commuting zones in the US to find that exposure to better neighborhoods leads children's future earnings to converge at a rate of 4% per year that they spend in that neighborhood to the future earnings of children who were already living there (Chetty & Hendren, 2015)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=" The authors leverage several econometric tools to control for potential confounders (like family effects) and identify this effect as the “causal effect of place” on a child's future outcomes." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' While identifying the underlying mechanisms driving these effects will require more research, the authors find a variety of neighborhood characteristics that positively correlate with upward mobility, including: i) lower poverty rates, ii) more social capital, iii) higher-quality schools (as measured by higher test scores), and iv) more college graduates (Chetty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=', 2018a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=" In a separate paper, researchers disaggregated the effects of neighborhoods on future outcomes by children's race and gender to find that black boys were much less likely to achieve upward mobility than white boys, even when they grew up in similar neighborhoods." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=" Interestingly, one of the features of neighborhoods with smaller gaps 9 between black and white boys' future mobility rates was the presence of more black fathers per capita (not necessarily a child's own father within the household)—pointing to the possible impact, again, of role model figures on facilitating better outcomes (Chetty et al." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=', 2018b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Finally, a recent collection of follow-on studies leveraged an unprecedented scale of Facebook data (70 billion friendships across 20 million individuals) to shed new empirical light on a long-standing topic: the belief that social capital, defined as friendships across socioeconomic divides, has a causal effect on intergenerational upward mobility (Chetty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=', 2022a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Chetty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=', 2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' These new studies illustrate how digital trace data and administrative records, together, can help illuminate potential pathways to improved life outcomes for youth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Thus, much like schools, neighborhoods may affect children through how they expose (or fail to expose) children to role models and mentors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' This possibility is underscored by another recent study, which highlighted how children who grow up in geographies with more inventors are more likely to become inventors themselves in the same technology classes that are prevalent in those geographies (Bell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Thus, children growing up in the Bay Area near Silicon Valley are more likely to invent new software/computer technologies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' children growing up in the Midwestern US are more likely to invent medical devices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' The authors argue that such differences point to a causal impact of exposure to innovation: children who are exposed to innovation are more likely to become inventors themselves when they grow up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Unfortunately, girls, children of color, and those living in low-income environments are often less likely to be exposed to inventors who share their backgrounds, reflecting longstanding structural inequity in society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' These recent findings suggest that boundaries between schools, families, and neighborhoods may actually be more fluid and interleaving than some researchers have previously suggested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=" If we think of schools as a part of many children's neighborhood experiences, and generalize the influence a family has to the impact their social network (which includes family, but also neighbors, peers, role models and others) has—it is clear that neighborhoods constitute a “meta-mechanism” that influence children's futures through a range of possible channels that include schools and family-like figures." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Looking ahead: exploring social factors in education with computational social science As many of the citations so far suggest, much of the existing computational work exploring social factors in education has been conducted by applied microeconomists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' However, there is a need and opportunity to draw more sociologists and philosophers of education into these debates—particularly to offer new perspectives on theories of social justice and the values that influence how we think about the role and purpose of education in an ever-evolving social context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Some have argued that rapidly-expanding datasets and new computational methods will bring about “the end of theory” (Anderson, 2008), yet the creativity and nuance of many of the studies we’ve discussed in this chapter should make it clear that asking meaningful questions will require much more than data mining alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Computational social scientists can partner with sociologists and philosophers to incorporate their rich theories to inform new empirical questions, while also illuminating how empirical inquiry can spark the development of novel theories and hypotheses (Blades, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Playing this intermediary role is sorely needed to more holistically investigate how social factors 10 shape educational and life outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Below, we outline some of the ways in which computational social scientists can help bridge these worlds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Measuring a broader set of inputs and outcomes One of the inherent limitations in exploring social factors that shape children’s educational and life outcomes is capturing the “right” data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' One simple way to think of such data is in terms of inputs and outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' For example, throughout this Chapter, we’ve shared studies that explore relationships between inputs like students or teachers, and outcomes like academic achievements (as measured by performance on standardized tests).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Despite the important role of social factors in shaping educational and life outcomes, few studies develop robust ways of measuring the strength of social ties (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' by capturing social network data between participants ), or other related measures that have been theoretically 4 linked to improved outcomes for youth and families, like social capital (Chetty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=', 2018a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Putnam, 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Thus, looking ahead, can we develop more robust ways of capturing and incorporating social network information into existing models?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' For example, perhaps by using data from existing social networking platforms (a la recent efforts by Chetty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=', 2022a and Chetty, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=', 2022b)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Or by creating new methods for capturing the (evolving) social networks of students and critical peer and adult figures in their lives?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Doing so may help offer more insights into the nature and kinds of social ties that are most associated with achievement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=" More ambitiously, future work could make even stronger connections with the long standing literature that makes far more visible the process and practices of schools—what is sometimes called the 'new sociology of education' (e." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Young, 1972)—and with those focused on critical pedagogy (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Friere, 1970).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Furthermore, in terms of outcomes, how might we broaden outcome measures beyond performance on tests to capture more about the impact social factors have on children’s conceptions of self, feelings about their place and purpose in the world and in relation to others, or other aspects of living a quality of life that may be correlated with—but still distinct from—easier-to-measure, inherently quantitative factors like test scores (or even longer-term measures like future income)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' These are difficult questions that will benefit from fresh, creative thinking from computational social scientists interested in developing new input and outcome measures from existing and new datasets—but that can only be achieved through meaningful engagement with other academic communities that have different philosophical, conceptual and often methodological approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Starting with the questions, not methods One of the benefits of established disciplines is that they often have built up, over decades or even centuries, established approaches and methods to different types of research questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' For example, linear regression models and their many variants are often the go-to choices for many applied microeconomists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' While such methodological focus can produce efficiencies in data analysis and sharing, it can also—over time—constrain the types of questions that even seem plausible to ask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' For example, linear regression is limited as a tool for natural language processing, where the words and phrases may have complex, non-linear dependencies in sentences;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' where the feature space may be much larger than the number of observations in the dataset itself (and hence, prone to overfitting);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' In 4 There are a few notable exceptions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' (Paluck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' 11 recognition of this, social scientists are developing a richer cadre of “Text as Data” methods (Gentzkow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' By starting with the questions and turning to methods as needed to explore and answer such questions as deeply and richly as possible, computational social scientists can help draw attention to a wider range of topics on the education research agenda—including those that may not directly identify causal relationships (which is of great interest to many applied microeconomists) but still help yield valuable insights that inform downstream causal analyses, design-based research, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' (Singer, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' A byproduct of such an approach may be a greater degree of methodological diversity applied to different research problems, and over time, an expansion of the questions researchers and practitioners even deem possible to ask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Unpacking the underlying mechanisms driving certain observed outcomes Much of the applied microeconomics literature on neighborhood effects or teacher effectiveness outline how much neighborhoods or teachers can increase children’s shorter and longer-term outcomes, but few unpack at a granular level what it is about neighborhoods, or teachers, or other social factors that make them more or less effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Indeed, looking back through the history of the sociology of education, we see a similar criticism of early work from economists and quantitative researchers in the 1960s and 70s exploring questions of equity and schooling, many of whom treated schools and associated institutions as a black box (Weiss, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' It is naive to think data and computational methods alone will enable researchers to answer these questions more deeply in the coming years;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' a rich body of qualitative research will continue to be indispensable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' This will encompass a continuation of research that is broadly considered more qualitative or quantitative in the traditional sense, but there may also be opportunities for novel computational work that spans across these boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Nevertheless, there is still a critical role of computational methods in this pursuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' For example, might we use techniques from computer vision applied to historical corpora of neighborhood-level Google Street View images (a la Naik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=', 2017) to better understand environmental predictors, and eventually, causal factors responsible for different rates of upward mobility across neighborhoods?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Can we access and merge datasets spanning notes and audio/video recordings describing teacher classroom observations (a la Kelly et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=', 2018), students’ written feedback, and other inputs to identify behaviors and practices that lead to larger gains in life outcomes for students?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' These and other questions may benefit from a fresh look from computational social scientists hailing from different disciplinary backgrounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' They will also require thoughtful considerations of data privacy and ethics, as new streams of data will bring with them new questions about how they should or should not be processed—and what their unintended consequences might be (Hakimi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Unpacking mechanisms is not just about providing richer research about particular phenomena, it is also about meaningfully engaging with researchers that have very different ontological and epistemological worldviews to further refine and develop the conceptual frames often seen in education data science (Eynon, 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Furthermore, unpacking mechanisms requires challenging our own notions of access and “success” in education, and not treating education as a straightforwardly positive and neutral ‘thing’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' What we have seen from the discussion above is that both the learning analytics/sciences and social factors perspectives offer a strong focus on distributional issues related to education.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' In other 12 words, there is a tendency to focus primarily on questions related to the resources required to ensure everyone has access to a good quality education (typically measured in quite narrow ways related to achievement on standardized tests) (Keddie, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' This may be, for example, through the use of more technology in the classroom, the use of grants or vouchers, using technology to provide more or better information to help with school choice, or trying to ensure that where a person can afford to live does not impact the quality of the school systems a young person can access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' These are all important issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' However, questions of equity and social justice—which are intrinsically linked to issues of quality education—have multiple dimensions (North, 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Beyond questions of distribution, we must also explore the cultural and political dimensions of injustice (Fraser, 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Specifically, questions of misrecognition (such as cultural domination and disrespect) that marginalised communities may experience at school, through, for example, the ways that young people are treated, and what is / what is not included in the curriculum (Power and Frandiji, 2010:388).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' And questions of misrepresentation may arise due to potential deficits of educational governance across school systems which make participation for some groups significantly challenging (Sayed, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Relatedly, we can enhance awareness of how some interventions designed to make schooling more equal can inadvertently reinforce the status quo or cause harm to different groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Research has shown, for example, how the common focus on distributive approaches to social justice in schools tends only to actually further stigmatise “the poor” rather than change economic structures to make society fairer (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' North, 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Leibowitz & Bozalek, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Achieving equity and justice in schools (and indeed in society more broadly) requires attention to the complex interplay between economic, cultural and political factors (Fraser, 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Computational social scientists may partner with sociologists, educationalists, and philosophers of education or familiarize themselves with related theories and methods, to bring a more nuanced understanding of the complex role that education plays in society (beyond delivering learning and qualification) and ways of theorising about social justice and education that would add to the existing research in this domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Bridging analysis and design As computational social scientists begin to more deeply unpack the underlying mechanisms responsible for observed effects, they will be uniquely suited to also help inform, and perhaps even lead, the design and development of interventions that seek to improve educational and life outcomes for youth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Currently, a wide range of researchers participate in the design and evaluation of education-related interventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Often, these interventions come in the form of applied microeconomists running randomized control trials like those highlighted earlier or in more international settings (Gibson & Sautmann, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' They also come, however, in the form of computing and design researchers using both human-centered and participatory design methods to develop frameworks and systems that seek to serve different segments of the population (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Costanza-Chock, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Not all computational social scientists may wish to participate in design-related activities that pick up where analyses of secondary data (or even preliminary causal analyses) leave off, but their methodological and domain insights may lend them to fill an important “in-between” 13 space that connects theory and methods from research domains to the practical challenges and considerations of intervention in field settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Engaging in these in-between spaces may help create new fruitful directions for research, and also, meaningful social change—with due deference and humility in light of how different cultures and groups of people may desire (or not desire) such interventions (Irani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=', 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Expanding the audience of research and evaluating/improving “how it lands” Learning analysts and engineers often see digital platform developers or certain groups of users (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' teachers and school administrators) as important audiences for their findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Applied micro economists and sociologists may largely target policymakers—often at state and national levels—as their primary audiences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' As computational social scientists select questions to work on, they also have an opportunity to think creatively about which audience(s) they wish to engage directly in dialogue with their work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Much like our point about methods above, letting the questions drive the thinking around which audiences are appropriate for the work—instead of assuming a priori that the audience must be a government policymaker, or platform designer, or some other known stakeholder—may help surface new ideas for people and systems that may have interest in learning about and building upon the research findings that computational social scientists help produce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Simply picking an audience, however, does not guarantee positive social change;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' there are significant barriers to communicating findings from research and science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' The COVID-19 pandemic has made this particularly clear (if it wasn’t already before): for example, school-based mask mandates were a hotly contentious issue across school districts in the US (Cottle, 2021), and more generally, vaccine hesitancy levels remain high (Sreedhar & Gopal, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' The ways in which individuals, organizations, policymakers, and governments as a whole process and act upon scientific findings is a complex function of individual motivations and preferences;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' sociopolitical forces;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' notions of fairness and morality;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' and several other factors (Haidt, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' There are also more tactical factors at play, like time scarcity (Rogers & Lasky-Fink, 2020) and the ability of decision-makers to identify the relevance of existing research to their own contexts (Nakajima, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' There is, therefore, a tremendous need for research that explores not only which policies and practices help improve educational outcomes, but also, how those practices can most effectively be communicated and positioned in order to increase the chances of their eventual implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' These communications-based research questions are certainly pertinent to education, but also other domains, and could benefit from the creative thinking and new questions that computational social scientists are well-positioned to put forth and test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Conclusion In this Chapter, we have presented two perspectives on how computational social scientists might engage with research pertaining to education.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' We have focused more on the social factors that shape educational outcomes—namely, schools, families, and neighborhoods—given the impact such factors have on influencing educational access, experiences, and outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' We believe it is an exciting time for computational social scientists to apply their interstitial interests and skill sets to questions in education—and to creatively come up with new questions that we can’t even imagine yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' 14 Further readings Interested readers may look to (Fischer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=', 2020) for a more in-depth review of the ways in which computational methods are being used to better understand data from digital learning environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Readers who wish to learn more about using behavioral science to foster greater parental engagement in schools may look to (Bergman, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Finally, for a broader view on trends in the use of data science and other computational methods in education, we refer readers to (McFarland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' 20659.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' 22 Young, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=', 1972.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Knowledge and control: New directions for the sociology of education.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' British Journal of Educational Studies, 20(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Xing, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' and Goggins, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' “Learning analytics in outer space: a Hidden Naïve Bayes model for automatic student off-task behavior detection.” In Proceedings of the Fifth International Conference on Learning Analytics and Knowledge, 176–183.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Zhao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=', Bhatt, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=', Cooper, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' and Shamma, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Flexible Learning with Semantic Visual Exploration and Sequence-Based Recommendation of MOOC Videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' Proceedings of ACM CHI Conference on Human Factors in Computing Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} +page_content=' 23' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE5T4oBgHgl3EQfeg9M/content/2301.05619v1.pdf'} diff --git a/btAzT4oBgHgl3EQfnf1c/content/tmp_files/2301.01581v1.pdf.txt b/btAzT4oBgHgl3EQfnf1c/content/tmp_files/2301.01581v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..be6bdf853fd30a317f35033ac853e9fa8f1e2d8d --- /dev/null +++ b/btAzT4oBgHgl3EQfnf1c/content/tmp_files/2301.01581v1.pdf.txt @@ -0,0 +1,941 @@ +arXiv:2301.01581v1 [cond-mat.stat-mech] 4 Jan 2023 +Universal to Non-Universal Transition of the statistics of Rare Events During the +Spread of Random Walks +R. K. Singh∗ and Stanislav Burov† +Department of Physics, Bar-Ilan University, Ramat-Gan 5290002, Israel +Particle hopping is a common feature in heterogeneous media. We explore such motion by using +the widely applicable formalism of the continuous time random walk and focus on the statistics of +rare events. Numerous experiments have shown that the decay of the positional probability density +function P(X, t), describing the statistics of rare events, exhibits universal exponential decay. We +show that such universality ceases to exist once the threshold of exponential distribution of particle +hops is crossed. While the mean hop is not diverging and can attain a finite value; the transition +itself is critical. The exponential universality of rare events arises due to the contribution of all the +different states occupied during the process. Once the reported threshold is crossed, a single large +event determines the statistics. In this realm, the big jump principle replaces the large deviation +principle, and the spatial part of the decay is unaffected by the temporal properties of rare events. +According to Wikipedia, ”Rare or extreme events are +events that occur with low frequency, and often refers to +infrequent events that have widespread impact and might +destabilize systems”. +Notable examples of rare events +include the stock market crash [1], earthquakes [2] and +cyclones [3]. The most recent on the list is the 2020 pan- +demic, which brought the whole world to a standstill. +The frequency and the scale of rare events in each field +are very important. If the rare events are not ”too rare” +and large enough, then the usual statistical behavior is +completely dictated by such rare events. A perfect exam- +ple of situations when rare events fundamentally modify +the nature of a physical process is sub-diffusion [4–10]. +While for normal diffusion, the mean squared displace- +ment (MSD) of a tracked particle grows linearly with +time, for sub-diffusion, the MSD grows in a non-linear +fashion, i.e. ⟨X2⟩ ∼ tα, while 0 < α < 1. Sub-diffusion +generally occurs when there are long time or space cor- +relations present. One of the sources for such correla- +tions can appear in the form of an extremely long so- +journ times, when the tracked particle is trapped in a +specific region in space [11]. When sub-diffusion occurs +in amorphous materials like glasses [12] or living cells +[13–15], it is quite common to observe trajectories that +are dominated by several (and sometimes even one) very +long trapping events [16]. +In many cases, the appearance of sub-diffusion is ac- +companied by unusual physical phenomena: aging [17– +19], weak ergodicity breaking [17, 20, 21] and non-self- +averaging [22–25]. Studies of theoretical models where +trapping is present, like the continuous time random walk +(CTRW) [26] and the quenched trap model (QTM) [27] +show that there is a critical transition to sub-diffusion, +aging, and non-ergodic behavior, when the mean trap- +ping time is diverging. +This transition is also accom- +panied by a transformation of the positional probability +density function (PDF). The universal Gaussian center +of the positional PDF turns into an α-stable L´evy type +[28, 29], i.e. shape that depends on a specific parameter +α that determines the trapping times. This transition +between a universal PDF that is determined by an ac- +cumulation of many events, and a non-universal PDF +that follows the properties of one single rare event is a +key feature associated with the the appearance of new +physical phenomena. Can transitions from universal to +non-universal PDFs symbolize an underlying phase tran- +sition that might lead to uncovered surprising physics, +like in the case of sub-diffusion? What are the proper- +ties of single rare events that can cause such a critical +transition in behavior? In this work, we focus on the ap- +pearance of the universal to the non-universal transition +of a positional PDF that, unlike the case of sub-diffusion, +appears for large positions |X| (the tails of the PDF). +Recently, in a huge number of experiments, exponen- +tial (and not Gaussian) decay of the PDF for large |X| +was spotted. Such decay of the PDF is termed Laplace +tails [30]. Notable examples include colloidal beads in +heterogeneous optical force field [31], zooplanktons under +crowding [32], glass-forming liquids [33], nanoparticles in +polymer melts [34], motion of particles at liquid-solid in- +terface [35], particle displacements close to glass and jam- +ming transitions [36, 37] etc. In [38] it was suggested that +there might exist a universal convergence to the Laplace +tails. +Indeed it was proven, by exploiting the CTRW +model that for a wide range of processes, the tails of the +PDF are determined by the accumulation of many events +and follow exponential decay (up to logarithmic correc- +tions) [39–41]. So by following the presented discussion, +we ask: Is there a transition from universal Laplace tails +to non-universal and process-specific tails? And if so, is +it accompanied by a critical transition like the diffusion +to subdiffusion transition? Are the tails of a PDF deter- +mined by one occurrence of one single event [42–46] or +they represent an accumulation of many realizations of +not-so-large but more frequent events [47–51]? +Similar to the case of subdiffusion to diffusion tran- +sition, we use the celebrated CTRW model, originally +exploited for explaining motion in amorphous materials +[12–15] and exponential decay of tails [30]. In this model, +a particle performs random jumps in space and waits for + +2 +a random amount of time between every two jumps. All +the jumps and waiting times are independent and iden- +tically distributed (IID) random variables. The distribu- +tion of a jump x is given by f(x), while each waiting time +τ has a distribution ψ(τ). The position of the process X +at time t is determined by the random number of jumps +Nt performed by time t, i.e. X = x1 +· · ·+xNt where xi, +is the size of a single jump. The positional PDF P(X, t) +is readily obtained in terms of the subordination equation +by conditioning on the number of jumps N, [4, 27, 52, 53] +P(X, t) = +∞ +� +N=0 +PN(X)Qt(N) +(1) +where PN(X) is the distribution of X for a given N(= Nt +for a fixed measurement time t) and Qt(N) is the distri- +bution of the number of jumps up to time t. The men- +tioned phenomena, like anomalous transport and aging, +appear for CTRWs with ψ(τ) ∼ τ −1−α (τ → ∞) when +α < 1. For such ψ(τ), the sum τ1+· · ·+τNt is dominated +(in the t → ∞ limit) by the maximal summand [54], as +can be observed from the behavior of +φα(t) = +�max{τ1, ..., τNt} +�Nt +i=1 τi +� +(2) +for large t. While the maxima of a set of random vari- +ables has been extensively studied [55–58], the definition +of φα(t) has the advantage that it ranges from infinites- +imal to finite values. This makes it appropriate as an +order parameter. We see in Fig. 1 (inset) that φα(t) sat- +urates to a finite value for α = 0.5, 0.8, 0.9 while for +α = 1.1, 1.2, 1.5 exhibits a decaying behavior at large +times. This difference in the properties of φα(t) follows +from the fact that for α < 1, ⟨τ⟩ = +� ∞ +0 +dτ τψ(τ) → ∞, +while it is finite for α > 1. This implies that as we move +from α > 1 to α < 1, the rare fluctuations in the sequence +of waiting times {τ1, ..., τNt} exhibit qualitatively differ- +ent behaviors. Consequently, the thermodynamic limit in +which the system behaves extensively (all waiting times +τi are of the same order of magnitude) ceases to exist for +α < 1. Can we see similar behavior if we focus on spa- +tial fluctuations? This question comes up naturally once +we realize that the system has access to the entire phase +space in the thermodynamic limit. The phase space can +be swept by focusing on large spatial fluctuations with- +out going to the t → ∞ limit. This means we take the +large |X| limit and expect the system to visit many states +during its evolution. +Motivated by this, let us look at a CTRW described by +Eq. (1) with jumps following f(x) ∼ e−|x|β with β > 0 +[59]. In analogy with φα(t) let us define +Fβ(X) = +�max{|x1|, ..., |xNt|} +�Nt +i=1 |xi| +� +. +(3) +We see in Fig. 1 that even for a well behaved ψ(τ) like +the exponential distribution, lim|X|→∞ Fβ(X) decreases +0.2 +0.4 +0.6 +0.8 +1.0 +10-1 +100 +101 +102 +Fβ(X) +X +β = 1/4 +1/3 +1/2 +0.8 +0.9 +1.0 +1.1 +1.25 +1.5 +10-2 +10-1 +100 +101 +105 +109 +t +φα(t) +FIG. 1: Fβ(X) for a CTRW with jumps x ∼ f(x) with mean +zero and variance one as a function of β. The waiting time +distribution ψ(τ) = e−τ and the measurement is done at t = +0.6 for trajectories reaching a position ±X. Inset shows the +behavior of φα(t) for ψ(τ) ∼ τ −1−α for large τ and α = +0.5, 0.8, 0.9, 1.1, 1.2, 1.5 from top to bottom. +monotonically for β ≥ 1 and takes a finite value for tra- +jectories with β < 1. In other words, +lim +|X|→∞ Fβ(X) = +� +0, +β ≥ 1 +cβ, +β < 1 +(4) +where cβ is a nonzero constant that depends on β. Al- +though cβ is seen to approach unity for β = 1/2 and +lower values, β = 0.9 seems to saturate to a value less +than one. While the transition of φα(t) is driven by the +divergence of the mean waiting time ⟨τ⟩, this is not the +case for the transition of Fβ(X). All the moments ⟨xn⟩ +are finite for β < 1. Nevertheless, the effective behav- +ior of limt→∞ φα(t) and lim|X|→∞ Fβ(X) is similar. As +a function of the parameters α/β, there is a transition +from a state defined by the accumulation of many events +to a state dominated by a single large event. +At this point, it is worth noticing that the role played +by φα(t) for describing temporal fluctuations is taken +over by Fβ(X) for spatial fluctuations. While the emer- +gence of a non-Gaussian center accompanying the transi- +tion of φα(t) is well understood [28, 29] the correspond- +ing behavior of P(X, t) accompanying the transition of +Fβ(X) is not known. +It was shown in Ref. [39] that +P(X, t) exhibits universal tails exhibiting exponential de- +cay for β > 1 and ψ(τ) analytic near τ = 0. Furthermore, +just like φα(t) → 0 marks the emergence of a univer- +sal Gaussian center following the central limit theorem, +Fβ(X) → 0 at |X| → ∞ characterizes the universal expo- +nential tails of P(X, t) [39]. On the other hand, a finite +value of φα(t) for α < 1 is reminiscent of the L´evy stable +PDF and is a reminder of the dynamical phase transition +(DPT) [60–63] characterized solely by the rare fluctua- + +3 +tions, but in the temporal domain. Does it mean we can +see a similar DPT at finite times if we focus on the tails +of P(X, t)? We now answer this question by exploring +the case of β ≤ 1. +For β = 1, the jumps follow Laplace distribution, that +is, f(x) = a +2 exp(−a|x|). This implies that the charac- +teristic function of a single jump, defined as the Fourier +transform ˆλ(k) = +� ∞ +−∞ dx eikxf(x) = +a2 +a2+k2 [26]. Un- +like the case for β > 1, ˆλ(k) for β = 1 has poles in the +complex plane. Furthermore, since the jumps are IID, +the distribution of a sum of N jumps in Fourier space +is ˆλN(k) and inverting it via contour integration [64] we +find that the distribution of position X after N jumps is +[65] +PN(X) = +ae−a|X| +22N−1Γ(N) +N−1 +� +m=0 +(2N − m − 2)! +m!(N − m − 1)!(2a|X|)m +≈ ae−a|X| +22NΓ(N) +� N +0 +dm exp[K(N, m)] +(5) +where K(N, m) = (2N − m) log(2N − m) − m log(m) − +(N − m) log(N − m) − (N − m) + m log(2a|X|). The last +line follows from Stirling’s approximation N! ≈ N Ne−N. +In order to evaluate the integral in (5) we use Laplace’s +method and locate m0 such that +dK +dm|m=m0 = 0. This +implies m0 = a|X| + N − +√ +a2X2 + N 2. +In the limit +a|X|/N ≫ 1, m0 ≈ N − +N 2 +2a|X| and then from (5) the +large deviation form is obtained +PN(X) ∼ exp +� +− NIN +�|X| +N +�� +, +(6) +with the rate function IN +� +|X| +N +� += − a|X| +N ++ log +� +a|X| +N +� +. +This implies that for large deviations, the distribution +of the sum possesses exponentially decaying tails with +logarithmic corrections. For ψ(τ) analytic near zero +ψ(τ) +τ→0 +∼ CAτ A + CA+1τ A+1 + CA+2τ A+2 + · · · , +(7) +where A is a non-negative integer, Qt(N) admits a large +deviation form Qt(N) +N→∞ +∼ +exp[−NIN(t)] with a uni- +versal rate function [66] IN(t) = − CA+1 +CA +t +N − (A + 1) +� +1 + +log +� +(CAΓ(A+1)) +1 +A+1 +A+1 +t +N +�� +. +Using PN(X) from (6) and +Qt(N) in (1) we have for large N [67] +P(X, t) ≈ +� ∞ +0 +dN exp[κ(N)] ≈ +� +2π +|κ′′(N0)| exp[κ(N0)], +(8) +where +κ(N) = −a|X| + N log +�a|X| +N +� +− Ct + N(A + 1) +� +1 + log +�d2 +N +�� +, +(9) +10-8 +10-6 +10-4 +10-2 +100 +-20 -10 + 0 + 10 20 +(a) +P(X,t) +10-8 +10-6 +10-4 +10-2 +100 +-20 -10 + 0 + 10 20 +(b) +10-8 +10-6 +10-4 +10-2 +100 +-20 -10 + 0 + 10 20 +(c) +P(X,t) +X +10-8 +10-6 +10-4 +10-2 +100 +-20 -10 + 0 + 10 20 +(d) +X +FIG. 2: Comparison of numerically estimated P(X, t) (red +circles) against the solution given in (11) (black dashed line). +The waiting time distributions are the following: (a) exponen- +tial mixture ψ(τ) = p1r1e−r1τ +p2r2e−r2τ with r1 = 1/4, r2 = +5/2, p1 = 1/4, p2 = 3/4 at t = 0.7; (b) gamma distribution +ψ(τ) = τ 3e−τ/6 at t = 0.8; (c) half Gaussian distribution +ψ(τ) = +� +2/πe−τ2/2 at t = 1.5; (d) power law distribution +ψ(τ) = 1/(1 + τ)2 at t = 0.4. +with C = − CA+1 +CA +and d2 = [CAΓ(A+1)] +1 +A+1 +A+1 +t and N0 is the +solution of κ′(N0) = 0. For large a|X|/N we have +N0 ≈ µ(a|X|) +1 +A+2 +(10) +with µ = d +A+1 +A+2 +2 +. Using κ′′(N0) = − A+2 +N0 in (8) we find +P(X, t) +∼ +|X|/t→∞ +� +2π +A + 2µ(a|X|) +1 +A+2 +exp +� +− t +� +C + a|X| +t +− +� +CAΓ(A + 2)a|X| +t +� +1 +A+2 �� +(11) +From Fig. 2 we see that the large deviation form of +P(X, t) evaluated in (11) agrees with numerically esti- +mated P(X, t) for different waiting time distributions. +In other words, P(X, t) possesses exponentially decaying +tails in the limit of large |X|/t when the distribution of +jumps is Laplace distributed. +P(X, t) derived in (11) holds for a wide class of waiting +time distributions analytic near zero. This further im- +plies that the rare fluctuations for a CTRW with Laplace +distributed jumps are described by the large deviation +principle [51]. In this regard, the case β = 1 is analogous +to the β > 1 case discussed in Ref. [39–41] if we restrict +our attention solely to the exponentially decaying fluc- +tuations of P(X, t). The analogy, however, ends here as +for β > 1 N0 increases linearly with |X| while for β = 1, +the growth is sublinear. +Furthermore, for β > 1 the +PDF exhibits exponentially decaying tails with logarith- +mic corrections [39] while for β = 1 the corrections are of + +4 +power-law type. Even though P(X, t) exhibits exponen- +tially decaying tails for both β > 1 and β = 1, different +forms of correction term for the two cases “hints” towards +a possible transition. Let us now explore the region β < 1 +to complete our understanding of this transition. +For β < 1 the distribution of jumps belongs to the +class of stretched exponential distributions [42] which +possesses heavy tails as � ∞ +0 +dx eλxf(x) = ∞ ∀ λ > 0 +and does not admit a large deviation form [51]. It is a +well-known result that the family of stretched exponen- +tial distributions satisfies the big jump principle [42] +P(x1 + · · · + xN ≥ X) +|X|→∞ +∼ +P(max{x1, ..., xN} ≥ X) +(12) +with the right hand side evaluating to 1 − +� +1 − +� ∞ +X dx f(x) +�N +for IID xi. Hence, from (1) we have +� ∞ +X +dX P(X, t) +|X|→∞ +∼ +1 − Gt +� +1 − +� ∞ +X +dx f(x) +� +(13) +where Gt(z) = �∞ +N=0 zNQt(N). Furthermore, for large +|X| we have +� ∞ +X dx f(x) ∼ 0, as a result we can analyze +P(X, t) in terms of the behavior of Gt(z) for z in the +neighborhood of unity. Now Gt(1−η) ≈ Gt(1)− ∂G +∂z |z=1η +for η small and ∂G +∂z = �∞ +N=1 NQt(N)zN−1. This implies +Gt +� +1 − +� ∞ +X dx f(x) +� +≈ 1 − ⟨Nt⟩ +� ∞ +X dx f(x) and from +here it follows that +P(X, t) +|X|→∞ +∼ +⟨Nt⟩f(X). +(14) +The above equation implies that the probability of be- +ing at a location X at time t equals the mean num- +ber of jumps ⟨Nt⟩ up to time t times the distribution +of a single jump f(X). With the distribution of a single +jump known, we only need to estimate the mean num- +ber of jumps ⟨Nt⟩ which in the Laplace domain reads +[26] ⟨ ˜ +Ns⟩ = +˜ +ψs +s(1− ˜ +ψs), where ˜ψs = +� ∞ +0 +dt e−stψ(t) is the +Laplace transform of ψ(τ). For ψ(τ) analytic near zero +(cf. (7)) we have in the limit t → 0 [68] +⟨Nt⟩ ≈ CAΓ(A + 1) +Γ(A + 2) +tA+1 + CA+1Γ(A + 2) +Γ(A + 3) +tA+2. +(15) +The reason to focus on the t → 0 limit is that it allows us +to address the rare fluctuations exhibited by the CTRW +at finite times, that is, |X|/t → ∞. +Specifically, in +many experiments that show Laplace decay of the PDF, +the non-Gaussian behavior was spotted for short enough +times [30]. For a long measurement time, the Gaussian +center eventually takes over [69]. Notwithstanding the +limited range of validity of (15), P(X, t) derived in (14) +holds at arbitrary times, and is in excellent agreement +with numerical simulations (see Fig. 3). +10-6 +10-4 +10-2 +100 +-800 +-600 +-400 +-200 + 0 + 200 + 400 + 600 + 800 +P(X,t) +X +e-|x|1/3 +, t = 0.4 +e-|x|1/2 +, t = 1.5 +FIG. 3: Numerically estimated P(X, t) for a CTRW with +jumps following a generalized Gaussian distribution with PDF +f(x) = +β +2Γ(1/β)e−|x|β with β = 1/3, for ψ(τ) = +� +2/πe−τ2/2 +(red circles) and β = 1/2 for ψ(τ) = +1 +(1+τ)2 (blue squares). +The black dashed lines represent the analytical form from +(14). +The result in (14) shows that the parameter range β ∈ +(0, 1) is markedly different from the region β ≥ 1. In ad- +dition, differences in the nature of the PDF of CTRW at +β = 1 and β > 1 imply towards the fact that the PDF of +a CTRW critically changes at β = 1, which is essentially +the same value for which the order parameter Fβ(X) +shows a critical transition. +The evaluation of P(X, t) +further corroborates our assertion of a universal to non- +universal transition as seen from the analysis of Fβ(X) +(see Fig. 1). The fact that lim|X|→∞ Fβ(X) = 0 for β ≥ 1 +is analogous to saying that P(X, t) ∼ exp[−tI(|X|/t)] +exists with a nontrivial rate function I(|X|/t) for ev- +ery β ≥ 1. +This rate function I(z) attains a linear +growth for large z and, therefore, the universal expo- +nential decay of the PDF, i.e., P(X, t) ∼ e−|X|. On the +other hand, for β < 1 we had seen from Fig. 1 that +lim|X|→∞ Fβ(X) = cβ > 0. +Eq. (14) shows that for +β < 1, the rate function I(|X|/t) is trivially zero. For +large |X|, the decay of the PDF is stretched exponen- +tial, i.e., P(X, t) ∼ e−|X|β, that specifically depends on +the parameter β. Notice that the transition of φα(t) in +the long time limit represents the transition from diffu- +sion to subdiffusion and is accompanied by the transition +of the PDF (in the |X|/t → 0 limit) from the universal +Gaussian form to the α-stable L´evy type that explicitly +depends on α. +Hopping dynamics which is an intrinsic feature of +CTRW, has been ubiquitously observed in polymer melts +[70], colloidal suspensions [71], rodlike particles through +smectic layers [72, 73], polymer glasses [74], binary mix- +tures [75], in one, two, and three spatial dimensions [76], +to mention a few. A characteristic feature of motion in +glassy materials [36–38, 77] and at the liquid-solid inter- +face [35, 78], where hopping dynamics is observed, has + +5 +been the exponential decay of the tails of the positional +PDF. Exponential decay, however, is the rule whenever +hopping dynamics is in play [39], making it a univer- +sal feature of transport in heterogeneous media. But in +some situations, like the case of particles with a constant +supply of energy (e.g., run and tumble particles), the par- +ticles can perform really long jumps during their explo- +ration of the heterogeneous media [79]. Our work shows +that a critical transition is expected for any system in- +volving such hops. This critical transition manifests itself +at the level of the positional PDF, where the universality +of Laplace tails ceases to exist. While the universal tails +are an outcome of an accumulation of many events and +the applicability of the large-deviation principle, the spe- +cific tails for β < 1 are determined by one single event, +that is, the big-jump principle. +Unlike a diffusion-to-subdiffusion transition which +takes place at long times and is accompanied by diver- +gences of the mean trapping time, the phase transition +reported in the present study is free from such diver- +gences. +The mean length of a hop can be finite, and +the transition is observed at finite times. Furthermore, +while the temporal properties of rare events leading to +subdiffusion affect the bulk, rare spatial events manifest +themselves mainly in the tails of the PDF. Interestingly, +the temporal features do not affect the spatial depen- +dence of the statistics of the rare events, that is, the tails +of P(X, t). +Acknowledgments: This work was supported by the Is- +rael Science Foundation Grant No. 2796/20. 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Exp. 2021, 103208 (2021). + +arXiv:2301.01581v1 [cond-mat.stat-mech] 4 Jan 2023 +Supplementary Material for Universal to Non-Universal Transition of the statistics of +Rare Events During the Spread of Random Walks +R. K. Singh∗ and Stanislav Burov† +Department of Physics, Bar-Ilan University, Ramat-Gan 5290002, Israel +ANALOGY BETWEEN TIME AND SPACE +While studying the diffusion to subdiffusion transition for continuous time random walks (CTRWs) with waiting +times following ψ(τ) ∼ τ −1−α (τ → ∞) we have used the quantity +φα(t) = +�max{τ1, ..., τNt} +�Nt +i=1 τi +� +(1) +and analyzed its long-time behavior. While considering the maxima of waiting times τi, bounded above by t, we do +not explicitly account for the backward reference time Bt = t − �Nt +i=1 τi [1] for a couple of reasons. First is that as +φα(t) is a function of t, occurrence of even a large Bt (expected for power law waiting times) would not change its +value because τi are sampled up to the last jump. Secondly, exclusion of Bt allows us to get away with the correlations +which will appear due to the fact that the time of measurement is fixed at t [1]. This has the further advantage of +putting time and space on equal footing as the quantity to study spatial fluctuations +Fβ(X) = +�max{|x1|, ..., |xNt|} +�Nt +i=1 |xi| +� +. +(2) +is analogous to φα(t) with τi replaced by |xi|. In terms of fluctuations, we focus on the absolute value of location +of the CTRW upto the last jump in Fβ(X). Similarly, φα(t) takes into account the waiting times only upto the last +jump taking place before the observation time t. The analogy of φα(t) and Fβ(X) sets the premise to study phase +transitions at finite times once we start looking at the rare fluctuations of a CTRW. +MEAN NUMBER OF JUMPS NEAR t = 0 +The mean number of jumps in Laplace space is ⟨ ˜Ns⟩ = +˜ +ψs +s(1− ˜ +ψs) [2] and for waiting time distribution analytic near zero +0 +1 +2 +0 +1 +2 +(a) + +0 +1 +2 +0 +1 +2 +(b) + +t +FIG. 1: Comparison of numerically estimated ⟨Nt⟩ against the solution in (3) for (a) half Gaussian distribution: ψ(τ) = +� +2 +π e−τ2/2, and (b) power law distribution ψ(τ) = +1 +(1+τ)2 . The symbols are numerical calculations and lines are ⟨Nt⟩ evaluated +from (3). + +2 +ψ(τ) +τ→0 +∼ CAτ A + CA+1τ A+1 + CA+2τ A+2 + · · · we have in time domain +⟨Nt⟩ ≈ CAΓ(A + 1) +Γ(A + 2) +tA+1 + CA+1Γ(A + 2) +Γ(A + 3) +tA+2 + CA+2Γ(A + 3) +Γ(A + 4) +tA+3 + C2 +AΓ2(A + 1) +Γ(2A + 3) t2A+2 ++ 2CACA+1Γ(A + 1)Γ(A + 2) +Γ(2A + 4) +t2A+3 + C3 +AΓ3(A + 1) +Γ(3A + 4) t3A+3. +(3) +We compare the approximate value of ⟨Nt⟩ evaluated from (3) against numerical calculations in Fig. 1 and find +that the approximate form in (3) captures the true behavior only at small times. The domain of the validity, however, +depends on the exact nature of the distribution. For example, when the distribution of waiting times is half Gaussian, +that is, ψ(τ) = +� +2 +πe−τ 2/2, we have A = 0, CA = +� +2 +π, CA+1 = 0, CA+2 = − +� +1 +2π and it is evident from Fig. +1 (a) +that the approximate form derived in (3) agrees with numerically estimated ⟨Nt⟩ upto t ≈ 1. On the other hand, +for the power law distribution ψ(τ) = +1 +(1+τ)2 we find that the usefulness of (3) is reduced to half the range, that is, +t ∈ (0, 1/2). The reason for this difference is that the small time behavior of ψ(τ) does not capture jumps taking +place at finite times. +∗ Electronic address: rksinghmp@gmail.com +† Electronic address: stasbur@gmail.com +[1] M. H¨oll, W. Wang, and E. Barkai, Phys. Rev. E 102, 042141 (2020). +[2] J. Klafter and I. M. Sokolov, First steps in random walks: from tools to applications (OUP Oxford, 2011). + diff --git a/btAzT4oBgHgl3EQfnf1c/content/tmp_files/load_file.txt b/btAzT4oBgHgl3EQfnf1c/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f181bd36c77780cd9d7b48f31c39726b49269169 --- /dev/null +++ b/btAzT4oBgHgl3EQfnf1c/content/tmp_files/load_file.txt @@ -0,0 +1,781 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf,len=780 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content='01581v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content='stat-mech] 4 Jan 2023 Universal to Non-Universal Transition of the statistics of Rare Events During the Spread of Random Walks R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' Singh∗ and Stanislav Burov† Department of Physics, Bar-Ilan University, Ramat-Gan 5290002, Israel Particle hopping is a common feature in heterogeneous media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' We explore such motion by using the widely applicable formalism of the continuous time random walk and focus on the statistics of rare events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' Numerous experiments have shown that the decay of the positional probability density function P(X, t), describing the statistics of rare events, exhibits universal exponential decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' We show that such universality ceases to exist once the threshold of exponential distribution of particle hops is crossed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' While the mean hop is not diverging and can attain a finite value;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' the transition itself is critical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' The exponential universality of rare events arises due to the contribution of all the different states occupied during the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' Once the reported threshold is crossed, a single large event determines the statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' In this realm, the big jump principle replaces the large deviation principle, and the spatial part of the decay is unaffected by the temporal properties of rare events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' According to Wikipedia, ”Rare or extreme events are events that occur with low frequency, and often refers to infrequent events that have widespread impact and might destabilize systems”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' Notable examples of rare events include the stock market crash [1], earthquakes [2] and cyclones [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' The most recent on the list is the 2020 pan- demic, which brought the whole world to a standstill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' The frequency and the scale of rare events in each field are very important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' If the rare events are not ”too rare” and large enough, then the usual statistical behavior is completely dictated by such rare events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' A perfect exam- ple of situations when rare events fundamentally modify the nature of a physical process is sub-diffusion [4–10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' While for normal diffusion, the mean squared displace- ment (MSD) of a tracked particle grows linearly with time, for sub-diffusion, the MSD grows in a non-linear fashion, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' ⟨X2⟩ ∼ tα, while 0 < α < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' Sub-diffusion generally occurs when there are long time or space cor- relations present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' One of the sources for such correla- tions can appear in the form of an extremely long so- journ times, when the tracked particle is trapped in a specific region in space [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' When sub-diffusion occurs in amorphous materials like glasses [12] or living cells [13–15], it is quite common to observe trajectories that are dominated by several (and sometimes even one) very long trapping events [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' In many cases, the appearance of sub-diffusion is ac- companied by unusual physical phenomena: aging [17– 19], weak ergodicity breaking [17, 20, 21] and non-self- averaging [22–25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' Studies of theoretical models where trapping is present, like the continuous time random walk (CTRW) [26] and the quenched trap model (QTM) [27] show that there is a critical transition to sub-diffusion, aging, and non-ergodic behavior, when the mean trap- ping time is diverging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' This transition is also accom- panied by a transformation of the positional probability density function (PDF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' The universal Gaussian center of the positional PDF turns into an α-stable L´evy type [28, 29], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' shape that depends on a specific parameter α that determines the trapping times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' This transition between a universal PDF that is determined by an ac- cumulation of many events, and a non-universal PDF that follows the properties of one single rare event is a key feature associated with the the appearance of new physical phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' Can transitions from universal to non-universal PDFs symbolize an underlying phase tran- sition that might lead to uncovered surprising physics, like in the case of sub-diffusion?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' What are the proper- ties of single rare events that can cause such a critical transition in behavior?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' In this work, we focus on the ap- pearance of the universal to the non-universal transition of a positional PDF that, unlike the case of sub-diffusion, appears for large positions |X| (the tails of the PDF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' Recently, in a huge number of experiments, exponen- tial (and not Gaussian) decay of the PDF for large |X| was spotted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' Such decay of the PDF is termed Laplace tails [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' Notable examples include colloidal beads in heterogeneous optical force field [31], zooplanktons under crowding [32], glass-forming liquids [33], nanoparticles in polymer melts [34], motion of particles at liquid-solid in- terface [35], particle displacements close to glass and jam- ming transitions [36, 37] etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' In [38] it was suggested that there might exist a universal convergence to the Laplace tails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' Indeed it was proven, by exploiting the CTRW model that for a wide range of processes, the tails of the PDF are determined by the accumulation of many events and follow exponential decay (up to logarithmic correc- tions) [39–41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' So by following the presented discussion, we ask: Is there a transition from universal Laplace tails to non-universal and process-specific tails?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' And if so, is it accompanied by a critical transition like the diffusion to subdiffusion transition?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' Are the tails of a PDF deter- mined by one occurrence of one single event [42–46] or they represent an accumulation of many realizations of not-so-large but more frequent events [47–51]?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' Similar to the case of subdiffusion to diffusion tran- sition, we use the celebrated CTRW model, originally exploited for explaining motion in amorphous materials [12–15] and exponential decay of tails [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' In this model, a particle performs random jumps in space and waits for 2 a random amount of time between every two jumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' All the jumps and waiting times are independent and iden- tically distributed (IID) random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' The distribu- tion of a jump x is given by f(x), while each waiting time τ has a distribution ψ(τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' The position of the process X at time t is determined by the random number of jumps Nt performed by time t, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' X = x1 +· · ·+xNt where xi, is the size of a single jump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' The positional PDF P(X, t) is readily obtained in terms of the subordination equation by conditioning on the number of jumps N, [4, 27, 52, 53] P(X, t) = ∞ � N=0 PN(X)Qt(N) (1) where PN(X) is the distribution of X for a given N(= Nt for a fixed measurement time t) and Qt(N) is the distri- bution of the number of jumps up to time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' The men- tioned phenomena, like anomalous transport and aging, appear for CTRWs with ψ(τ) ∼ τ −1−α (τ → ∞) when α < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' For such ψ(τ), the sum τ1+· · ·+τNt is dominated (in the t → ∞ limit) by the maximal summand [54], as can be observed from the behavior of φα(t) = �max{τ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=', τNt} �Nt i=1 τi � (2) for large t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' While the maxima of a set of random vari- ables has been extensively studied [55–58], the definition of φα(t) has the advantage that it ranges from infinites- imal to finite values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' This makes it appropriate as an order parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' We see in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' 1 (inset) that φα(t) sat- urates to a finite value for α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content='8, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content='9 while for α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content='1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content='2, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content='5 exhibits a decaying behavior at large times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' This difference in the properties of φα(t) follows from the fact that for α < 1, ⟨τ⟩ = � ∞ 0 dτ τψ(τ) → ∞, while it is finite for α > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' This implies that as we move from α > 1 to α < 1, the rare fluctuations in the sequence of waiting times {τ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=', τNt} exhibit qualitatively differ- ent behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' Consequently, the thermodynamic limit in which the system behaves extensively (all waiting times τi are of the same order of magnitude) ceases to exist for α < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' Can we see similar behavior if we focus on spa- tial fluctuations?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' This question comes up naturally once we realize that the system has access to the entire phase space in the thermodynamic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' The phase space can be swept by focusing on large spatial fluctuations with- out going to the t → ∞ limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' This means we take the large |X| limit and expect the system to visit many states during its evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' Motivated by this, let us look at a CTRW described by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' (1) with jumps following f(x) ∼ e−|x|β with β > 0 [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' In analogy with φα(t) let us define Fβ(X) = �max{|x1|, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=', |xNt|} �Nt i=1 |xi| � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' (3) We see in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' 1 that even for a well behaved ψ(τ) like the exponential distribution, lim|X|→∞ Fβ(X) decreases 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content='0 10-1 100 101 102 Fβ(X) X β = 1/4 1/3 1/2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content='5 10-2 10-1 100 101 105 109 t φα(t) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' 1: Fβ(X) for a CTRW with jumps x ∼ f(x) with mean zero and variance one as a function of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' The waiting time distribution ψ(τ) = e−τ and the measurement is done at t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content='6 for trajectories reaching a position ±X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' Inset shows the behavior of φα(t) for ψ(τ) ∼ τ −1−α for large τ and α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content='8, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content='9, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content='1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content='2, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content='5 from top to bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' monotonically for β ≥ 1 and takes a finite value for tra- jectories with β < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' In other words, lim |X|→∞ Fβ(X) = � 0, β ≥ 1 cβ, β < 1 (4) where cβ is a nonzero constant that depends on β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' Al- though cβ is seen to approach unity for β = 1/2 and lower values, β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content='9 seems to saturate to a value less than one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' While the transition of φα(t) is driven by the divergence of the mean waiting time ⟨τ⟩, this is not the case for the transition of Fβ(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' All the moments ⟨xn⟩ are finite for β < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' Nevertheless, the effective behav- ior of limt→∞ φα(t) and lim|X|→∞ Fβ(X) is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' As a function of the parameters α/β, there is a transition from a state defined by the accumulation of many events to a state dominated by a single large event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' At this point, it is worth noticing that the role played by φα(t) for describing temporal fluctuations is taken over by Fβ(X) for spatial fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' While the emer- gence of a non-Gaussian center accompanying the transi- tion of φα(t) is well understood [28, 29] the correspond- ing behavior of P(X, t) accompanying the transition of Fβ(X) is not known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' It was shown in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' [39] that P(X, t) exhibits universal tails exhibiting exponential de- cay for β > 1 and ψ(τ) analytic near τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' Furthermore, just like φα(t) → 0 marks the emergence of a univer- sal Gaussian center following the central limit theorem, Fβ(X) → 0 at |X| → ∞ characterizes the universal expo- nential tails of P(X, t) [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' On the other hand, a finite value of φα(t) for α < 1 is reminiscent of the L´evy stable PDF and is a reminder of the dynamical phase transition (DPT) [60–63] characterized solely by the rare fluctua- 3 tions, but in the temporal domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' Does it mean we can see a similar DPT at finite times if we focus on the tails of P(X, t)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' We now answer this question by exploring the case of β ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' For β = 1, the jumps follow Laplace distribution, that is, f(x) = a 2 exp(−a|x|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' This implies that the charac- teristic function of a single jump, defined as the Fourier transform ˆλ(k) = � ∞ −∞ dx eikxf(x) = a2 a2+k2 [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' Un- like the case for β > 1, ˆλ(k) for β = 1 has poles in the complex plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' Furthermore, since the jumps are IID, the distribution of a sum of N jumps in Fourier space is ˆλN(k) and inverting it via contour integration [64] we find that the distribution of position X after N jumps is [65] PN(X) = ae−a|X| 22N−1Γ(N) N−1 � m=0 (2N − m − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' (N − m − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' (2a|X|)m ≈ ae−a|X| 22NΓ(N) � N 0 dm exp[K(N, m)] (5) where K(N, m) = (2N − m) log(2N − m) − m log(m) − (N − m) log(N − m) − (N − m) + m log(2a|X|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' The last line follows from Stirling’s approximation N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' ≈ N Ne−N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' In order to evaluate the integral in (5) we use Laplace’s method and locate m0 such that dK dm|m=m0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' This implies m0 = a|X| + N − √ a2X2 + N 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' In the limit a|X|/N ≫ 1, m0 ≈ N − N 2 2a|X| and then from (5) the large deviation form is obtained PN(X) ∼ exp � − NIN �|X| N �� , (6) with the rate function IN � |X| N � = − a|X| N + log � a|X| N � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' This implies that for large deviations, the distribution of the sum possesses exponentially decaying tails with logarithmic corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' For ψ(τ) analytic near zero ψ(τ) τ→0 ∼ CAτ A + CA+1τ A+1 + CA+2τ A+2 + · · · , (7) where A is a non-negative integer, Qt(N) admits a large deviation form Qt(N) N→∞ ∼ exp[−NIN(t)] with a uni- versal rate function [66] IN(t) = − CA+1 CA t N − (A + 1) � 1 + log � (CAΓ(A+1)) 1 A+1 A+1 t N �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' Using PN(X) from (6) and Qt(N) in (1) we have for large N [67] P(X, t) ≈ � ∞ 0 dN exp[κ(N)] ≈ � 2π |κ′′(N0)| exp[κ(N0)], (8) where κ(N) = −a|X| + N log �a|X| N � − Ct + N(A + 1) � 1 + log �d2 N �� , (9) 10-8 10-6 10-4 10-2 100 20 -10 0 10 20 (a) P(X,t) 10-8 10-6 10-4 10-2 100 20 -10 0 10 20 (b) 10-8 10-6 10-4 10-2 100 20 -10 0 10 20 (c) P(X,t) X 10-8 10-6 10-4 10-2 100 20 -10 0 10 20 (d) X FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' 2: Comparison of numerically estimated P(X, t) (red circles) against the solution given in (11) (black dashed line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' The waiting time distributions are the following: (a) exponen- tial mixture ψ(τ) = p1r1e−r1τ +p2r2e−r2τ with r1 = 1/4, r2 = 5/2, p1 = 1/4, p2 = 3/4 at t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content='7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' (b) gamma distribution ψ(τ) = τ 3e−τ/6 at t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content='8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' (c) half Gaussian distribution ψ(τ) = � 2/πe−τ2/2 at t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content='5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' (d) power law distribution ψ(τ) = 1/(1 + τ)2 at t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' with C = − CA+1 CA and d2 = [CAΓ(A+1)] 1 A+1 A+1 t and N0 is the solution of κ′(N0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' For large a|X|/N we have N0 ≈ µ(a|X|) 1 A+2 (10) with µ = d A+1 A+2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' Using κ′′(N0) = − A+2 N0 in (8) we find P(X, t) ∼ |X|/t→∞ � 2π A + 2µ(a|X|) 1 A+2 exp � − t � C + a|X| t − � CAΓ(A + 2)a|X| t � 1 A+2 �� (11) From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' 2 we see that the large deviation form of P(X, t) evaluated in (11) agrees with numerically esti- mated P(X, t) for different waiting time distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' In other words, P(X, t) possesses exponentially decaying tails in the limit of large |X|/t when the distribution of jumps is Laplace distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' P(X, t) derived in (11) holds for a wide class of waiting time distributions analytic near zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' This further im- plies that the rare fluctuations for a CTRW with Laplace distributed jumps are described by the large deviation principle [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' In this regard, the case β = 1 is analogous to the β > 1 case discussed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' [39–41] if we restrict our attention solely to the exponentially decaying fluc- tuations of P(X, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' The analogy, however, ends here as for β > 1 N0 increases linearly with |X| while for β = 1, the growth is sublinear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' Furthermore, for β > 1 the PDF exhibits exponentially decaying tails with logarith- mic corrections [39] while for β = 1 the corrections are of 4 power-law type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' Even though P(X, t) exhibits exponen- tially decaying tails for both β > 1 and β = 1, different forms of correction term for the two cases “hints” towards a possible transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' Let us now explore the region β < 1 to complete our understanding of this transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' For β < 1 the distribution of jumps belongs to the class of stretched exponential distributions [42] which possesses heavy tails as � ∞ 0 dx eλxf(x) = ∞ ∀ λ > 0 and does not admit a large deviation form [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' It is a well-known result that the family of stretched exponen- tial distributions satisfies the big jump principle [42] P(x1 + · · · + xN ≥ X) |X|→∞ ∼ P(max{x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=', xN} ≥ X) (12) with the right hand side evaluating to 1 − � 1 − � ∞ X dx f(x) �N for IID xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' Hence, from (1) we have � ∞ X dX P(X, t) |X|→∞ ∼ 1 − Gt � 1 − � ∞ X dx f(x) � (13) where Gt(z) = �∞ N=0 zNQt(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' Furthermore, for large |X| we have � ∞ X dx f(x) ∼ 0, as a result we can analyze P(X, t) in terms of the behavior of Gt(z) for z in the neighborhood of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' Now Gt(1−η) ≈ Gt(1)− ∂G ∂z |z=1η for η small and ∂G ∂z = �∞ N=1 NQt(N)zN−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' This implies Gt � 1 − � ∞ X dx f(x) � ≈ 1 − ⟨Nt⟩ � ∞ X dx f(x) and from here it follows that P(X, t) |X|→∞ ∼ ⟨Nt⟩f(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' (14) The above equation implies that the probability of be- ing at a location X at time t equals the mean num- ber of jumps ⟨Nt⟩ up to time t times the distribution of a single jump f(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' With the distribution of a single jump known, we only need to estimate the mean num- ber of jumps ⟨Nt⟩ which in the Laplace domain reads [26] ⟨ ˜ Ns⟩ = ˜ ψs s(1− ˜ ψs), where ˜ψs = � ∞ 0 dt e−stψ(t) is the Laplace transform of ψ(τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' For ψ(τ) analytic near zero (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' (7)) we have in the limit t → 0 [68] ⟨Nt⟩ ≈ CAΓ(A + 1) Γ(A + 2) tA+1 + CA+1Γ(A + 2) Γ(A + 3) tA+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' (15) The reason to focus on the t → 0 limit is that it allows us to address the rare fluctuations exhibited by the CTRW at finite times, that is, |X|/t → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' Specifically, in many experiments that show Laplace decay of the PDF, the non-Gaussian behavior was spotted for short enough times [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' For a long measurement time, the Gaussian center eventually takes over [69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' Notwithstanding the limited range of validity of (15), P(X, t) derived in (14) holds at arbitrary times, and is in excellent agreement with numerical simulations (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' 10-6 10-4 10-2 100 800 600 400 200 0 200 400 600 800 P(X,t) X e-|x|1/3 , t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content='4 e-|x|1/2 , t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content='5 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' 3: Numerically estimated P(X, t) for a CTRW with jumps following a generalized Gaussian distribution with PDF f(x) = β 2Γ(1/β)e−|x|β with β = 1/3, for ψ(τ) = � 2/πe−τ2/2 (red circles) and β = 1/2 for ψ(τ) = 1 (1+τ)2 (blue squares).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' The black dashed lines represent the analytical form from (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' The result in (14) shows that the parameter range β ∈ (0, 1) is markedly different from the region β ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' In ad- dition, differences in the nature of the PDF of CTRW at β = 1 and β > 1 imply towards the fact that the PDF of a CTRW critically changes at β = 1, which is essentially the same value for which the order parameter Fβ(X) shows a critical transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' The evaluation of P(X, t) further corroborates our assertion of a universal to non- universal transition as seen from the analysis of Fβ(X) (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' The fact that lim|X|→∞ Fβ(X) = 0 for β ≥ 1 is analogous to saying that P(X, t) ∼ exp[−tI(|X|/t)] exists with a nontrivial rate function I(|X|/t) for ev- ery β ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' This rate function I(z) attains a linear growth for large z and, therefore, the universal expo- nential decay of the PDF, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=', P(X, t) ∼ e−|X|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' On the other hand, for β < 1 we had seen from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' 1 that lim|X|→∞ Fβ(X) = cβ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' (14) shows that for β < 1, the rate function I(|X|/t) is trivially zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' For large |X|, the decay of the PDF is stretched exponen- tial, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=', P(X, t) ∼ e−|X|β, that specifically depends on the parameter β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' Notice that the transition of φα(t) in the long time limit represents the transition from diffu- sion to subdiffusion and is accompanied by the transition of the PDF (in the |X|/t → 0 limit) from the universal Gaussian form to the α-stable L´evy type that explicitly depends on α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' Hopping dynamics which is an intrinsic feature of CTRW, has been ubiquitously observed in polymer melts [70], colloidal suspensions [71], rodlike particles through smectic layers [72, 73], polymer glasses [74], binary mix- tures [75], in one, two, and three spatial dimensions [76], to mention a few.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' A characteristic feature of motion in glassy materials [36–38, 77] and at the liquid-solid inter- face [35, 78], where hopping dynamics is observed, has 5 been the exponential decay of the tails of the positional PDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' Exponential decay, however, is the rule whenever hopping dynamics is in play [39], making it a univer- sal feature of transport in heterogeneous media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' But in some situations, like the case of particles with a constant supply of energy (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=', run and tumble particles), the par- ticles can perform really long jumps during their explo- ration of the heterogeneous media [79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' Our work shows that a critical transition is expected for any system in- volving such hops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' This critical transition manifests itself at the level of the positional PDF, where the universality of Laplace tails ceases to exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' While the universal tails are an outcome of an accumulation of many events and the applicability of the large-deviation principle, the spe- cific tails for β < 1 are determined by one single event, that is, the big-jump principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' Unlike a diffusion-to-subdiffusion transition which takes place at long times and is accompanied by diver- gences of the mean trapping time, the phase transition reported in the present study is free from such diver- gences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' The mean length of a hop can be finite, and the transition is observed at finite times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' Furthermore, while the temporal properties of rare events leading to subdiffusion affect the bulk, rare spatial events manifest themselves mainly in the tails of the PDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' Interestingly, the temporal features do not affect the spatial depen- dence of the statistics of the rare events, that is, the tails of P(X, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' Acknowledgments: This work was supported by the Is- rael Science Foundation Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' 2796/20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' RKS thanks the Israel Academy of Sciences and Humanities (IASH) and the Council of Higher Education (CHE) Fellowship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' ∗ Electronic address: rksinghmp@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content='com † Electronic address: stasbur@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content='com [1] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' Amihud, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' 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Robertson, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' Greene, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' Natl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' 103, 1221 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' [15] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' Wang, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' Austin, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' Cox, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' 97, 048302 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' [16] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' Jeon, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' Tejedor, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' Burov, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' Barkai, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' Selhuber- Unkel, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' Berg-Sørensen, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' Oddershede, and R.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' I France 2, 1705 (1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' [18] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' Akimoto, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' Barkai, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' Saito, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} 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Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' 94, 240602 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' [22] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' Pronin, Physica A: Statistical Mechanics and its Ap- plications 596, 127180 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' [23] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' Shafir and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' Burov, J.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' Uchaikin, Physics-Uspekhi 46, 821 (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' [29] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' Garoni and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' Frankel, J.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' Ciarlo, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' Pesce, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' Greco, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' Sasso, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' 126, 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' 2021, 103208 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content='01581v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content='stat-mech] 4 Jan 2023 Supplementary Material for Universal to Non-Universal Transition of the statistics of Rare Events During the Spread of Random Walks R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' Singh∗ and Stanislav Burov† Department of Physics, Bar-Ilan University, Ramat-Gan 5290002, Israel ANALOGY BETWEEN TIME AND SPACE While studying the diffusion to subdiffusion transition for continuous time random walks (CTRWs) with waiting times following ψ(τ) ∼ τ −1−α (τ → ∞) we have used the quantity φα(t) = �max{τ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=', τNt} �Nt i=1 τi � (1) and analyzed its long-time behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' While considering the maxima of waiting times τi, bounded above by t, we do not explicitly account for the backward reference time Bt = t − �Nt i=1 τi [1] for a couple of reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' First is that as φα(t) is a function of t, occurrence of even a large Bt (expected for power law waiting times) would not change its value because τi are sampled up to the last jump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' Secondly, exclusion of Bt allows us to get away with the correlations which will appear due to the fact that the time of measurement is fixed at t [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' This has the further advantage of putting time and space on equal footing as the quantity to study spatial fluctuations Fβ(X) = �max{|x1|, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=', |xNt|} �Nt i=1 |xi| � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' (2) is analogous to φα(t) with τi replaced by |xi|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' In terms of fluctuations, we focus on the absolute value of location of the CTRW upto the last jump in Fβ(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' Similarly, φα(t) takes into account the waiting times only upto the last jump taking place before the observation time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' The analogy of φα(t) and Fβ(X) sets the premise to study phase transitions at finite times once we start looking at the rare fluctuations of a CTRW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' MEAN NUMBER OF JUMPS NEAR t = 0 The mean number of jumps in Laplace space is ⟨ ˜Ns⟩ = ˜ ψs s(1− ˜ ψs) [2] and for waiting time distribution analytic near zero 0 1 2 0 1 2 (a) 0 1 2 0 1 2 (b) t FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' 1: Comparison of numerically estimated ⟨Nt⟩ against the solution in (3) for (a) half Gaussian distribution: ψ(τ) = � 2 π e−τ2/2, and (b) power law distribution ψ(τ) = 1 (1+τ)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' The symbols are numerical calculations and lines are ⟨Nt⟩ evaluated from (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' 2 ψ(τ) τ→0 ∼ CAτ A + CA+1τ A+1 + CA+2τ A+2 + · · · we have in time domain ⟨Nt⟩ ≈ CAΓ(A + 1) Γ(A + 2) tA+1 + CA+1Γ(A + 2) Γ(A + 3) tA+2 + CA+2Γ(A + 3) Γ(A + 4) tA+3 + C2 AΓ2(A + 1) Γ(2A + 3) t2A+2 + 2CACA+1Γ(A + 1)Γ(A + 2) Γ(2A + 4) t2A+3 + C3 AΓ3(A + 1) Γ(3A + 4) t3A+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' (3) We compare the approximate value of ⟨Nt⟩ evaluated from (3) against numerical calculations in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' 1 and find that the approximate form in (3) captures the true behavior only at small times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' The domain of the validity, however, depends on the exact nature of the distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' For example, when the distribution of waiting times is half Gaussian, that is, ψ(τ) = � 2 πe−τ 2/2, we have A = 0, CA = � 2 π, CA+1 = 0, CA+2 = − � 1 2π and it is evident from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' 1 (a) that the approximate form derived in (3) agrees with numerically estimated ⟨Nt⟩ upto t ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' On the other hand, for the power law distribution ψ(τ) = 1 (1+τ)2 we find that the usefulness of (3) is reduced to half the range, that is, t ∈ (0, 1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' The reason for this difference is that the small time behavior of ψ(τ) does not capture jumps taking place at finite times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' ∗ Electronic address: rksinghmp@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content='com † Electronic address: stasbur@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content='com [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' H¨oll, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' Wang, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' Barkai, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' E 102, 042141 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' [2] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' Klafter and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} +page_content=' Sokolov, First steps in random walks: from tools to applications (OUP Oxford, 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf'} diff --git a/btFPT4oBgHgl3EQfxDUf/vector_store/index.pkl b/btFPT4oBgHgl3EQfxDUf/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..b7f226b77237643e823bd8f9d4395b11ddfa3120 --- /dev/null +++ b/btFPT4oBgHgl3EQfxDUf/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:21809b18461f54675447e2aed6c23679beac3cacffef74ee2ba3dd158f5549fe +size 552700 diff --git a/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf b/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..6cab327cc177ef53c395aaeeb9aa638fc3abb719 --- /dev/null +++ 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FOR SELF-MAPS OF THE +SPHERE +H´ECTOR BARGE AND LUIS HERN´ANDEZ-CORBATO +Abstract. Let Sm = {x2 +0+x2 +1+· · ·+x2 +m = 1} and P = {x0 = x1 = 0}∩Sm. Suppose that f is +a self–map of Sm such that f −1(P) = P and |deg(f|P )| < |deg(f)|. Then, the number of fixed +points of f n grows at least exponentially with base |d| > 1, where d = deg(f)/deg(f|P ) ∈ Z. +1. Introduction +In [13], Shub raised the question on whether algebraic intersections numbers bound assymp- +totically from below geometrical intersection numbers for C1 maps. For a self-map f : M → M +on a manifold, a particular case of the previous question comes down to whether the number +#Fix(fn) of points fixed by fn and the Lefschetz numbers L(fn) satisfy +lim sup 1 +n log(#Fix(fn)) ≥ lim sup 1 +n log |L(fn)| +This inequality is known as the growth rate inequality and bounds from below the base of +the exponential growth rate of periodic points. If the sequence of Lefschetz numbers L(fn) +is unbounded, Lefschetz-Dold theorem together with a result of Shub and Sullivan [15] (that +states that the index sequence of a fixed point for a C1 map is bounded) imply that there are +infinitely many periodic points. The growth rate inequality appeared again as an open problem +in the Proceedings of the ICM 2006 [14]. It is wide open in dimensions greater than 1. +The Lefschetz number of a self-map on a sphere can be computed from the topological degree +as L(f) = 1 ± deg(f), so the growth rate inequality for self-maps of the sphere can be rewritten +in terms of the degree as follows: +(1) +lim sup +n→+∞ +1 +n log(#Fix(fn)) ≥ log |deg(f)| +Shub’s original paper [13] proved (1) for rational maps on S2. However, even for a C1 map +of degree 2 on S2, it is not known whether the growth rate inequality holds. Note that the C1 +assumption is crucial, a degree-2 north-south map on S2 has only 2 fixed points and no other +periodic point. +Recently, the growth rate inequality has been proved in several instances for maps on the +sphere. In dimension 2, the sharp bound of log |deg(f)| was obtained when f preserves some +singular foliations [12, 10, 4] or under hypothesis of dynamical nature [5, 7, 6, 8]. In higher +dimensions the results are scarce. Weaker bounds for the growth rate of periodic points when +the map preserves some foliation and some mild hypotheses is satisfied were obtained in [2, 3]. +In this article we prove a weak form of the growth rate inequality for maps on Sm = {x2 +0 + +x2 +1 + · · · + x2 +m = 1} ⊂ Rm+1. Suppose f : Sm → Sm leaves the codimension–2 sphere P = {x0 = +x1 = 0} completely invariant, that is, f−1(P) = P. Then, the degree of f is equal to the product +of two factors: the degree of the restriction of f to P and a “transversal” degree denoted d ∈ Z. +The latter can be interpreted in terms of the action induced by f on the homology group or the +fundamental group of Sm − P. +2020 Mathematics Subject Classification. Primary 37C25, 37E99; Secondary 55M20. +Key words and phrases. Growth rate inequality, periodic point, topological degree. +The authors are partially supported by the Spanish Ministerio de Ciencia e Innovaci´on (grants PGC2018- +098321-B-I00 and PID2021-126124NB-I00). +1 +arXiv:2301.08543v1 [math.DS] 20 Jan 2023 + +Theorem 1. Let f : Sm → Sm be a map such that f−1(P) = P and deg(f) ̸= 0 and let +d = deg(f)/deg(f|P ). Then, #Fix(fn) + #Fix(fn +|P ) ≥ |dn − 1|. In particular, +lim inf +n→+∞ +1 +n log(#Fix(fn)) ≥ log |d| +In dimension m = 2, P is a 0–sphere and deg(f|P ) can only take the values −1, 0, 1. We +deduce the following corollary from Theorem 1, that was previously stated in [6]. +Corollary 2. Suppose f : S2 → S2 and there are two points {p, p′} such that f−1({p, p′}) = +{p, p′}. Then, +lim inf +n→+∞ +1 +n log(#Fix(fn)) ≥ log |deg(f)| +In higher dimensions we obtain a weak bound for the growth rate: +Corollary 3. Under the hypothesis of Theorem 1, if m = 3 or, more generally, if the growth +rate inequality holds in Sm−2 then +lim inf +n→+∞ +1 +n#Fix(fn) ≥ 1 +2 log |deg(f)| +The result follows from the first inequality in Theorem 1 and the fact that max{|d|, |deg(f|P )|} ≥ +� +|deg(f)|. +Theorem 1 is deduced from Theorem 5, which is stated in the ensuing section. The proof is +contained in the last section of the article. It requires a detailed local analysis of the map in the +normal direction to P, presented in Section 4, and uses an argument from topological degree +theory, which is quickly reviewed in Section 3. +2. Setting +Let S = Sm = {(x0, . . . , xm) : x2 +0 + · · · + x2 +m = 1} be the standard m–sphere in Rm+1 +and P = {x0 = x1 = 0} ∩ S be an m − 2–dimensional sphere which we will refer to as the +polar sphere. The complement S − P is diffeomorphic to S1 × Dm−1, where Dm−1 denotes the +(m − 1)–dimensional open unit disk, via the map +(2) +(x0, . . . , xm) �−→ +� (x0, x1) +||(x0, x1)||, (x2, . . . , xm) +� +On the other hand, P ′ = {x2 = · · · = xm = 0} ∩ S is a 1–sphere such that S is the join of P +and P ′. The set S − P ′ is diffeomorphic to D2 × Sm−2 by +(3) +(x0, . . . , xm) �−→ +� +(x0, x1), +(x2, . . . , xm) +||(x2, . . . , xm)|| +� +Equations (2) and (3) define coordinate charts for S − P and S − P ′, respectively. If we +replace (x2, . . . , xm) by (0, 0, x2, . . . , xm) in (3), we obtain a diffeomorphism between S − P ′ +and D2 × P. This description will be used extensively throughout this note, as it provides a +product structure for the system of neighborhoods of P defined by the inequalities x2 +0 + x2 +1 < r +for 0 < r < 1: they are diffeomorphic to D2(r) × P. +Similarly, the inequalities x2 +0 + x2 +1 > 1 − r′, 0 < r′ < 1, define neighborhoods of P ′ in S − P +diffeomorphic by (2) to S1 × Dm−1(r′). Note that the radial coordinates of D2 and Dm−1 are +related by r + r′ = 1. For any 0 < r < 1, +(4) +S +∼= D2(r) × P +⊔ S1 × D +m−1(1 − r) +Evidently, the fundamental group of S − P is isomorphic to Z. Using the coordinates from +(3), for any r ∈ (0, 1) and p ∈ P, it is easy to see that π1(S − P) is generated by a loop γ that +makes a positive turn around the origin in the 2-disk D2(r) × {p}. Let us consider the lift of +S − P that trivializes [γ], pr: � +S − P → S − P. +2 + +In our results, f is a self–map of S for which P is completely invariant, f−1(P) = P. Since +S − P is invariant under f, f can be lifted to � +S − P. Let τ be a generator of the group of deck +transformations of the cover. Any lift F of f satisfies +(5) +F τ = τ d F +where d ∈ Z is the transversal degree defined by f∗[γ] = d · [γ] in π1(S − P) (see (12) later). As +we will prove in Subsection 3.1, deg(f) = d · deg(f|P ). +A fixed point of F projects onto a fixed point of f in S − P. +The following lemma is +a consequence of (5) and establishes a relation between fixed points of different lifts. +It is +standard in Nielsen theory (cf. [9, Lemma 4.1.10]). +Lemma 4. If two different lifts F, G = τ mF of f satisfy pr(Fix(F)) ∩ pr(Fix(G)) ̸= ∅, then +d ̸= 1 and |d − 1| divides m. +By the previous lemma, we can bound from below #{Fix(f) ∩ (S − P)} by the number of +lifts of f among {F, τF, . . . , τ |d−1|−1F} that have fixed points. +Theorem 5. Suppose that d ̸= 0, 1. There are at most 2 #Fix(f|P ) fixed point free maps among +{F, τF, . . . , τ |d−1|−1F}. +Theorem 5 and Lemma 4 immediately imply that +#{Fix(f) ∩ (S − P)} ≥ |d − 1| − 2 #{Fix(f|P )} +Replace f by fn in this inequality to deduce Theorem 1. Note that the transversal degree +associated to fn is dn. +3. Topological degree +The topological degree of a map g: M → L between closed connected and oriented manifolds +of dimension n ≥ 1 roughly counts with multiplicity the number of preimages of a point. For +a complete account on degree theory we refer to [11]. We give below a precise definition of the +degree in algebraic topological terms. Recall that the orientation of a closed connected manifold +M is given by a fundamental class [M], i.e. a generator of the reduced homology group �Hn(M). +Reduced and unreduced homology groups only differ at dimension 0, the reason to choose here +the reduced groups shall become clear later. The image [M]x of the fundamental class [M] +under the projection �Hn(M) → Hn(M, M − {x}) is the local orientation at x ∈ M, and it is a +generator of Hn(M, M − {x}). +Definition 6. Let g: M → L be a map between n–dimensional closed orientable manifolds such +that �Hn(M) ∼= �Hn(L) ∼= Z and [M], [L] be fundamental classes of M, L, respectively. The degree +of g, denoted deg(g), is the integer that satisfies +(6) +g∗([M]) = deg(g) · [L], +The reason to choose the hypothesis �Hn(M) ∼= Z in the definition is that it is satisfied by +closed connected oriented manifolds of dimension n ≥ 1 (spaces that appear in the standard +definition of topological degree) and also by 0–spheres (such as the polar sphere in S2, that is +relevant in this paper). Incidentally, note that the degree of a map between 0–spheres can only +take the values −1, 0, 1. +By duality, the degree can be alternatively defined using reduced cohomology groups. +If +ωM, ωL are generators of the reduced n-th cohomology group of M, L, we have that g∗(ωL) = +deg(g) ωM. +3 + +3.1. Decomposition of the degree. Suppose now that f : M → M is a self-map and N is a +completely invariant submanifold (i.e. f−1(N) = N). Under some topological hypothesis, the +degree of f is equal to the product of two factors: the degree of the restriction of f to N and +an integer d that accounts for the winding around N in the transversal direction, which we call +transversal degree. +(7) +deg(f) = d · deg(f|N) +Lemma 7. Let M be a closed connected and orientable n-dimensional manifold and f : M → M +a continuous map with deg(f) ̸= 0. Suppose that N ⊂ M is a closed orientable submanifold +of codimension k ≥ 2 that is completely invariant, i.e. +f−1(N) = N, and �Hn−k(N) ∼= Z. +Then deg(f|N) divides deg(f). Moreover, if Hk(M) ∼= 0 or if k = n there is a non trivial class +β ∈ Hk−1(M − N) such that f∗(β) = d · β, where d = deg(f) / deg(f|N). +Proof. Observe that, since N is completely invariant, the map f can be considered as a map +of pairs f : (M, M − N) → (M, M − N). Let [M] ∈ Hn(M) a fundamental class in M. The +homomorphism +(8) +⌢ [M] : Hn−k(N) → Hk(M, M − N) +consisting in capping each (unreduced) cohomology class in Hn−k(N) with the fundamental +class [M] is an isomorphism (see [1, Theorem 8.3, pg. 351]). +Let ω be a generator of the reduced cohomology group �Hn−k(N) (note that ω also generates +Hn−k(N) unless k = n). Then, by naturality of the cap product [1, Theorem 5.2, pg. 336] it +follows +(9) +f∗(f∗ +|N(ω) ⌢ [M]) = ω ⌢ f∗([M]). +If we examine the left-hand side of (9) we get +(10) +f∗(f∗ +|N(ω) ⌢ [M]) = f∗((deg(f|N) · ω) ⌢ [M]) = deg(f|N) · f∗(ω ⌢ [M]). +On the other hand, +(11) +ω ⌢ f∗([M]) = ω ⌢ (deg(f) · [M]) = deg(f) · (ω ⌢ [M]). +Hence, from (9), (10) and (11) it follows that deg(f|N) | deg(f) and the quotient is an integer +d that satisfies f∗(ω ⌢ [M]) = d · (ω ⌢ [M]) in Hk(M, M − N). +The second statement follows immediately from the naturality of the long exact sequence +of homology of (M, M − N) in the case Hk(M) is trivial. We can take β to be the image of +ω ⌢ [M] by the boundary morphism Hk(M, M −N) → Hk−1(M −N). In fact, by exactness the +existence of β is guaranteed as long as ω ⌢ [M] does not belong to the image of p: Hk(M) → +Hk(M, M − N). In the case k = n, N = {x, y} is a 0–sphere, Hk(M) is generated by the +fundamental class [M] and p([M]) = [M]x + [M]y. Then, the preimage of p([M]) under the +duality isomorphism (8) is the (unreduced) 0–cohomology class represented by the constant +map on N equal to 1 and, in particular, is different from ω. Then, ω ⌢ [M] /∈ im(p) and the +result follows. +□ +Heuristically, the transversal degree d counts how many times the image of the boundary of +a small neighborhood of N wraps around N. The interpretation is much clearer in the case +M = S = Sm and N = P, the polar sphere of codimension k = 2. Note that H2(S) is trivial +for all m > 2 and if m = 2 = k, P is a 0–sphere and the lemma still applies. From (2) we get +that H1(S − P) ∼= Z and we deduce that +(f|S−P )∗ : H1(S − P) → H1(S − P) +is conjugate to the multiplication by d in Z. Evidently, the same description applies to the action +induced in the fundamental group of S − P as well: if γ is a loop that generates π1(S − P) then +(12) +f∗[γ] = d · [γ]. +4 + +3.2. Vector fields and fixed points. The final step in the proof of Theorem 5 uses an argu- +ment from topological degree theory. In order to keep the article self-contained, we formulate +and prove the elementary results which are needed. +Let U be an open subset of Rm and B ⊂ U be diffeomorphic to D +m. Any non-singular vector +field v on ∂B defines a map j : ∂B → Sm−1 by jv(x) = v(x)/||v(x)||. +Lemma 8. +(i) If v points inwards B then jv is not nulhomotopic (i.e., not homotopic to +the constant map). +(ii) If v, w never point to the same direction (that is, jv(x) ̸= jw(x) for all x) and jv is not +nulhomotopic then jw is not nulhomotopic. +Further, suppose that ∂B is decomposed as the union of two hemispheres E+, E−, that is +(E+, E−) is diffeomorphic to (H+, H−), where H+ and H− denote the uppper and lower hemi- +sphere of Sm−1. +(iii) If v points inwards on E+ ∩ E−, jv(E+) ⊂ H+ and jv(E−) ⊂ H− then jv is not nulho- +motopic. +Proof. (i) Clearly, jv is conjugate to a self-map of Sm−1 that is homotopic to the antipodal map. +The conclusion follows from the fact that the antipodal map on Sm−1 is not nulhomotopic +(otherwise, we could construct an homotopy from the identity map to a constant map by +composing with the antipodal map). +(ii) jw is homotopic to j−v. We can now use an argument as in (i) to conclude. +(iii) If σ is the reflection through the equator on Sm−1, σ ◦ jv is conjugate to the antipodal +map. Again, we conclude that it is not nulhomotopic. +□ +Given a map h: U → Rm, we define a vector field vh(x) = h(x) − x. Singularities of vh +correspond to fixed points of h, so when we work with jvh we tacitly assume h has no fixed +points on the boundary of B. One of the central ideas of topological degree theory is that it +is possible to detect fixed points of h inside B just by studying vh on ∂B or, more precisely, +the homotopy class of jvh. Indeed, if Fix(h) ∩ B = ∅ then vh has no singularities in B and, +using a foliation of B by spheres that converge to an interior point, it is possible to construct +an homotopy from jvh to a constant map. In other words, +Lemma 9. If jvh is not nulhomotopic, there exists a fixed point of h in B. +Let us point out that one of the first results in topological degree theory is that the reverse +implication is true up to homotopy. +If jvh is nulhomotopic, it is possible to construct an +homotopy between h and h′ relative to ∂B such that h′ has no fixed points in B. +4. Local analysis at fixed points in P +Recall the setting from Section 2. P has a basis of neighborhoods diffeomorphic by (3) to +D2(r) × Sm−2. +The map f induces, by projection onto the first factor, a dynamics in the +2-dimensional normal direction around P. We obtain a family of C1 maps fp : D2(s) → D2, +p ∈ P, for some fixed small s > 0. +The smoothness of f poses a restriction on the behavior of fp for a fixed point p ∈ P because +of the following reason: a C1 map is injective in a neighborhood of a repelling fixed point. +This fact follows from the inverse function theorem and the fact that the repelling condition +implies that the eigenvalues of the jacobian matrix at the fixed point lie outside the unit disk +and, in particular, away from zero. We shall prove later that if any fp is injective then the +transversal degree satisfies |d| ≤ 1 and Theorem 1 becomes trivial. Accordingly, we focus on the +case |d| > 1 in which fp is not injective for any p ∈ P and, in particular, the Jacobian matrix +Ap of fp at the origin in the 2–dimensional normal direction is singular. Therefore, there are +only two dynamically different cases stated in terms of the spectral radius of Ap, either it is +smaller than 1 and the origin is an attractor for fp or it is greater or equal than 1 and there is +an attracting cone region. +We proceed now to study the dynamics of planar maps such as fp. Later, we apply the local +picture to describe the behavior of f in the normal direction to P. +5 + +4.1. Planar results. Suppose g ∈ R2 → R2 is a C1 map that fixes the origin, g(0) = 0. Denote +by A the Jacobian matrix of g at 0. By the definition of differentiability at the origin, for every +ϵ > 0 there exists δ > 0 such that +||g(u) − Au|| +||u|| +< ϵ, +for all u such that 0 < ||u|| < δ +(13) +where we have used the identification T0R2 ∼= R2 and || · || is a norm in R2. Recall that all +the norms in a finite dimensional vector space are equivalent. The spectral radius of A, ρ(A), +largely determines the behavior of g in a neighborhood of 0. +Lemma 10. For every c > ρ(A) there exists a norm || · || in R2 such that +||Au|| < c ||u|| for every u ∈ R2 − {0}. +Proof. If A is diagonalizable over R, we can take the ℓ1–norm associated to a basis B composed +of eigenvectors, that is, ||u|| = ||(u1, u2)B|| := |u1|+|u2|. If the eigenvalues of A are not real, the +ℓ2–norm associated to an orthogonal basis satisfies the conclusion. Finally, if the eigenvalues of +A are equal but A is not diagonalizable, let e0 be an eigenvector and e1 ̸= 0 not collinear to e0. +Then, we can take the ℓ1–norm associated to the basis {Ke0, e1} for large enough K > 0. +□ +An immediate consequence of the previous lemma and (13) is that if ρ(A) < 1 then the origin +is a local attractor for g. +Corollary 11. Suppose ρ(A) < 1 and let ϵ ∈ (0, 1 − ρ(A)), then there exists δ > 0 and a norm +in R2 such that ||g(u)|| < (1 − ϵ)||u|| whenever 0 < ||u|| < δ. +In the case there is an eigenvalue λ with |λ| ≥ 1, we can locate the region where the inequality +||g(u)|| < ||u|| does not hold. +Lemma 12. Suppose the eigenvalues of A are {0, λ} with |λ| ≥ 1. Denote B = {e0, eλ} a basis +composed of eigenvectors, || · || the ℓ1–norm associated to B and +C(α) = {u = (u0, uλ)B ∈ R2 − {0} : |u0/uλ| < α}. +For every ϵ ∈ (0, 1/2) there exists δ > 0 such that if 0 < ||u|| < δ and +(i) if u /∈ C +� +|λ|+2ϵ−1 +1−2ϵ +� +then ||g(u)|| < (1 − ϵ)||u||. +(ii) if u ∈ C +� +|λ|−3ϵ +3ϵ +� +then |(g(u))λ| > ϵ||u|| and, in particular, g(u) /∈ ⟨e0⟩. (⟨v⟩ denotes the +subspace spanned by v) +Proof. Apply (13) to || · || and ϵ. For (i), if u = (u0, uλ)B with ||u|| < δ +(1 − ϵ)||u|| = (1 − ϵ)|u0| + (1 − ϵ)|uλ| ≥ ϵ|u0| + (|λ| + ϵ)|uλ| = ||Au|| + ϵ||u|| > ||g(u)|| +To deduce (ii) we use that ||g(u)|| + ϵ||u|| ≥ ||Au||: +|(g(u))λ| = ||g(u)|| − |(g(u))0| ≥ |λ||uλ| − 2ϵ||u|| = (|λ| − 2ϵ)|uλ| − 2ϵ|u0| > ϵ||u|| > 0. +□ +4.2. Analysis in the normal direction. Since f(P) = P, f restricts to a continuous map +f : D2(s) × P −→ D2 × P +for some s > 0, where we extensively use the coordinates introduced in (3). The Jacobian +matrix of f in a point p ∈ P has the following form: +Jfp = +�Ap +0 +∗ +(Jf|P )p +� +where the basis for the tangent space at p is ordered according to the local product structure +around P: first the (2-dimensional) normal space to P and then the ((m − 2)–dimensional) +6 + +tangent space to P. Alternatively, Ap can be defined as the Jacobian at 0 of the following +composition +fp : D2(s) → D2(s) × {p} �→ D2(s) × P +f +−−→ D2 × P +proj +−−→ D2 +(14) +Lemma 13. If |d| > 1 then Ap is singular for every p ∈ P. +Proof. Suppose that Ap is regular. Then, fp restricts to a diffeomorphism between D2(r) and +V = fp(D2(r)), for small r > 0. In particular, if γ is a generator of π1(D2(r) − {0}), then fp(γ) +is a generator of π1(V − {0}). Think of γ as a loop in (D2(r) − {0}) × {p} and choose it small +so that f(γ) ⊂ D2 × Uf(p), where Uf(p) is contractible in P. It follows that both γ and f(γ) +generate π1(S − P) and, by (12), we deduce that d = ±1, a contradiction. +□ +Therefore, either Corollary 11 or Lemma 12 apply to fp when |d| > 1. For a given p ∈ P, we +extend the results by continuity to fq for q close to p. +Proposition 14. Suppose that |d| > 1. For every p ∈ P, there exists a neighborhood D2(δ)×Up +of p in S and a norm || · || in R2 such that: +(i) If ρ(Ap) < 1, for all q ∈ Up and all u ∈ D2(δ), +||fq(u)|| ≤ ||u||. +(ii) Otherwise, there is a basis {e0, e1} of R2 and α ∈ R+ such that for every q ∈ Up +– if u /∈ C(α) and u ∈ D2(δ) then ||fq(u)|| ≤ ||u||. +– fq(C(α) ∩ D2(δ)) ∩ ⟨e0⟩ = ∅. +Proof. Let || · || be the norm from Corollary 11 or Lemma 12 depending on which alternative, +(i) or (ii), applies. +For (i), apply Corollary 11 to any ϵ ∈ (0, 1 − ρ(A)) to obtain || · || and δ such that ||fp(u)|| ≤ +(1 − ϵ)||u|| holds whenever ||u|| < δ. In order to extend the conclusion to fq, for q close to p, +we use the smoothness of f. +Since fq(u) = +�� 1 +0 Dfq|tu dt +� +u, we have that +(15) +||fq(u) − fp(u)|| ≤ +�� 1 +0 +||Dfq|tu − Dfp|tu|| dt +� +||u|| < γq||u|| +where γq = maxv∈D2(δ)||Dfq|v − Dfp|v||. +Since f is C1, γq → 0 as q → p. +Let Up be a +neighborhood of p in P such that |γq| < ϵ for all q ∈ Up. Then we conclude that ||fq(u)|| ≤ ||u|| +for all q ∈ Up and ||u|| < δ. +For (ii), denote λ be the non–zero eigenvalue of Ap. Firstly, take ϵ > 0 small enough so that +|λ|+2ϵ−1 +1−2ϵ +< |λ|−3ϵ +3ϵ +and set α = |λ|+2ϵ−1 +1−2ϵ +. Apply Lemma 12 to obtain ||·||, δ and {e0, e1}. The conclusions for q = p +follow immediately from the lemma. To extend the results to a neighborhood Up of p we use +(15) and proceed exactly as in (i). The argument above can be used verbatim to prove the first +item. For the second item, |(fq(u))λ| ≥ |(fp(u))λ| − ϵ||u|| > 0. Finally, note that the norm from +Lemma 12 may not be the ℓ2-norm so we might need to shrink δ to δ′ so that the standard +2-disk D2(δ′) fits inside the disk defined by ||u|| < δ. +□ +Below, in the proof of Theorem 5, only the fixed points p of P for which the second alterna- +tive in Proposition 14 applies require special attention. The local description obtained above +provides a cone C(α) above each q close to p that contains the repelling sector (if any) and +whose image misses the attracting direction spanned by e0. +5. Proof of Theorem 5 +Recall (see (2)) that S − P is diffeomorphic to S1 × Dm−1. Consider the cover +7 + +(16) +� +S − P ∼= R × Dm−1 −→ S1 × Dm−1 ∼= S − P +defined as the standard cover R → S1, t �→ e2πit in the first factor and as the identity in the +second factor. +Our aim is to prove that except for a few cases, every lift of f to � +S − P has a fixed point +in BM = [−M, M] × Dm−1(1 − δ) for large M and small δ > 0. The argument is based on +topological degree theory. The key observation is that, for most of the lifts F, the vector field +vF (x) = F(x) − x in R × Dm−1 never points to the same direction as the coordinate vector field +∂/∂r(x), whose definition will be recalled next, on the boundary of R × Dm−1(1 − δ). Then, we +can apply Lemmas 8 and 9 to conclude that F has a fixed point inside BM for large M > 0. +From (3), we deduce that S − P − P ′ is diffeomorphic to D2 +0 × P, where the subscript in D2 +0 +indicates that the disk is punctured at the origin. The lift of S − P − P ′ to the cover (16) is +therefore diffeomorphic to R × (0, 1) × P, where the second factor, (0, 1), corresponds to the +radial coordinate r of D2 +0 and also to 1 − r′, where r′ is the radial coordinate of Dm−1 in (16) +(cf. the discussion before (4)). The coordinate r and, more precisely, the vector field ∂/∂r that +it defines in the cover play a central role in the discussion. On the lateral face of the cylinder +BM, ∂/∂r points inwards. Note that, alternatively, if we use the coordinates from (2) instead +of those of (3), we see that the lift of S − P − P ′ is diffeomorphic to R × Dm−1 +0 +. +The lift of the partition (4) of S − P for r = δ still displays a product structure: +� +S − P = R × (0, δ) × P ⊔ R × Dm−1(1 − δ) +The factor R corresponds to the angular coordinate in the normal bundle of P in the first term +and to the lift of S1 (that parametrizes P ′) in the second instance. The projection onto the +first factor pr: � +S − P → R conjugates a generator τ of the group of deck transformations of the +cover and the translation by 1 in R. +Recall from the statement of Theorem 5 that d ̸= 0, 1. For an arbitrary lift F of f, by (5) we +have that Fτ = τ dF so +pr(x) = pr(y) + 1 +⇒ +pr(F(x)) = pr(F(y)) + d. +for all x, y ∈ � +S − P. Therefore, for a fixed δ > 0, if M is sufficiently large and if d > 1 then +pr(F(x)) ≤ −M − 1 +for every x ∈ {−M} × Dm−1(1 − δ) and +pr(F(x)) ≥ M + 1 +for every x ∈ {M} × Dm−1(1 − δ) +(17) +whereas if d ≤ −1 then +pr(F(x)) ≥ −M + 1 +for every x ∈ {−M} × Dm−1(1 − δ) and +pr(F(x)) ≤ M − 1 +for every x ∈ {M} × Dm−1(1 − δ) +(18) +These inequalities imply that on the left and right faces (as in Figure 1) of the solid cylinder +BM the vector field vF points outwards when d > 1 and inwards when d ≤ −1. +Let us focus now on the lateral face of BM and suppose further that d ̸= −1. Let p ∈ Fix(f|P ) +and consider the neighborhood D2(δ) × Up of p and the norm || · || from Proposition 14. Denote +Vp the lift of D2 +0(δ) × Up to the cover (note that the disk is punctured) and suppose vF and +∂/∂r point to the same direction at x ∈ Vp. This implies that the projection of x and F(x) to +D2 × P, have the form (u, q) and (au, q) for some a > 1 and q ∈ Up. In particular, fq(u) = au, +which automatically implies ||fq(u)|| > ||u||. Thus, the second alternative of Proposition 14 +applies to p, so there exists a basis {e0, e1} and α ∈ R+ such that u ∈ C(α) ∩ D2(δ) and +fq(C(α) ∩ D2(δ)) ∩ ⟨e0⟩ = ∅ for every q ∈ Up. The last property can be stated equivalently as +(19) +f((C(α) ∩ D2(δ)) × Up) ∩ (⟨e0⟩ × Up) = ∅ +We now lift these elements to the cover. +The cone region C(α) × Up lifts to a sequence +of domains On = (I + n +2 ) × (0, 1) × Up, indexed by n ∈ Z, where we are now employing the +8 + +x +τx +τ −1x +On +On+1 +On+2 +On−1 +On−2 +Ek +Ek+1 +Ek−1 +Ek+2 +BM +F(x) +F(τ −1x) +F(τx) +Figure 1. +Picture of a piece of BM for d = 2. The darker pieces and thicker +segments in the horizontal strip are the intersection of the domains On and the +strips En with the lateral face of BM, defined by r = δ. Arrows illustrate vF at +τ −1x, x, τx. +coordinates of R × (0, 1) × P, the lift of D2 +0 × P, and I is an interval in R of length < 1/2. See +Figure 1. Similarly, the strip ⟨e0⟩ × Up lifts to a sequence of strips En = {�e0 + n +2 } × (0, 1) × Up +for some �e0 ∈ R. Restricted to Vp, the strips En and the domains On are pairwise disjoint and +are placed alternately. Evidently, τ(En) = En+2 and τ(On) = On+2. +Denote Oδ +n = On ∩ (R × (0, δ) × Up), the subset of On defined by 0 < r < δ. The condition +(19) implies that F(Oδ +n) does not meet Em for any m ∈ Z. This imposes a serious restriction +on the number of lifts for which the image of Oδ +n intersects itself. +Lemma 15. There are at most two elements G of {F, τF, . . . , τ |d−1|−1F} such that +G(Oδ +n) ∩ On ̸= ∅ +for some n ∈ Z. +Proof. Recall that for any lift G, G(On+2) = Gτ(On) = τ dG(On) so if G(Oδ +n) lies in between +Ek and Ek+1 then G(Oδ +n+2) lies in between Ek+2d and Ek+2d+1. Suppose that G, G′ = τ mG are +two lifts of f such that G(Oδ +n) ∩ On ̸= ∅ and G′(Oδ +n+2s) ∩ On+2s ̸= ∅ for some s ̸= 0. It follows +that n + 2s = m + n + 2sd, so |d − 1| divides m. +In sum, there is at most one lift in {F, τF, . . . , τ |d−1|−1F} such that the intersection in the +statement is non-empty for some even n and at most one lift such that the intersection is +non-empty for some odd n. +□ +Incidentally, note that the previous lemma trivially holds when d = −1. As a consequence of +the discussion above we deduce: +Lemma 16. Let p ∈ Fix(f|P ) and d ̸= 0, 1. There exists δp > 0, Up neighborhood of p in P +such that there are at most two lifts G among {F, τF, . . . , τ |d−1|−1F} for which the vector field +vG(x) = G(x) − x points to the same direction as ∂/∂r at some point of Vp. +In the second part of the proof, we show that the lifts for which vG and ∂/∂r do not point +to the same direction at Vp, for any p ∈ Fix(f|P ), always have a fixed point on BM. In view of +the previous corollary this assertion concludes the proof. +We have already described what happens in a neighborhood ∪Up of the set of fixed points in +P. Since f|P has no fixed point on P − ∪Up, by continuity, there exists δ1 such that every point +in D2(δ1) × (P − ∪Up) is displaced tangentially to P by f, that is, if f((v, q)) = (fq(v), q′) then +q ̸= q′. As a consequence, for any lift G of f, vG and ∂/∂r do not point to the same direction +on R × (0, δ1) × (P − ∪Up). +Take δ > 0 smaller than all δp and δ1. The results from the previous paragraph and Lemma +16 imply that there are at least |d − 1| − 2#Fix(f|P ) lifts among {F, τF, . . . , τ |d−1|−1F} such +that vG and ∂/∂r do not point to the same direction on the region {r = δ} = R × {δ} × P. Say +G is one of them. +Take M large enough so that (17) holds for G and δ. Smooth out a small neighborhood of +the edges of the cylinder BM to obtain a convex domain B′ +M diffeomorphic to a closed ball. We +now define a vector field w on ∂B′ +M to apply the results from Section 3. +Case d > 1. Let w be the normal unitary vector on ∂B′ +M that point inwards. Note that w +coincides with ∂/∂r on the lateral face of B′ +M. In the rest of ∂B′ +M the inequalities (17) hold +9 + +provided the smoothing region is small enough. By the choice of G we conclude that w never +points to the same direction as vG. +Since B′ +M is diffeomorphic to a ball and w points inwards on its boundary, we can apply +Lemma 8 (i) and (ii) to deduce that jvG is not nulhomotopic. Then, Lemma 9 concludes that +G has a fixed point inside B′ +M, as desired. +Case d ≤ −1. Define w as the unitary vector that points inwards on the lateral face of +B′ +M and as the unitary vector that points outwards on the pieces of ∂B′ +M that are part of the +left and right faces of BM. 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Ingenieros Inform´aticos, Universidad Polit´ecnica de Madrid, 28660 Madrid, Espa˜na +Email address: h.barge@upm.es +Facultad de Ciencias Matem´aticas, Universidad Complutense de Madrid, 28040 Madrid, Espa˜na +Email address: luishcorbato@mat.ucm.es +10 + diff --git a/dtFAT4oBgHgl3EQfZB2n/content/tmp_files/load_file.txt b/dtFAT4oBgHgl3EQfZB2n/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0274154e75fc2c5660b5ee395609a92701c0466d --- /dev/null +++ b/dtFAT4oBgHgl3EQfZB2n/content/tmp_files/load_file.txt @@ -0,0 +1,470 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf,len=469 +page_content='ON THE GROWTH RATE INEQUALITY FOR SELF-MAPS OF THE SPHERE H´ECTOR BARGE AND LUIS HERN´ANDEZ-CORBATO Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Let Sm = {x2 0+x2 1+· · ·+x2 m = 1} and P = {x0 = x1 = 0}∩Sm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Suppose that f is a self–map of Sm such that f −1(P) = P and |deg(f|P )| < |deg(f)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Then, the number of fixed points of f n grows at least exponentially with base |d| > 1, where d = deg(f)/deg(f|P ) ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Introduction In [13], Shub raised the question on whether algebraic intersections numbers bound assymp- totically from below geometrical intersection numbers for C1 maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' For a self-map f : M → M on a manifold, a particular case of the previous question comes down to whether the number #Fix(fn) of points fixed by fn and the Lefschetz numbers L(fn) satisfy lim sup 1 n log(#Fix(fn)) ≥ lim sup 1 n log |L(fn)| This inequality is known as the growth rate inequality and bounds from below the base of the exponential growth rate of periodic points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' If the sequence of Lefschetz numbers L(fn) is unbounded, Lefschetz-Dold theorem together with a result of Shub and Sullivan [15] (that states that the index sequence of a fixed point for a C1 map is bounded) imply that there are infinitely many periodic points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' The growth rate inequality appeared again as an open problem in the Proceedings of the ICM 2006 [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' It is wide open in dimensions greater than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' The Lefschetz number of a self-map on a sphere can be computed from the topological degree as L(f) = 1 ± deg(f), so the growth rate inequality for self-maps of the sphere can be rewritten in terms of the degree as follows: (1) lim sup n→+∞ 1 n log(#Fix(fn)) ≥ log |deg(f)| Shub’s original paper [13] proved (1) for rational maps on S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' However, even for a C1 map of degree 2 on S2, it is not known whether the growth rate inequality holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Note that the C1 assumption is crucial, a degree-2 north-south map on S2 has only 2 fixed points and no other periodic point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Recently, the growth rate inequality has been proved in several instances for maps on the sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' In dimension 2, the sharp bound of log |deg(f)| was obtained when f preserves some singular foliations [12, 10, 4] or under hypothesis of dynamical nature [5, 7, 6, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' In higher dimensions the results are scarce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Weaker bounds for the growth rate of periodic points when the map preserves some foliation and some mild hypotheses is satisfied were obtained in [2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' In this article we prove a weak form of the growth rate inequality for maps on Sm = {x2 0 + x2 1 + · · · + x2 m = 1} ⊂ Rm+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Suppose f : Sm → Sm leaves the codimension–2 sphere P = {x0 = x1 = 0} completely invariant, that is, f−1(P) = P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Then, the degree of f is equal to the product of two factors: the degree of the restriction of f to P and a “transversal” degree denoted d ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' The latter can be interpreted in terms of the action induced by f on the homology group or the fundamental group of Sm − P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Primary 37C25, 37E99;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Secondary 55M20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Growth rate inequality, periodic point, topological degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' The authors are partially supported by the Spanish Ministerio de Ciencia e Innovaci´on (grants PGC2018- 098321-B-I00 and PID2021-126124NB-I00).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content='08543v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content='DS] 20 Jan 2023 Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Let f : Sm → Sm be a map such that f−1(P) = P and deg(f) ̸= 0 and let d = deg(f)/deg(f|P ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Then, #Fix(fn) + #Fix(fn |P ) ≥ |dn − 1|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' In particular, lim inf n→+∞ 1 n log(#Fix(fn)) ≥ log |d| In dimension m = 2, P is a 0–sphere and deg(f|P ) can only take the values −1, 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' We deduce the following corollary from Theorem 1, that was previously stated in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Suppose f : S2 → S2 and there are two points {p, p′} such that f−1({p, p′}) = {p, p′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Then, lim inf n→+∞ 1 n log(#Fix(fn)) ≥ log |deg(f)| In higher dimensions we obtain a weak bound for the growth rate: Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Under the hypothesis of Theorem 1, if m = 3 or, more generally, if the growth rate inequality holds in Sm−2 then lim inf n→+∞ 1 n#Fix(fn) ≥ 1 2 log |deg(f)| The result follows from the first inequality in Theorem 1 and the fact that max{|d|, |deg(f|P )|} ≥ � |deg(f)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Theorem 1 is deduced from Theorem 5, which is stated in the ensuing section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' The proof is contained in the last section of the article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' It requires a detailed local analysis of the map in the normal direction to P, presented in Section 4, and uses an argument from topological degree theory, which is quickly reviewed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Setting Let S = Sm = {(x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' , xm) : x2 0 + · · · + x2 m = 1} be the standard m–sphere in Rm+1 and P = {x0 = x1 = 0} ∩ S be an m − 2–dimensional sphere which we will refer to as the polar sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' The complement S − P is diffeomorphic to S1 × Dm−1, where Dm−1 denotes the (m − 1)–dimensional open unit disk, via the map (2) (x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' , xm) �−→ � (x0, x1) ||(x0, x1)||, (x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' , xm) � On the other hand, P ′ = {x2 = · · · = xm = 0} ∩ S is a 1–sphere such that S is the join of P and P ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' The set S − P ′ is diffeomorphic to D2 × Sm−2 by (3) (x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' , xm) �−→ � (x0, x1), (x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' , xm) ||(x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' , xm)|| � Equations (2) and (3) define coordinate charts for S − P and S − P ′, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' If we replace (x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' , xm) by (0, 0, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' , xm) in (3), we obtain a diffeomorphism between S − P ′ and D2 × P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' This description will be used extensively throughout this note, as it provides a product structure for the system of neighborhoods of P defined by the inequalities x2 0 + x2 1 < r for 0 < r < 1: they are diffeomorphic to D2(r) × P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Similarly, the inequalities x2 0 + x2 1 > 1 − r′, 0 < r′ < 1, define neighborhoods of P ′ in S − P diffeomorphic by (2) to S1 × Dm−1(r′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Note that the radial coordinates of D2 and Dm−1 are related by r + r′ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' For any 0 < r < 1, (4) S ∼= D2(r) × P ⊔ S1 × D m−1(1 − r) Evidently, the fundamental group of S − P is isomorphic to Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Using the coordinates from (3), for any r ∈ (0, 1) and p ∈ P, it is easy to see that π1(S − P) is generated by a loop γ that makes a positive turn around the origin in the 2-disk D2(r) × {p}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Let us consider the lift of S − P that trivializes [γ], pr: � S − P → S − P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' 2 In our results, f is a self–map of S for which P is completely invariant, f−1(P) = P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Since S − P is invariant under f, f can be lifted to � S − P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Let τ be a generator of the group of deck transformations of the cover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Any lift F of f satisfies (5) F τ = τ d F where d ∈ Z is the transversal degree defined by f∗[γ] = d · [γ] in π1(S − P) (see (12) later).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' As we will prove in Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content='1, deg(f) = d · deg(f|P ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' A fixed point of F projects onto a fixed point of f in S − P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' The following lemma is a consequence of (5) and establishes a relation between fixed points of different lifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' It is standard in Nielsen theory (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' [9, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content='10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' If two different lifts F, G = τ mF of f satisfy pr(Fix(F)) ∩ pr(Fix(G)) ̸= ∅, then d ̸= 1 and |d − 1| divides m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' By the previous lemma, we can bound from below #{Fix(f) ∩ (S − P)} by the number of lifts of f among {F, τF, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' , τ |d−1|−1F} that have fixed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Suppose that d ̸= 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' There are at most 2 #Fix(f|P ) fixed point free maps among {F, τF, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' , τ |d−1|−1F}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Theorem 5 and Lemma 4 immediately imply that #{Fix(f) ∩ (S − P)} ≥ |d − 1| − 2 #{Fix(f|P )} Replace f by fn in this inequality to deduce Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Note that the transversal degree associated to fn is dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Topological degree The topological degree of a map g: M → L between closed connected and oriented manifolds of dimension n ≥ 1 roughly counts with multiplicity the number of preimages of a point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' For a complete account on degree theory we refer to [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' We give below a precise definition of the degree in algebraic topological terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Recall that the orientation of a closed connected manifold M is given by a fundamental class [M], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' a generator of the reduced homology group �Hn(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Reduced and unreduced homology groups only differ at dimension 0, the reason to choose here the reduced groups shall become clear later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' The image [M]x of the fundamental class [M] under the projection �Hn(M) → Hn(M, M − {x}) is the local orientation at x ∈ M, and it is a generator of Hn(M, M − {x}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Let g: M → L be a map between n–dimensional closed orientable manifolds such that �Hn(M) ∼= �Hn(L) ∼= Z and [M], [L] be fundamental classes of M, L, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' The degree of g, denoted deg(g), is the integer that satisfies (6) g∗([M]) = deg(g) · [L], The reason to choose the hypothesis �Hn(M) ∼= Z in the definition is that it is satisfied by closed connected oriented manifolds of dimension n ≥ 1 (spaces that appear in the standard definition of topological degree) and also by 0–spheres (such as the polar sphere in S2, that is relevant in this paper).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Incidentally, note that the degree of a map between 0–spheres can only take the values −1, 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' By duality, the degree can be alternatively defined using reduced cohomology groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' If ωM, ωL are generators of the reduced n-th cohomology group of M, L, we have that g∗(ωL) = deg(g) ωM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Decomposition of the degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Suppose now that f : M → M is a self-map and N is a completely invariant submanifold (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' f−1(N) = N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Under some topological hypothesis, the degree of f is equal to the product of two factors: the degree of the restriction of f to N and an integer d that accounts for the winding around N in the transversal direction, which we call transversal degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' (7) deg(f) = d · deg(f|N) Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Let M be a closed connected and orientable n-dimensional manifold and f : M → M a continuous map with deg(f) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Suppose that N ⊂ M is a closed orientable submanifold of codimension k ≥ 2 that is completely invariant, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' f−1(N) = N, and �Hn−k(N) ∼= Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Then deg(f|N) divides deg(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Moreover, if Hk(M) ∼= 0 or if k = n there is a non trivial class β ∈ Hk−1(M − N) such that f∗(β) = d · β, where d = deg(f) / deg(f|N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Observe that, since N is completely invariant, the map f can be considered as a map of pairs f : (M, M − N) → (M, M − N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Let [M] ∈ Hn(M) a fundamental class in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' The homomorphism (8) ⌢ [M] : Hn−k(N) → Hk(M, M − N) consisting in capping each (unreduced) cohomology class in Hn−k(N) with the fundamental class [M] is an isomorphism (see [1, Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content='3, pg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' 351]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Let ω be a generator of the reduced cohomology group �Hn−k(N) (note that ω also generates Hn−k(N) unless k = n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Then, by naturality of the cap product [1, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content='2, pg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' 336] it follows (9) f∗(f∗ |N(ω) ⌢ [M]) = ω ⌢ f∗([M]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' If we examine the left-hand side of (9) we get (10) f∗(f∗ |N(ω) ⌢ [M]) = f∗((deg(f|N) · ω) ⌢ [M]) = deg(f|N) · f∗(ω ⌢ [M]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' On the other hand, (11) ω ⌢ f∗([M]) = ω ⌢ (deg(f) · [M]) = deg(f) · (ω ⌢ [M]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Hence, from (9), (10) and (11) it follows that deg(f|N) | deg(f) and the quotient is an integer d that satisfies f∗(ω ⌢ [M]) = d · (ω ⌢ [M]) in Hk(M, M − N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' The second statement follows immediately from the naturality of the long exact sequence of homology of (M, M − N) in the case Hk(M) is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' We can take β to be the image of ω ⌢ [M] by the boundary morphism Hk(M, M −N) → Hk−1(M −N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' In fact, by exactness the existence of β is guaranteed as long as ω ⌢ [M] does not belong to the image of p: Hk(M) → Hk(M, M − N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' In the case k = n, N = {x, y} is a 0–sphere, Hk(M) is generated by the fundamental class [M] and p([M]) = [M]x + [M]y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Then, the preimage of p([M]) under the duality isomorphism (8) is the (unreduced) 0–cohomology class represented by the constant map on N equal to 1 and, in particular, is different from ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Then, ω ⌢ [M] /∈ im(p) and the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' □ Heuristically, the transversal degree d counts how many times the image of the boundary of a small neighborhood of N wraps around N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' The interpretation is much clearer in the case M = S = Sm and N = P, the polar sphere of codimension k = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Note that H2(S) is trivial for all m > 2 and if m = 2 = k, P is a 0–sphere and the lemma still applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' From (2) we get that H1(S − P) ∼= Z and we deduce that (f|S−P )∗ : H1(S − P) → H1(S − P) is conjugate to the multiplication by d in Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Evidently, the same description applies to the action induced in the fundamental group of S − P as well: if γ is a loop that generates π1(S − P) then (12) f∗[γ] = d · [γ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Vector fields and fixed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' The final step in the proof of Theorem 5 uses an argu- ment from topological degree theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' In order to keep the article self-contained, we formulate and prove the elementary results which are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Let U be an open subset of Rm and B ⊂ U be diffeomorphic to D m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Any non-singular vector field v on ∂B defines a map j : ∂B → Sm−1 by jv(x) = v(x)/||v(x)||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' (i) If v points inwards B then jv is not nulhomotopic (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=', not homotopic to the constant map).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' (ii) If v, w never point to the same direction (that is, jv(x) ̸= jw(x) for all x) and jv is not nulhomotopic then jw is not nulhomotopic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Further, suppose that ∂B is decomposed as the union of two hemispheres E+, E−, that is (E+, E−) is diffeomorphic to (H+, H−), where H+ and H− denote the uppper and lower hemi- sphere of Sm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' (iii) If v points inwards on E+ ∩ E−, jv(E+) ⊂ H+ and jv(E−) ⊂ H− then jv is not nulho- motopic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' (i) Clearly, jv is conjugate to a self-map of Sm−1 that is homotopic to the antipodal map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' The conclusion follows from the fact that the antipodal map on Sm−1 is not nulhomotopic (otherwise, we could construct an homotopy from the identity map to a constant map by composing with the antipodal map).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' (ii) jw is homotopic to j−v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' We can now use an argument as in (i) to conclude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' (iii) If σ is the reflection through the equator on Sm−1, σ ◦ jv is conjugate to the antipodal map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Again, we conclude that it is not nulhomotopic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' □ Given a map h: U → Rm, we define a vector field vh(x) = h(x) − x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Singularities of vh correspond to fixed points of h, so when we work with jvh we tacitly assume h has no fixed points on the boundary of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' One of the central ideas of topological degree theory is that it is possible to detect fixed points of h inside B just by studying vh on ∂B or, more precisely, the homotopy class of jvh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Indeed, if Fix(h) ∩ B = ∅ then vh has no singularities in B and, using a foliation of B by spheres that converge to an interior point, it is possible to construct an homotopy from jvh to a constant map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' In other words, Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' If jvh is not nulhomotopic, there exists a fixed point of h in B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Let us point out that one of the first results in topological degree theory is that the reverse implication is true up to homotopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' If jvh is nulhomotopic, it is possible to construct an homotopy between h and h′ relative to ∂B such that h′ has no fixed points in B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Local analysis at fixed points in P Recall the setting from Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' P has a basis of neighborhoods diffeomorphic by (3) to D2(r) × Sm−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' The map f induces, by projection onto the first factor, a dynamics in the 2-dimensional normal direction around P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' We obtain a family of C1 maps fp : D2(s) → D2, p ∈ P, for some fixed small s > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' The smoothness of f poses a restriction on the behavior of fp for a fixed point p ∈ P because of the following reason: a C1 map is injective in a neighborhood of a repelling fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' This fact follows from the inverse function theorem and the fact that the repelling condition implies that the eigenvalues of the jacobian matrix at the fixed point lie outside the unit disk and, in particular, away from zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' We shall prove later that if any fp is injective then the transversal degree satisfies |d| ≤ 1 and Theorem 1 becomes trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Accordingly, we focus on the case |d| > 1 in which fp is not injective for any p ∈ P and, in particular, the Jacobian matrix Ap of fp at the origin in the 2–dimensional normal direction is singular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Therefore, there are only two dynamically different cases stated in terms of the spectral radius of Ap, either it is smaller than 1 and the origin is an attractor for fp or it is greater or equal than 1 and there is an attracting cone region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' We proceed now to study the dynamics of planar maps such as fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Later, we apply the local picture to describe the behavior of f in the normal direction to P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' 5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Planar results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Suppose g ∈ R2 → R2 is a C1 map that fixes the origin, g(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Denote by A the Jacobian matrix of g at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' By the definition of differentiability at the origin, for every ϵ > 0 there exists δ > 0 such that ||g(u) − Au|| ||u|| < ϵ, for all u such that 0 < ||u|| < δ (13) where we have used the identification T0R2 ∼= R2 and || · || is a norm in R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Recall that all the norms in a finite dimensional vector space are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' The spectral radius of A, ρ(A), largely determines the behavior of g in a neighborhood of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' For every c > ρ(A) there exists a norm || · || in R2 such that ||Au|| < c ||u|| for every u ∈ R2 − {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' If A is diagonalizable over R, we can take the ℓ1–norm associated to a basis B composed of eigenvectors, that is, ||u|| = ||(u1, u2)B|| := |u1|+|u2|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' If the eigenvalues of A are not real, the ℓ2–norm associated to an orthogonal basis satisfies the conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Finally, if the eigenvalues of A are equal but A is not diagonalizable, let e0 be an eigenvector and e1 ̸= 0 not collinear to e0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Then, we can take the ℓ1–norm associated to the basis {Ke0, e1} for large enough K > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' □ An immediate consequence of the previous lemma and (13) is that if ρ(A) < 1 then the origin is a local attractor for g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Corollary 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Suppose ρ(A) < 1 and let ϵ ∈ (0, 1 − ρ(A)), then there exists δ > 0 and a norm in R2 such that ||g(u)|| < (1 − ϵ)||u|| whenever 0 < ||u|| < δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' In the case there is an eigenvalue λ with |λ| ≥ 1, we can locate the region where the inequality ||g(u)|| < ||u|| does not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Suppose the eigenvalues of A are {0, λ} with |λ| ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Denote B = {e0, eλ} a basis composed of eigenvectors, || · || the ℓ1–norm associated to B and C(α) = {u = (u0, uλ)B ∈ R2 − {0} : |u0/uλ| < α}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' For every ϵ ∈ (0, 1/2) there exists δ > 0 such that if 0 < ||u|| < δ and (i) if u /∈ C � |λ|+2ϵ−1 1−2ϵ � then ||g(u)|| < (1 − ϵ)||u||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' (ii) if u ∈ C � |λ|−3ϵ 3ϵ � then |(g(u))λ| > ϵ||u|| and, in particular, g(u) /∈ ⟨e0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' (⟨v⟩ denotes the subspace spanned by v) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Apply (13) to || · || and ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' For (i), if u = (u0, uλ)B with ||u|| < δ (1 − ϵ)||u|| = (1 − ϵ)|u0| + (1 − ϵ)|uλ| ≥ ϵ|u0| + (|λ| + ϵ)|uλ| = ||Au|| + ϵ||u|| > ||g(u)|| To deduce (ii) we use that ||g(u)|| + ϵ||u|| ≥ ||Au||: |(g(u))λ| = ||g(u)|| − |(g(u))0| ≥ |λ||uλ| − 2ϵ||u|| = (|λ| − 2ϵ)|uλ| − 2ϵ|u0| > ϵ||u|| > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Analysis in the normal direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Since f(P) = P, f restricts to a continuous map f : D2(s) × P −→ D2 × P for some s > 0, where we extensively use the coordinates introduced in (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' The Jacobian matrix of f in a point p ∈ P has the following form: Jfp = �Ap 0 ∗ (Jf|P )p � where the basis for the tangent space at p is ordered according to the local product structure around P: first the (2-dimensional) normal space to P and then the ((m − 2)–dimensional) 6 tangent space to P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Alternatively, Ap can be defined as the Jacobian at 0 of the following composition fp : D2(s) → D2(s) × {p} �→ D2(s) × P f −−→ D2 × P proj −−→ D2 (14) Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' If |d| > 1 then Ap is singular for every p ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Suppose that Ap is regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Then, fp restricts to a diffeomorphism between D2(r) and V = fp(D2(r)), for small r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' In particular, if γ is a generator of π1(D2(r) − {0}), then fp(γ) is a generator of π1(V − {0}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Think of γ as a loop in (D2(r) − {0}) × {p} and choose it small so that f(γ) ⊂ D2 × Uf(p), where Uf(p) is contractible in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' It follows that both γ and f(γ) generate π1(S − P) and, by (12), we deduce that d = ±1, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' □ Therefore, either Corollary 11 or Lemma 12 apply to fp when |d| > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' For a given p ∈ P, we extend the results by continuity to fq for q close to p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Proposition 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Suppose that |d| > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' For every p ∈ P, there exists a neighborhood D2(δ)×Up of p in S and a norm || · || in R2 such that: (i) If ρ(Ap) < 1, for all q ∈ Up and all u ∈ D2(δ), ||fq(u)|| ≤ ||u||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' (ii) Otherwise, there is a basis {e0, e1} of R2 and α ∈ R+ such that for every q ∈ Up – if u /∈ C(α) and u ∈ D2(δ) then ||fq(u)|| ≤ ||u||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' – fq(C(α) ∩ D2(δ)) ∩ ⟨e0⟩ = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Let || · || be the norm from Corollary 11 or Lemma 12 depending on which alternative, (i) or (ii), applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' For (i), apply Corollary 11 to any ϵ ∈ (0, 1 − ρ(A)) to obtain || · || and δ such that ||fp(u)|| ≤ (1 − ϵ)||u|| holds whenever ||u|| < δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' In order to extend the conclusion to fq, for q close to p, we use the smoothness of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Since fq(u) = �� 1 0 Dfq|tu dt � u, we have that (15) ||fq(u) − fp(u)|| ≤ �� 1 0 ||Dfq|tu − Dfp|tu|| dt � ||u|| < γq||u|| where γq = maxv∈D2(δ)||Dfq|v − Dfp|v||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Since f is C1, γq → 0 as q → p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Let Up be a neighborhood of p in P such that |γq| < ϵ for all q ∈ Up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Then we conclude that ||fq(u)|| ≤ ||u|| for all q ∈ Up and ||u|| < δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' For (ii), denote λ be the non–zero eigenvalue of Ap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Firstly, take ϵ > 0 small enough so that |λ|+2ϵ−1 1−2ϵ < |λ|−3ϵ 3ϵ and set α = |λ|+2ϵ−1 1−2ϵ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Apply Lemma 12 to obtain ||·||, δ and {e0, e1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' The conclusions for q = p follow immediately from the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' To extend the results to a neighborhood Up of p we use (15) and proceed exactly as in (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' The argument above can be used verbatim to prove the first item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' For the second item, |(fq(u))λ| ≥ |(fp(u))λ| − ϵ||u|| > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Finally, note that the norm from Lemma 12 may not be the ℓ2-norm so we might need to shrink δ to δ′ so that the standard 2-disk D2(δ′) fits inside the disk defined by ||u|| < δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' □ Below, in the proof of Theorem 5, only the fixed points p of P for which the second alterna- tive in Proposition 14 applies require special attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' The local description obtained above provides a cone C(α) above each q close to p that contains the repelling sector (if any) and whose image misses the attracting direction spanned by e0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Proof of Theorem 5 Recall (see (2)) that S − P is diffeomorphic to S1 × Dm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Consider the cover 7 (16) � S − P ∼= R × Dm−1 −→ S1 × Dm−1 ∼= S − P defined as the standard cover R → S1, t �→ e2πit in the first factor and as the identity in the second factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Our aim is to prove that except for a few cases, every lift of f to � S − P has a fixed point in BM = [−M, M] × Dm−1(1 − δ) for large M and small δ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' The argument is based on topological degree theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' The key observation is that, for most of the lifts F, the vector field vF (x) = F(x) − x in R × Dm−1 never points to the same direction as the coordinate vector field ∂/∂r(x), whose definition will be recalled next, on the boundary of R × Dm−1(1 − δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Then, we can apply Lemmas 8 and 9 to conclude that F has a fixed point inside BM for large M > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' From (3), we deduce that S − P − P ′ is diffeomorphic to D2 0 × P, where the subscript in D2 0 indicates that the disk is punctured at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' The lift of S − P − P ′ to the cover (16) is therefore diffeomorphic to R × (0, 1) × P, where the second factor, (0, 1), corresponds to the radial coordinate r of D2 0 and also to 1 − r′, where r′ is the radial coordinate of Dm−1 in (16) (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' the discussion before (4)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' The coordinate r and, more precisely, the vector field ∂/∂r that it defines in the cover play a central role in the discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' On the lateral face of the cylinder BM, ∂/∂r points inwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Note that, alternatively, if we use the coordinates from (2) instead of those of (3), we see that the lift of S − P − P ′ is diffeomorphic to R × Dm−1 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' The lift of the partition (4) of S − P for r = δ still displays a product structure: � S − P = R × (0, δ) × P ⊔ R × Dm−1(1 − δ) The factor R corresponds to the angular coordinate in the normal bundle of P in the first term and to the lift of S1 (that parametrizes P ′) in the second instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' The projection onto the first factor pr: � S − P → R conjugates a generator τ of the group of deck transformations of the cover and the translation by 1 in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Recall from the statement of Theorem 5 that d ̸= 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' For an arbitrary lift F of f, by (5) we have that Fτ = τ dF so pr(x) = pr(y) + 1 ⇒ pr(F(x)) = pr(F(y)) + d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' for all x, y ∈ � S − P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Therefore, for a fixed δ > 0, if M is sufficiently large and if d > 1 then pr(F(x)) ≤ −M − 1 for every x ∈ {−M} × Dm−1(1 − δ) and pr(F(x)) ≥ M + 1 for every x ∈ {M} × Dm−1(1 − δ) (17) whereas if d ≤ −1 then pr(F(x)) ≥ −M + 1 for every x ∈ {−M} × Dm−1(1 − δ) and pr(F(x)) ≤ M − 1 for every x ∈ {M} × Dm−1(1 − δ) (18) These inequalities imply that on the left and right faces (as in Figure 1) of the solid cylinder BM the vector field vF points outwards when d > 1 and inwards when d ≤ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Let us focus now on the lateral face of BM and suppose further that d ̸= −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Let p ∈ Fix(f|P ) and consider the neighborhood D2(δ) × Up of p and the norm || · || from Proposition 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Denote Vp the lift of D2 0(δ) × Up to the cover (note that the disk is punctured) and suppose vF and ∂/∂r point to the same direction at x ∈ Vp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' This implies that the projection of x and F(x) to D2 × P, have the form (u, q) and (au, q) for some a > 1 and q ∈ Up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' In particular, fq(u) = au, which automatically implies ||fq(u)|| > ||u||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Thus, the second alternative of Proposition 14 applies to p, so there exists a basis {e0, e1} and α ∈ R+ such that u ∈ C(α) ∩ D2(δ) and fq(C(α) ∩ D2(δ)) ∩ ⟨e0⟩ = ∅ for every q ∈ Up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' The last property can be stated equivalently as (19) f((C(α) ∩ D2(δ)) × Up) ∩ (⟨e0⟩ × Up) = ∅ We now lift these elements to the cover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' The cone region C(α) × Up lifts to a sequence of domains On = (I + n 2 ) × (0, 1) × Up, indexed by n ∈ Z, where we are now employing the 8 x τx τ −1x On On+1 On+2 On−1 On−2 Ek Ek+1 Ek−1 Ek+2 BM F(x) F(τ −1x) F(τx) Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Picture of a piece of BM for d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' The darker pieces and thicker segments in the horizontal strip are the intersection of the domains On and the strips En with the lateral face of BM, defined by r = δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Arrows illustrate vF at τ −1x, x, τx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' coordinates of R × (0, 1) × P, the lift of D2 0 × P, and I is an interval in R of length < 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' See Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Similarly, the strip ⟨e0⟩ × Up lifts to a sequence of strips En = {�e0 + n 2 } × (0, 1) × Up for some �e0 ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Restricted to Vp, the strips En and the domains On are pairwise disjoint and are placed alternately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Evidently, τ(En) = En+2 and τ(On) = On+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Denote Oδ n = On ∩ (R × (0, δ) × Up), the subset of On defined by 0 < r < δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' The condition (19) implies that F(Oδ n) does not meet Em for any m ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' This imposes a serious restriction on the number of lifts for which the image of Oδ n intersects itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Lemma 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' There are at most two elements G of {F, τF, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' , τ |d−1|−1F} such that G(Oδ n) ∩ On ̸= ∅ for some n ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Recall that for any lift G, G(On+2) = Gτ(On) = τ dG(On) so if G(Oδ n) lies in between Ek and Ek+1 then G(Oδ n+2) lies in between Ek+2d and Ek+2d+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Suppose that G, G′ = τ mG are two lifts of f such that G(Oδ n) ∩ On ̸= ∅ and G′(Oδ n+2s) ∩ On+2s ̸= ∅ for some s ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' It follows that n + 2s = m + n + 2sd, so |d − 1| divides m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' In sum, there is at most one lift in {F, τF, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' , τ |d−1|−1F} such that the intersection in the statement is non-empty for some even n and at most one lift such that the intersection is non-empty for some odd n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' □ Incidentally, note that the previous lemma trivially holds when d = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' As a consequence of the discussion above we deduce: Lemma 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Let p ∈ Fix(f|P ) and d ̸= 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' There exists δp > 0, Up neighborhood of p in P such that there are at most two lifts G among {F, τF, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' , τ |d−1|−1F} for which the vector field vG(x) = G(x) − x points to the same direction as ∂/∂r at some point of Vp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' In the second part of the proof, we show that the lifts for which vG and ∂/∂r do not point to the same direction at Vp, for any p ∈ Fix(f|P ), always have a fixed point on BM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' In view of the previous corollary this assertion concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' We have already described what happens in a neighborhood ∪Up of the set of fixed points in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Since f|P has no fixed point on P − ∪Up, by continuity, there exists δ1 such that every point in D2(δ1) × (P − ∪Up) is displaced tangentially to P by f, that is, if f((v, q)) = (fq(v), q′) then q ̸= q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' As a consequence, for any lift G of f, vG and ∂/∂r do not point to the same direction on R × (0, δ1) × (P − ∪Up).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Take δ > 0 smaller than all δp and δ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' The results from the previous paragraph and Lemma 16 imply that there are at least |d − 1| − 2#Fix(f|P ) lifts among {F, τF, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' , τ |d−1|−1F} such that vG and ∂/∂r do not point to the same direction on the region {r = δ} = R × {δ} × P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Say G is one of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Take M large enough so that (17) holds for G and δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Smooth out a small neighborhood of the edges of the cylinder BM to obtain a convex domain B′ M diffeomorphic to a closed ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' We now define a vector field w on ∂B′ M to apply the results from Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Case d > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Let w be the normal unitary vector on ∂B′ M that point inwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Note that w coincides with ∂/∂r on the lateral face of B′ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' In the rest of ∂B′ M the inequalities (17) hold 9 provided the smoothing region is small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' By the choice of G we conclude that w never points to the same direction as vG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Since B′ M is diffeomorphic to a ball and w points inwards on its boundary, we can apply Lemma 8 (i) and (ii) to deduce that jvG is not nulhomotopic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Then, Lemma 9 concludes that G has a fixed point inside B′ M, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Case d ≤ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Define w as the unitary vector that points inwards on the lateral face of B′ M and as the unitary vector that points outwards on the pieces of ∂B′ M that are part of the left and right faces of BM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Complete the definition of w on ∂B′ M by an interpolation that guarantees that in the smoothing region and close to the left face, {−M} × Dm−1(1 − δ), w never points in the positive direction (increasing first coordinate), while close to the right face, {M} × Dm−1(1 − δ), w never points in the negative direction (decreasing first coordinate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Again, by the choice of G, w and vG never point to the same direction and, by Lemma 8 (iii) and (ii) and Lemma 9 we conclude that G has a fixed point in B′ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' References [1] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Bredon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Topology and geometry, volume 139 of Graduate Texts in Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content=' Springer-Verlag, New York, 1993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} +page_content='es 10' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQfZB2n/content/2301.08543v1.pdf'} diff --git a/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf b/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..7105d94b36c44a62dc6aec0d2dc1dbc35c1c43f1 --- /dev/null +++ b/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:299dcce14979a2ebf4f56919a9b10f53963875a4bd538df72308ad62e3e423e0 +size 185900 diff --git a/eNE0T4oBgHgl3EQf5gKT/vector_store/index.pkl b/eNE0T4oBgHgl3EQf5gKT/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..920bc34b366bb75fbd8432b1995ed1a785fe30e9 --- /dev/null +++ b/eNE0T4oBgHgl3EQf5gKT/vector_store/index.pkl @@ -0,0 +1,3 @@ +version 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Yu2,3 +Ishan Misra1 +1FAIR, Meta AI +2UC Berkeley / ICSI +3University of Michigan +Code: https://github.com/facebookresearch/CutLER +natural images +paintings +sketches +clip arts +videos +traffic images +Domains +Sample +Results +37.5 +6.7 +Prev. SOTA +CutLER +Watercolor +31.7 +10 +UVO +30.4 +9.9 +Comic +21.1 +7.9 +Clipart +18.4 +7.7 +KITTI +21.6 +8.1 +Objects365 +36.9 +15.9 +VOC +22.4 +9.7 +COCO 20K +21.9 +9.6 +COCO +8.4 +3.8 +LVIS +17.3 +9.9 +OpenImages +Datasets +AP50 +Figure 1. Zero-shot unsupervised object detection and instance segmentation using our CutLER model, which is trained without +human supervision. We evaluate the model using the standard detection APbox +50 . CutLER gives a strong performance on a variety of +benchmarks spanning diverse image domains - video frames, paintings, clip arts, complex scenes, etc. Compared to the previous state- +of-the-art method, FreeSOLO [47] with a backbone of ResNet101, CutLER with a backbone of ResNet50 provides strong gains on all +benchmarks, increasing performance by more than 2× on 10 of the 11 benchmarks. We evaluate [47] with its official code and checkpoint. +Abstract +We propose Cut-and-LEaRn (CutLER), a simple ap- +proach for training unsupervised object detection and seg- +mentation models. +We leverage the property of self- +supervised models to ‘discover’ objects without supervision +and amplify it to train a state-of-the-art localization model +without any human labels. CutLER first uses our proposed +MaskCut approach to generate coarse masks for multiple +objects in an image, and then learns a detector on these +masks using our robust loss function. We further improve +performance by self-training the model on its predictions. +Compared to prior work, CutLER is simpler, compatible +with different detection architectures, and detects multiple +objects. CutLER is also a zero-shot unsupervised detec- +tor and improves detection performance AP50 by over 2.7× +on 11 benchmarks across domains like video frames, paint- +ings, sketches, etc. With finetuning, CutLER serves as a low- +shot detector surpassing MoCo-v2 by 7.3% APbox and 6.6% +APmask on COCO when training with 5% labels. +1. Introduction +Object localization is a critical task in computer vision +that enables AI systems to perceive, reason, plan and act in +an object-centric manner. Training models for localization +require special annotations like object boxes, masks, local- +ized points, etc. which are both difficult and resource inten- +sive to collect. Without accounting for overhead, annotating +∼164K images in the COCO dataset [32] with masks for +just 80 classes took more than 28K human hours of annota- +tion time. In this work, we study unsupervised object detec- +tion and instance segmentation models that can be trained +without any human labels. Our key insight is that simple +probing and training mechanisms can amplify the innate lo- +calization ability of self-supervised models [7], leading to +state-of-the-art unsupervised zero-shot detectors. +Our method Cut-and-LEaRn (CutLER) consists of three +simple, architecture- and data-agnostic mechanisms. Con- +sistent with prior self-supervised learning methods [7–9, +26], CutLER is trained exclusively on unlabeled ImageNet +data without needing additional training data, but contrary +to these methods, CutLER can be directly employed to per- +form complex segmentation and detection tasks over a wide +range of domains. First, we propose MaskCut that can au- +tomatically produce multiple initial coarse masks for each +image, using the pretrained self-supervised features. Sec- +ond, we propose a simple loss dropping strategy to train +detectors using the coarse masks while being robust to ob- +jects missed by MaskCut. Finally, we observe that despite +1 +arXiv:2301.11320v1 [cs.CV] 26 Jan 2023 + +Naranjd-OrangeLeaveitJUSTINEMARKOWSKlearning from these coarse masks, the detectors ‘clean’ the +ground truth and produce masks (and boxes) that are bet- +ter than the coarse masks used to train them. Therefore, +we further show that multiple rounds of self-training on the +models’ own predictions allow it to evolve from capturing +the similarity of local pixels to capturing the global geome- +try of the object, thus producing finer segmentation masks. +Prior work shows that a self-supervised vision trans- +former (ViT) [15] can automatically learn patch-wise fea- +tures that detect a single salient object in an image [7,38,43, +44,50]. However, unlike CutLER, such salient object detec- +tion methods only locate a single, usually the most promi- +nent, object and cannot be used for real world images con- +taining multiple objects. While some recent methods, e.g., +FreeSOLO [47] and DETReg [3], also aim at unsupervised +multi-object detection (or multi-object discovery), they rely +on a particular detection architecture, e.g., SOLO-v2 [48] +or DDETR [5,54]. Additionally, apart from self-supervised +features trained on ImageNet [12], the current state-of-the- +art methods FreeSOLO and MaskDistill [42] also require +‘in-domain’ unlabeled data for model training. +In contrast, CutLER works with various detection archi- +tectures and can be trained solely on ImageNet, without +requiring in-domain unlabeled data. Thus, during model +training, CutLER does not see any images from any target +dataset and yields a zero-shot model capable of detecting +and segmenting multiple objects in diverse domains. +Features of CutLER. 1) Simplicity: CutLER is simple to +train and agnostic to the choice of detection and backbone +architectures. Thus, it can be integrated effortlessly into +existing object detection and instance segmentation works. +2) Zero-shot detector: CutLER trained solely on ImageNet +shows strong zero-shot performance on 11 different bench- +marks where it outperforms prior work trained with addi- +tional in-domain data. We double the APbox +50 performance +on 10 of these benchmarks, as shown in Fig. 1, and even +outperform supervised detectors on the UVO video instance +segmentation benchmark. 3) Robustness: CutLER exhibits +strong robustness against domain shifts when tested on im- +ages from different domains such as video frames, sketches, +paintings, clip arts, etc. 4) Pretraining for supervised de- +tection: CutLER can also serve as a pretrained model for +training fully supervised object detection and instance seg- +mentation models and improves performance on COCO, in- +cluding on few-shot object detection benchmarks. +2. Related Work +Self-supervised feature learning involves inferring the +patterns within the large-scale unlabeled data without us- +ing human-annotated labels. +Contrastive learning based +[8, 26, 34, 52] methods learn such representations that sim- +ilar samples or various augmentations of the same instance +are close to each other, while dissimilar instances are far +DINO LOST TokenCut FreeSOLO Ours +detect multiple objects + + + + + +zero-shot detector + + + + + +compatible with various +detection architectures +- + +- + + +pretrained model for +supervised detection + + + + + +Table 1. We compare previous methods on unsupervised object +detection, including DINO [7], LOST [38], TokenCut [50] and +FreeSOLO [47], with our CutLER in term of key properties. Our +CutLER is the only method with all these desired properties. +apart. +Similarity-based self-supervised learning methods +[10, 23] learn representations via minimizing the distance +between different augmentations of the same instance and +use only positive sample pairs. Clustering-based feature +learning [1, 6, 46, 53, 55] automatically discovers the nat- +ural grouping of data in the latent representation space. Re- +cently, [2,25] have shown that masked autoencoders, which +learn representations via masking out a large random subset +of image patches and reconstructing the missing pixels or +patches [2, 13, 14, 25], are scalable self-supervised learners +for computer vision [25]. +In contrast to these unsupervised representation learn- +ing efforts, our work aims to automatically discover natural +pixel groupings and locate instances within each image. +Unsupervised object detection and instance segmen- +tation. The main comparisons to previous works are listed +in Table 1 and are elaborated as follows: +DINO [7] observes that the underlying semantic segmen- +tation of images can emerge from the self-supervised Vision +Transformer (ViT) [15], which does not appear explicitly in +either supervised ViT or ConvNets [7, 56]. Based on this +observation, LOST [38] and TokenCut [50] leverage self- +supervised ViT features and propose to segment one single +salient object [11,38,50] from each image based on a graph +that is constructed with DINO’s patch features. +These previous works either can not detect more than +one object from each image, e.g., DINO and TokenCut, or +can not improve the quality of features for better transfer +to downstream detection and segmentation tasks, e.g., To- +kenCut and LOST. Unlike these works, CutLER can locate +multiple objects and serve as a pretrained model for label- +efficient and fully-supervised learning. +FreeSOLO [47] achieves unsupervised instance segmen- +tation by extracting coarse object masks in an unsuper- +vised manner, followed by mask refinement through a self- +training procedure. While FreeSOLO’s FreeMask stage can +generate multiple coarse masks per image, the quality of +these masks is often rather low [47]. MaskDistill [42] dis- +tills class-agnostic initial masks from the affinity graph pro- +duced by a self-supervised DINO [7]. However, it utilizes +one single mask per image in the distillation stage, which +2 + +Lcls + Lbox + Lmask +Lexp +unlabeled +data +MaskCut +ViT +Detector +self-training +Ldrop +Figure 2. Overview of CutLER. We propose a simple yet effec- +tive method to train an object detection and instance segmentation +model without using any supervision. We first propose MaskCut to +extract initial coarse masks from the features of a self-supervised +ViT. We then learn a detector using our loss dropping strategy that +is robust to objects missed by MaskCut. We further improve the +model using multiple rounds of self-training. +greatly limits the model’s ability to detect multiple objects. +By contrast, the initial masks generated by our Mask- +Cut are usually better in quality and quantity than the ini- +tial masks used by [42, 47]. Therefore, CutLER achieves +2×∼4× higher APbox and APmask than FreeSOLO [47] +and MaskDistill [42] on almost all experimented detection +and segmentation benchmarks, even when FreeSOLO and +MaskDistill are trained and tested on the same domain. +3. Method +We tackle the problem of unsupervised object detection +and segmentation with a simple cut-and-learn pipeline. Our +method builds upon insights from recent work [7,50], show- +ing that self-supervised representations can discover ob- +jects. While these methods often find a single object per +image, we propose a simple approach that can discover +multiple objects and significantly improves segmentation +and detection performance. The overview of our cut-and- +learn pipeline is illustrated in Fig. 2. +First, we propose +MaskCut that generates multiple binary masks per image +using self-supervised features from DINO [7] (Sec. 3.2). +Second, we show a dynamic loss dropping strategy, called +DropLoss, that can learn a detector from MaskCut’s ini- +tial masks while encouraging the model to explore objects +missed by MaskCut (Sec. 3.3); Third, we further improve +the performance of our method through multiple rounds of +self-training (Sec. 3.4). +3.1. Preliminaries +Normalized Cuts (NCut) treats the image segmentation +problem as a graph partitioning task [37]. We construct a +fully connected undirected graph via representing each im- +age as a node. Each pair of nodes is connected by edges +with weights Wij that measure the similarity of the con- +nected nodes. NCut minimizes the cost of partitioning the +graph into two sub-graphs, i.e., a bipartition, by solving a +generalized eigenvalue system +(D − W)x = λDx +(1) +for finding the eigenvector x that corresponds to the second +smallest eigenvalue λ, where D is a N×N diagonal matrix +with d(i) = � +j Wij and W is a N×N symmetrical matrix. +DINO and TokenCut. +DINO [7] finds that the self- +supervised ViT can automatically learn a certain degree +of perceptual grouping of image patches. TokenCut [50] +leverages the DINO features for NCut and obtaining fore- +ground/background segments in an image. The authors use +the similarity of the patches in the DINO feature space as +the similarity weight Wij in NCut. Specifically, follow- +ing multiple recent methods [38, 42, 50], we use the cosine +similarity of ‘key’ features from the last attention layer of +DINO-pretrained model, i.e., Wij = +KiKj +∥Ki∥2∥Kj∥2 where Ki +is the ‘key’ feature of patch i, and solve Eq. (1) for finding +the second smallest eigenvector x. +A limitation of TokenCut is that it only computes a sin- +gle binary mask for an image and thus only finds one object +per image. Although we can use the other N −2 smallest +eigenvectors to locate more than one instance, this signifi- +cantly degrades the performance for multi-object discovery, +as demonstrated in Sec. 5. +3.2. MaskCut for Discovering Multiple Objects +As we discussed in Sec. 3.1, vanilla NCut is limited to +discovering a single object in an image. We propose Mask- +Cut that extends NCut to discover multiple objects per im- +age by iteratively applying NCut to a masked similarity ma- +trix (illustrated in Fig. 3). After getting the bipartition xt +from NCut at stage t, we get two disjoint groups of patches +and construct a binary mask M t, where +M t +ij = +� +1, +if M t +ij ≥ mean(xt) +0, +otherwise. +(2) +To determine which group corresponds to the foreground, +we make use of two criteria: +1) intuitively, the fore- +ground patches should be more prominent than background +patches [7, 43, 50]. Therefore, the foreground mask should +contain the patch corresponding to the maximum absolute +value in the second smallest eigenvector M t; 2) we in- +corporate a simple but empirically effective object-centric +prior [33]: the foreground set should contain less than two +of the four corners. We reverse the partitioning of the fore- +ground and background, i.e., M t +ij = 1−M t +ij, if the criteria +1 is not satisfied while the current foreground set contains +two corners or the criteria 2 is not satisfied. In practice, we +also set all Wij <τ ncut to 1e−5 and Wij ≥τ ncut to 1. +To get a mask for the (t+1)th object, we update the node +similarity W t+1 +ij +via masking out these nodes corresponding +to the foreground in previous stages: +W t+1 +ij += (Ki +�t +s=1 ˆ +M s +ij)(Kj +�t +s=1 ˆ +M s +ij) +∥Ki∥2∥Kj∥2 +(3) +3 + +NCut +patch-wise +affinity matrix +masked +affinity matrix +… +mask 1 +mask 2 +patchified input +pseudo masks +masked +affinity matrix +n × n +n2 × n2 +n2 × n2 +n2 × n2 +1 +4 +7 +2 +5 +8 +3 +6 +9 +ViT +NCut +Figure 3. MaskCut can discover multiple object masks in an image without supervision. We build upon [7, 50] and create a patch-wise +similarity matrix for the image using a self-supervised DINO [7] model’s features. We apply Normalized Cuts [37] to this matrix and +obtain a single foreground object mask of the image. We then mask out the affinity matrix values using the foreground mask and repeat the +process, which allows MaskCut to discover multiple object masks in a single image. In this pipeline illustration, we set n=3. +where ˆ +M s +ij = 1−M s +ij. Using the updated W t+1 +ij +, we re- +peat Eqs. (1) and (2) to get a mask M t+1. We repeat this +process t times and set t=3 by default. +3.3. DropLoss for Exploring Image Regions +A standard detection loss penalizes predicted regions +ri that do not overlap with the ‘ground-truth’. +Since +the ‘ground-truth’ masks given by MaskCut may miss in- +stances, the standard loss does not enable the detector to +discover new instances not labeled in the ‘ground-truth’. +Therefore, we propose to ignore the loss of predicted re- +gions ri that have a small overlap with the ‘ground-truth’. +More specifically, during training, we drop the loss for each +predicted region ri that has a maximum overlap of τ IoU with +any of the ‘ground-truth’ instances: +Ldrop(ri) = 1(IoUmax +i +> τ IoU)Lvanilla(ri) +(4) +where IoUmax +i +denotes the maximum IoU with all ‘ground- +truth’ for ri and Lvanilla refers to the vanilla loss function of +detectors. Ldrop does not penalize the model for detecting +objects missed in the ‘ground-truth’ and thus encourages +the exploration of different image regions. In practice, we +use a low threshold τ IoU = 0.01. +3.4. Multi-Round Self-Training +Empirically, we find that despite learning from the coarse +masks obtained by MaskCut, detection models ‘clean’ the +ground truth and produce masks (and boxes) that are better +than the initial coarse masks used for training. The detectors +refine mask quality, and our DropLoss strategy encourages +them to discover new object masks. Thus, we leverage this +property and use multiple rounds of self-training to improve +the detector’s performance. +We use the predicted masks and proposals with a confi- +dence score over 0.75−0.5t from the tth-round as the addi- +tional pseudo annotations for the (t + 1)th-round of self- +training. To de-duplicate the predictions and the ground +truth from round t, we filter out ground-truth masks with +an IoU > 0.5 with the predicted masks. We found that +three rounds of self-training are sufficient to obtain good +performance. Each round steadily increases the number of +‘ground-truth’ samples used to train the model. +3.5. Implementation Details +Training data. +We only use the images from the Ima- +geNet [12] dataset (1.3 million images) for all parts of the +CutLER model and do not use any type of annotations either +for training or any supervised pretrained models. +MaskCut. We use MaskCut with three stages on images +resized to 480×480 pixels and compute a patch-wise affin- +ity matrix using the ViT-B/8 [15] DINO [7] model. We use +Conditional Random Field (CRF) [30] to post-process the +masks and compute their bounding boxes. +Detector. +While CutLER is agnostic to the underlying +detector, we use popular Mask R-CNN [27] and Cascade +Mask R-CNN [4] for all experiments, and use Cascade +Mask R-CNN by default, unless otherwise noted. We train +the detector on ImageNet with initial masks and bounding +boxes for 160K iterations with a batch size of 16. When +training the detectors with a ResNet-50 backbone [28], we +initialize the model with the weights of a self-supervised +pretrained DINO [7] model. We explored other pre-trained +models, including MoCo-v2 [9], SwAV [6], and CLD [46], +and found that they gave similar detection performance. We +also leverage the copy-paste augmentation [16, 19] during +the model training process. Rather than using the vanilla +copy-paste augmentation, to improve the model’s ability to +segment small objects, we randomly downsample the mask +with a scalar uniformly sampled between 0.3 and 1.0. We +then optimize the detector for 160K iterations using SGD +with a learning rate of 0.005, which is decreased by 5 after +80K iterations, and a batch size of 16. We apply a weight +decay of 5×10−5 and a momentum of 0.9. +Self-training. +We initialize the detection model in each +stage using the weights from the previous stage. We op- +timize the detector using SGD with a learning rate of 0.01 +for 80K iterations. Since the self-training stage can provide +4 + +Datasets → +Avg. +COCO +COCO20K +VOC +LVIS +UVO +Clipart +Comic +Watercolor +KITTI +Objects365 OpenImages +Metrics → +AP50 AR AP50 AR AP50 AR AP50 AR AP50 AR AP50 AR AP50 AR AP50 AR AP50 AR AP50 AR AP50 AR AP50 +AR +Prev. SOTA [47] +9.0 13.4 +9.6 12.6 +9.7 12.6 15.9 21.3 +3.8 +6.4 10.0 14.2 +7.9 15.1 +9.9 16.3 +6.7 16.2 +7.7 +7.1 +8.1 10.2 +9.9 +14.9 +CutLER +24.3 35.5 21.9 32.7 22.4 33.1 36.9 44.3 +8.4 21.8 31.7 42.8 21.1 41.3 30.4 38.6 37.5 44.6 18.4 27.5 21.6 34.2 17.3 +29.6 +vs. prev. SOTA ++15.3 +22.1 +12.3 +20.1 +12.7 +20.5 +21.0 +23.0 +4.6 +15.4 +21.7 +28.6 +13.2 +26.2 +20.5 +22.3 +30.8 +28.4 +10.7 +20.4 +13.5 +24.0 +7.4 ++14.7 +Table 2. State-of-the-art zero-shot unsupervised object detection performance on 11 different datasets spanning a variety of domains. +We report class-agnostic multi-object detection performance and the averaged results for 11 datasets using APbox +50 and ARbox +100. Our CutLER +is trained in an unsupervised manner solely on ImageNet. While the previous SOTA method [47] is typically fine-tuned on extra data, e.g., +∼241k unlabeled COCO images, CutLER significantly outperforms it. Results of [47] are produced with official code and checkpoint. +a sufficient number of pseudo-masks for model training, we +don’t use the DropLoss during the self-training stages. +We provide more details on model implementation and +training in Appendix A.1. +4. Experiments +We evaluate CutLER on various detection and segmen- +tation benchmarks. In Sec. 4.1, we show that CutLER can +discover objects without any supervision on completely un- +seen images. Despite being evaluated in a zero-shot manner +on eleven benchmarks, CutLER outperforms prior methods +that use in-domain training data. Sec. 4.2 shows that fine- +tuning CutLER further improves detection performance, +outperforming prior work like MoCo-V2 and FreeSOLO. +4.1. Unsupervised Zero-shot Evaluations +We conduct extensive experiments on eleven different +datasets, covering various object categories, image styles, +video frames, resolutions, camera angles, etc. to verify the +effectiveness of CutLER as a universal unsupervised object +detection and segmentation method. We describe the differ- +ent datasets used for zero-shot evaluation in detail in Ap- +pendix A.2. CutLER is trained solely using images from +ImageNet and evaluated in a zero-shot manner on all down- +stream datasets without finetuning on any labels or data. +Evaluating unsupervised object detectors poses two +unique challenges. First, since the model is trained with- +out any notion of semantic classes, it cannot be evaluated +using the class-aware detection setup. +Thus, like prior +work [3,38,48] we use the class-agnostic detection evalua- +tion. Second, object detection datasets often only annotate +a subset of the objects in the images. For example, while +COCO and LVIS use the same images, COCO only labels +80 object classes, and LVIS labels 1203 object classes. In +this partially labeled setup, Average Recall (AR) is a valu- +able metric for unsupervised detection as it does not penal- +ize the models for detecting novel objects unlabeled in the +dataset. Thus, we additionally report AR for all datasets. +Zero-shot detection on 11 benchmarks. We evaluate Cut- +LER on a variety of datasets and report the detection perfor- +mance using APbox +50 and ARbox +100 metrics in Fig. 1 and Table 2. +CutLER uses a smaller model size and less training data +than prior work. Compared to the previous SOTA approach, +ground truth +CutLER (ours) +FreeSOLO +Figure 4. +Compared to the previous state-of-the-art [47], our +CutLER can better discriminate instances (e.g. person and skis in +col. 1), discover more objects (e.g. apple and raisins in col. 2), +and produce higher quality segmentation masks even for small ob- +jects (e.g. kite in col. 3); compared to human annotations, CutLER +can locate novel instances that are overlooked by human annota- +tors, such as the streetlight and clock tower in col. 4. Qualitative +comparisons between previous SOTA methods (row 1) and our +CutLER (row 2) on COCO, as well as ground truth annotations by +human annotators (row 3), are visualized. +FreeSOLO [47] with a backbone of ResNet101, CutLER, +with the smaller ResNet50 backbone, significantly outper- +forms it in each of these benchmarks spanning various im- +age distributions, more than doubling performance on 10 of +them. Also note that, FreeSOLO requires FreeMask pre- +training using approximately 1.3M ImageNet images and +model fine-tuning using additional data in test benchmarks. +We observe that on different domains, e.g. watercolor or +frames from videos (UVO dataset), CutLER improves per- +formance by over 4× and 2×, respectively. Fig. 1 shows +some qualitative examples of CutLER’s predictions. +Detailed comparisons on COCO20K and COCO. Table 3 +presents detailed detection and segmentation evaluations +(also referred to as ‘multi-object’ discovery) on two pop- +ular benchmarks: COCO val2017 [32] and COCO 20K, +which contains a subset of 20K images of COCO [38, 47]. +CutLER consistently surpasses prior works by a large mar- +5 + +Methods +Pretrain +Detector +Init. +COCO 20K +COCO val2017 +APbox +50 APbox +75 APbox APmask +50 +APmask +75 +APmask +APbox +50 APbox +75 APbox APmask +50 +APmask +75 +APmask +non zero-shot methods +LOST [38] +IN+COCO FRCNN +DINO +- +- +- +2.4 +1.0 +1.1 +- +- +- +- +- +- +MaskDistill [42] IN+COCO MRCNN +MoCo +- +- +- +6.8 +2.1 +2.9 +- +- +- +- +- +- +FreeSOLO∗ [47] IN+COCO SOLOv2 DenseCL +9.7 +3.2 +4.1 +9.7 +3.4 +4.3 +9.6 +3.1 +4.2 +9.4 +3.3 +4.3 +zero-shot methods +DETReg [3] +IN +DDETR +SwAV +- +- +- +- +- +- +3.1 +0.6 +1.0 +8.8 +1.9 +3.3 +DINO [7] +IN +- +DINO +1.7 +0.1 +0.3 +- +- +- +- +- +- +- +- +- +TokenCut [50] +IN +- +DINO +- +- +- +- +- +- +5.8 +2.8 +3.0 +4.8 +1.9 +2.4 +CutLER (ours) +IN +MRCNN +DINO +21.8 11.1 +10.1 +18.6 +9.0 +8.0 +21.3 11.1 +10.2 +18.0 +8.9 +7.9 +CutLER (ours) +IN +Cascade +DINO +22.4 12.5 +11.9 +19.6 +10.0 +9.2 +21.9 11.8 +12.3 +18.9 +9.7 +9.2 +vs. prev. SOTA ++12.7 +9.3 ++7.8 ++9.9 ++6.6 ++4.9 ++12.3 +8.7 ++8.1 ++9.5 ++6.4 ++4.9 +Table 3. Unsupervised object detection and instance segmentation on COCO 20K and COCO val2017. We report the detection and +segmentation metrics and note the pretraining data (Pretrain), detectors, and backbone initialization (Init.). Methods in the top half of the +table train on extra unlabeled images from the downstream datasets, while zero-shot methods in the bottom half only train on ImageNet. +Despite using an older detector, CutLER outperforms all prior works on all evaluation metrics. ∗: results obtained with the official code and +checkpoint. IN, Cascade, MRCNN, and FRCNN denote ImageNet, Cascade Mask R-CNN, Mask R-CNN, and Faster R-CNN, respectively. +Methods +AP50 +AP75 +AP +APS +APM +APL +rOSD [43] +13.1 +- +4.3 +- +- +- +LOD [44] +13.9 +- +4.5 +- +- +- +LOST [38] +19.8 +- +6.7 +- +- +- +FreeSOLO∗ [47] +15.9 +3.6 +5.9 +0.0 +2.0 +9.3 +CutLER (ours) +36.9 +19.2 +20.2 +1.3 +6.5 +32.2 +vs. prev. SOTA ++17.1 ++15.6 ++13.5 ++1.3 ++4.5 ++22.9 +Table 4. Zero-shot unsupervised object detection on VOC. ∗: re- +produced results with official code and checkpoint. +gin (often gets 2∼3× higher AP) on both the segmentation +and detection tasks. Although CutLER is not trained on any +images from COCO, it surpasses existing methods trained +on COCO by more than 10% in terms of APmask +50 +and APbox +50 . +Fig. 4 shows the qualitative comparisons between [47] +and our CutLER on COCO val2017, along with human +annotations. Surprisingly, CutLER can often detect novel +instances that human annotators miss. +We present detailed comparisons on COCO 20K, COCO +val2017 and LVIS [24] benchmarks in Appendix A.3. +Detailed comparisons on UVO and VOC. For a com- +prehensive comparison with existing unsupervised multi- +object detection methods, we report the results for UVO +val [45] and VOC trainval07 [17]. Table 4 shows that +CutLER yields significant performance gains over previous +SOTA, obtaining over 3× higher AP, with the most consid- +erable improvement coming from APL. On UVO, Table 5 +shows that CutLER more than quadruples the AP of previ- +ous SOTA and almost triples the APbox +50 . Our APmask +50 +is even +4.8% higher than the fully-supervised SOLOv2 [48] trained +on LVIS with 100% annotations, significantly narrowing the +gap between supervised and unsupervised learning. +Methods +APbox +50 APbox +75 APboxAPmask +50 APmask +75 APmask +fully-supervised methods: +SOLO-v2 (w/ COCO) [48] +- +- +- +38.0 +20.9 +21.4 +Mask R-CNN (w/ COCO) [27] +- +- +- +31.0 +14.2 +15.9 +SOLO-v2 (w/ LVIS) [48] +- +- +- +14.8 +5.9 +7.1 +unsupervised methods: +FreeSOLO∗ [47] +10.0 +1.8 +3.2 +9.5 +2.0 +3.3 +CutLER (ours) +31.7 14.1 16.1 +22.8 +8.0 +10.1 +vs. prev. SOTA ++21.7+12.3+12.9 +13.3 +6.0 ++6.8 +Table 5. Zero-shot unsupervised object detection and instance +segmentation on the UVO val video benchmark. CutLER out- +performs prior unsupervised methods and achieves better perfor- +mance than the supervised SOLO-v2 model trained on the LVIS +dataset. ∗: reproduced results with official code and checkpoint. +4.2. Label-Efficient and Fully-Supervised Learning +We now evaluate CutLER as a pretraining method for +training object detection and instance segmentation mod- +els. While CutLER can discover objects without any su- +pervision, finetuning it on a target dataset aligns the model +output to the same set of objects labeled in the dataset. +Setup. We use CutLER to initialize a standard Cascade +Mask R-CNN [4] detector with a ResNet50 [28]. Prior work +uses more advanced detectors, SOLOv2 [48] used in [47] +and DDETR [54] used in [3], that perform better. However, +we choose Cascade Mask R-CNN for its simplicity and +show in Sec. 5 that CutLER’s performance improves with +stronger detectors. We train the detector on the COCO [32] +dataset using the bounding box and instance mask labels. +To evaluate label efficiency, we subsample the training set to +create subsets with varying proportions of labeled images. +We train the detector, initialized with CutLER, on each of +these subsets. As a baseline, we follow the settings from +MoCo-v2 [9] and train the same detection architecture ini- +tialized with a MoCo-v2 ResNet50 model, given its strong +6 + + Instance Segmentation: AP (%) +9 +11 +13 +15 +17 +19 +21 +23 +25 +27 +29 +31 +33 +35 +37 +39 +1 +2 +5 10 20 30 40 50 60 80100 +Object Detection: AP (%) +11 +13 +15 +17 +19 +21 +23 +25 +27 +29 +31 +33 +35 +37 +39 +41 +43 +45 +1 +2 +5 10 20 30 40 50 60 80100 +% of Labeled Data +% of Labeled Data +CutLER +MoCo-v2 +FreeSOLO +CutLER +MoCo-v2 +DETReg +7.3% +6.6% +Figure 5. Finetuning CutLER for low-shot and fully super- +vised detection and instance segmentation. We fine-tune a Cas- +cade Mask R-CNN model initialized with CutLER or MoCo-v2 on +varying amounts of labeled data on the COCO dataset. We use the +same schedule as the self-supervised pretrained MoCo-v2 coun- +terpart and report the detection and instance segmentation perfor- +mance. CutLER consistently outperforms the MoCo-v2 baseline: +in the low-shot setting with 1% labels and the fully supervised set- +ting using 100% labels. CutLER also outperforms FreeSOLO [47] +and DETReg [3] on this benchmark despite using an older detec- +tion architecture. Results with Mask R-CNN are in the appendix. +performance on object detection tasks. Both MoCo-v2 and +our models are trained for the 1× schedule using Detec- +tron2 [51], except for extremely low-shot settings with 1% +or 2% labels. Following previous works [47], when train- +ing with 1% or 2% labels, we train both MoCo-v2 and our +model for 3,600 iterations with a batch size of 16. +Results. Fig. 5 shows the results of fine-tuning the detec- +tor on different subsets of COCO. When tested with low- +shot settings, e.g., 2% and 5% labeled data, our approach +achieves 5.4% and 7.3% higher APbox than the MoCo-v2 +baseline, respectively. Even when training with full anno- +tations, CutLER still consistently gives more than 2% im- +provements, outperforming MoCo-v2 for both object detec- +tion and segmentation. More impressively, CutLER out- +performs prior SOTA methods - FreeSOLO [47] and DE- +TReg [3] despite using an older detection architecture. +5. Ablations +We analyze the design decisions in CutLER. We use sim- +ilar settings to Sec. 4 and train CutLER only on ImageNet. +We use the Cascade Mask R-CNN detection architecture +and evaluate our model primarily on the COCO and UVO +unsupervised detection benchmarks. All ablation studies +are conducted without self-training unless otherwise noted. +Importance of each component. We analyze the main +components of CutLER and report their relative contribu- +tion in Table 6. We report results on the popular COCO [32] +dataset and a densely annotated video instance segmenta- +Methods +UVO +COCO +APmask +50 +APmask +APmask +50 +APmask +TokenCut [50] +- +- +4.9 +2.0 +Base +14.6 +5.4 +13.5 +5.7 ++ MaskCut +19.3 +8.1 +15.8 +7.7 ++ DropLoss +20.9 +9.0 +16.6 +8.2 ++ copy-paste [16,19] +21.5 +9.9 +17.7 +8.8 ++ self-train (CutLER) +22.8 +10.1 +18.9 +9.7 +Table 6. Ablation study on the contribution of each component. +Results reported on COCO and video segmentation dataset UVO. +Methods +APbox +50 APbox ARbox +100 APmask +50 APmask ARmask +100 +TokenCut (1 eigenvec.) +5.2 +2.6 +5.0 +4.9 +2.0 +4.4 +TokenCut (3 eigenvec.) +4.7 +1.7 +8.1 +3.6 +1.2 +6.9 +MaskCut (t = 3) +6.0 +2.9 +8.1 +4.9 +2.2 +6.9 +CutLER +21.9 12.3 +32.7 +18.9 +9.7 +27.1 +Table 7. CutLER achieves much higher results even when com- +pared to a modified TokenCut that can produce more than one +mask per image. Compared to TokenCut, MaskCut gets a higher +recall without reducing precision. We report results on COCO. +tion dataset UVO [45]. We also report the performance of +running TokenCut [50] on the COCO dataset. Next, we use +TokenCut’s official codes to generate masks on ImageNet +and use them for training a Cascade Mask R-CNN [4]. This +base model provides substantial gains over just using To- +kenCut on COCO. We add each of our proposed compo- +nents to this strong base model. Using MaskCut increases +APmask +50 +and APmask by 4.7% and 2.7%, respectively. Also, +the improvements to APmask +50 +is larger for densely annotated +dataset UVO, i.e. 4.7% vs. 2.7%. These results prove that +MaskCut’s ability to segment multiple instances per image +is vital for densely annotated datasets. Adding DropLoss +brings another 1.6% and 0.9% improvements to APmask +50 +for +UVO and COCO, respectively. Multi-round of self-training +increases the quantity and quality of pseudo-masks, leading +to 1.3% improvements. These results show that each simple +proposed component is critical for strong performance. +Comparison with TokenCut. TokenCut [50] is also a +zero-shot segmentation method. However, it only segments +a single instance per image, as discussed in Sec. 3.1. In +order to generate more than one segmentation mask per +image, we use a modified TokenCut by using more of the +smaller eigenvectors and combining all produced masks. +Table 7 shows the object detection performance on COCO’s +validation set for vanilla TokenCut, our modified Token- +Cut and CutLER. Although using more eigenvectors in- +creases the recall ARbox +100, it significantly reduces the pre- +cision APbox. CutLER not only improves the average recall +ARbox +100 by 4× but also surpasses TokenCut’s average preci- +sion APbox by 4.8×, i.e. 480% relative improvements. +7 + +Size → 240 360 480 640 +APmask +50 +15.1 16.6 17.7 17.9 +(a) Image size. +τ ncut → +0 +0.1 0.15 0.2 +0.3 +APmask +50 +17.1 17.5 17.7 17.6 17.5 +(b) τ ncut for MaskCut. +N → +2 +3 +4 +APmask +50 +16.9 17.7 17.7 +(c) # masks per image. +τ IoU → +0 +0.01 0.1 +0.2 +APmask +50 +17.4 17.7 14.4 12.7 +(d) τ IoU for DropLoss. +Table 8. Ablations for MaskCut and DropLoss used for training CutLER. We report CutLER’s detection and instance segmentation +performance on COCO val2017, without adding the self-training stage. (a) We vary the size of the image used for MaskCut. (b) We +vary the threshold τ ncut in MaskCut, which controls the sparsity of the affinity matrix used for Normalized Cuts. (c) We vary the number of +masks extracted using MaskCut and train different CutLER models. (d) We vary τ IoU in DropLoss, i.e., the maximum overlap between the +predicted regions and the ground truth beyond which the loss for the predicted regions is ignored. Default settings are highlighted in gray. +UVO +COCO +APmask +50 +APmask +APmask +75 +APmask +50 +APmask +APmask +75 +1 round +20.6 +9.0 +7.0 +17.7 +8.8 +8.0 +2 rounds +22.2 +9.6 +7.5 +18.5 +9.5 +8.8 +3 rounds +22.8 +10.1 +8.0 +18.9 +9.7 +9.2 +4 rounds +22.8 +10.2 +8.2 +18.9 +9.8 +9.3 +Table 9. Number of self-training rounds used in CutLER. We +find that 3 rounds of self-training are sufficient. Self-training pro- +vides larger gains for the densely labeled UVO dataset. +round 1 +round 2 +round 3 +2.58 M +2.77 M +3.03 M +# of masks +Figure 6. Multiple rounds of self-training can improve the pseudo- +masks in terms of quality and quantity. We show qualitative visu- +alizations and the number of pseudo-masks for all three rounds. +Design choices in MaskCut and DropLoss and their im- +pact on the final localization performance is presented in Ta- +ble 8. We first study the effect of the image size used by +MaskCut for generating the initial masks. As expected, Ta- +ble 8a shows that MaskCut benefits from using higher reso- +lution images presumably as it provides a higher resolution +similarity between pixels. We pick a resolution of 480px +for a better trade-off between the speed of MaskCut and +its performance. In Table 8b, we study the effect of the +threshold used in MaskCut for producing a binary W ma- +trix (Sec. 3.2). Overall, CutLER seems to be robust to the +threshold values. We understand the impact of the num- +ber of masks per image generated by MaskCut in Table 8c. +Increasing the number improves the performance of the re- +sulting CutLER models. This shows that MaskCut gener- +ates high-quality masks that directly impact the overall per- +formance. Finally, in Table 8d, we vary the IOU threshold +used for DropLoss. With a high threshold, we ignore the +loss for a higher number of predicted regions while encour- +aging the model to explore. 0.01 works best for the trade-off +between exploration and detection performance. +Mask R-CNN Cascade Mask R-CNN +ViTDet +APbox +50 / APbox +20.3 / 10.6 +20.8 / 11.5 +21.5 / 11.8 +APmask +50 / APmask +17.2 / 8.5 +17.7 / 8.8 +18.0 / 9.0 +Table 10. CutLER with different detection architectures. We +report results on COCO and observe that CutLER is agnostic to the +detection architecture and improves performance using stronger +detection architectures such as ViTDet with a backbone of ViT-B. +Pre-train +CutLER +APbox +50 APbox +75 APbox APmask +50 +APmask +75 +APmask +IN1K +IN1K +20.8 +10.8 +11.5 +17.7 +8.0 +8.8 +YFCC1M YFCC1M 19.4 +10.4 +10.9 +16.3 +7.4 +8.1 +IN1K +YFCC1M 14.9 +7.6 +8.2 +12.1 +5.4 +5.9 +YFCC1M IN1K +14.8 +7.2 +8.0 +11.8 +5.2 +5.8 +Table 11. Impact of datasets used to pre-train DINO and train +CutLER. CutLER’s detection performance is similar when pre- +training both DINO and CutLER with the same dataset: the object- +centric ImageNet dataset or the non-object-centric YFCC dataset. +Self-training and its impact on the final performance is an- +alyzed in Table 9. Self-training consistently improves per- +formance across the UVO and COCO benchmarks and all +metrics. UVO, which has dense object annotations, benefits +more from the multi-round of self-training. By default, Cut- +LER uses 3 rounds of self-training. Fig. 6 shows qualitative +examples of how self-training improves both the quality of +predictions and the number of objects predicted. +Generalization to different detection architectures. We +use different detector architectures for training CutLER and +measure their performance in Table 10. We observe that +CutLER works with various architectures, and its perfor- +mance is improved with stronger architectures. +Impact of the pretraining dataset. We now study the im- +pact of the dataset used for 1) pretraining the self-supervised +DINO model and 2) training the CutLER model. The com- +monly used ImageNet dataset has a well-known object- +centric bias [12] which may affect the unsupervised detec- +tion performance. Thus, we also use YFCC [40], a non- +object-centric dataset. We control for the number of images +in both ImageNet and YFCC for a fair comparison and use +them for training DINO and CutLER. As Table 11 shows, +CutLER’s performance on COCO is robust to the choice of +object-centric or non-object-centric datasets as long as the +8 + +4-2007same dataset is used to train DINO and CutLER. This shows +the generalization of CutLER to different data distributions. +However, training DINO and CutLER with different data +leads to worse performance, suggesting the importance of +using the same image distribution for learning both DINO +and CutLER models. +6. Summary +Object localization is a fundamental task in computer vi- +sion. In this paper, we have shown that a simple yet effec- +tive cut-and-learn approach can achieve extraordinary per- +formance on challenging object detection and instance seg- +mentation tasks without needing to train with human an- +notations. 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When training +the detectors with a ResNet-50 backbone [28], we initialize +the model with the weights of a self-supervised pretrained +DINO [7] model. We explored other pre-trained models, in- +cluding MoCo-v2 [9], SwAV [6], and CLD [46], and found +that they give similar detection performance. Therefore, we +initialize model weights with DINO by default. +We also leverage the copy-paste augmentation [16, 19] +during the model training process. Rather than using the +vanilla copy-paste augmentation to improve the model’s +ability to segment small objects, we randomly downsam- +ple the mask with a scalar uniformly sampled between 0.3 +and 1.0. We then optimize the detector for 160K iterations +using SGD with a learning rate of 0.005, which is decreased +by 5 after 80K iterations and a batch size of 16. We apply +a weight decay of 5×10−5 and a momentum of 0.9. +For the multi-round of self-training, in each stage, we +initialize the detection model using the weights from the +previous stage. We optimize the detector using SGD with +a learning rate of 0.01 for 80K iterations. Since the self- +training stage can provide a sufficient number of pseudo- +masks for model training, we don’t use the exploration loss +during the self-training stage. +A.2. Datasets used for zero-shot evaluation +COCO and COCO20K [32] is a large-scale object detec- +tion and instance segmentation dataset, containing about +115K and 5K images in the training and validation split, re- +spectively. Additionally, COCO has an unannotated split of +123K images. We test our model in a class-agnostic manner +on COCO val2017 and COCO 20K, without fine-tuning +on any images in COCO. COCO 20K is a subset of the +COCO trainval2014 [32], containing 19817 randomly +sampled images, used as a benchmark in [38, 43, 50]. We +report class-agnostic COCO style averaged precision and +averaged recall for object detection and segmentation tasks. +Pascal VOC [17] is another popular benchmark for object +dtetection. We evaluate our model on its trainval07 +split in COCO style evaluation matrics. +UVO [45]. Unidentified Video Objects (UVO) is an exhaus- +tively annotated dataset for video object detection and in- +stance segmentation. We evaluate our model on UVO val +by frame-by-frame inference and report results in COCO +style evaluation matrics. +LVIS [24] collected 2.2 million high-quality instance seg- +mentation masks for over 1000 entry-level object cate- +gories, which naturally constitutes the long-tailed data dis- +tribution. We report class-agnostic object detection and in- +stance segmentation results on LVIS val split, containing +about 5K images. +CrossDomain [29] contains three subsets of watercolor, cli- +part, and comics, in which objects are depicted in water- +color, sketch and painting styles, respectively. We evaluate +our model on all annotated images from these three datasets, +i.e., traintest. +Objects365 V2 [36] presents a supervised object detection +benchmark with a focus on diverse objects in the wild. We +evaluate CutLER on the 80K images from its val split. +OpenImages V6 [31] unifies image classification, object de- +tection, and instance segmentation, visual relationship de- +tection, etc. in one dataset. We evaluate CutLER on its 42K +images from the val split. +KITTI [18] presents a dataset captured from cameras +mounted on mobile vehicles used for autonomous driv- +ing research. We evaluate CutLER on 7521 images from +KITTI’s trainval split. +We provide the summary of these datasets used for zero- +shot evaluation in Table 12. +A.3. Additional results for zero-shot detection & +segmentation +In this section, we use official COCO API and pro- +vide more results with standard COCO metrics, including +AP across various IoU thresholds - AP (averaged over IoU +thresholds from 0.5 to 0.95 with a step size of 0.05), AP50 +(IoU@0.5) and AP75 (IoU@0.75), and AP across scales - +APS (small objects), APM (medium objects) and APL (large +objects). We provide detailed results on all these bench- +marks listed in Table 12 and report these results in Ta- +ble 13. We report the performance of object detection for all +datasets. In addition, for those datasets that provide annota- +tions for instance segmentation, we also present the perfor- +mance of the instance segmentation task. It is worth noting +that on these datasets without segmentation labels, CutLER +can still predict instance segmentation masks, but since we +do not have ground truth masks to be compared, we cannot +evaluate the results. +A.4. CutLER vs. Selective Search +Selective Search [41] is a popular unsupervised object +discovery method, used in many early state-of-the-art de- +tectors such as R-CNN [22] and Fast R-CNN [21]. How- +ever, generating possible object locations with sliding win- +dows greatly reduces inference speed (please refer to [41] +for more details on selective search). +We compare Cut- +LER’s performance to selective search in Fig. 7 and ob- +serve that CutLER provides a significant improvement in +both precision and recall, which indicates that CutLER is a +12 + +datasets +domain +testing data +#images +instance segmentation label +COCO [32] +natural images +val2017 split +5,000 + +COCO20K [32] +natural images +a subset of COCO +20,000 + +UVO [45] +video frames +val split +7,356 + +LVIS [24] +natural images +val split +19,809 + +KITTI [18] +traffic images +trainval split +7,521 + +Pascal VOC [17] +natural images +trainval07 split +9,963 + +Clipart [29] +clip arts +traintest split +1,000 + +Watercolor [29] +paintings +traintest split +2,000 + +Comic [29] +sketches +traintest split +2,000 + +Objects365-V2 [36] +natural images +val split +80,000 + +OpenImages-V6 [31] +natural images +val split +41,620 + +Table 12. Summary of datasets used for zero-shot evaluation. +Datasets +APbox +50 APbox +75 APbox APbox +S APbox +M APbox +L ARbox +1 +ARbox +10 ARbox +100 APmask +50 +APmask +75 +APmask APmask +S +APmask +M +APmask +L +ARmask +1 +ARmask +10 +ARmask +100 +COCO +21.9 +11.8 +12.3 +3.7 +12.7 +29.6 +6.8 +19.6 +32.8 +18.9 +9.2 +9.7 +2.4 +8.8 +24.3 +5.8 +16.5 +27.1 +COCO20K +22.4 +11.9 +12.5 +4.1 +12.7 +29.5 +6.8 +19.7 +33.1 +19.6 +9.2 +10.0 +2.8 +8.9 +24.3 +5.8 +16.6 +27.4 +UVO +31.7 +14.1 +16.1 +3.7 +11.3 +25.3 +6.8 +24.5 +42.5 +31.6 +14.1 +16.1 +3.7 +11.3 +25.3 +4.6 +18.0 +32.2 +LVIS +8.4 +3.9 +4.5 +2.7 +9.1 +15.1 +2.4 +9.2 +21.8 +6.7 +3.2 +3.5 +1.9 +6.1 +12.5 +2.1 +7.9 +18.7 +KITTI +18.4 +6.7 +8.5 +0.5 +5.6 +19.2 +6.2 +16.6 +27.8 +- +- +- +- +- +- +- +- +- +Pascal VOC 36.9 +19.2 +20.2 +1.3 +6.5 +32.2 +16.5 +32.8 +44.0 +- +- +- +- +- +- +- +- +- +Clipart +21.1 +6.0 +8.7 +1.1 +5.8 +11.6 +6.6 +27.0 +40.7 +- +- +- +- +- +- +- +- +- +Watercolor +37.5 +10.9 +15.7 +0.1 +1.1 +20.0 +19.4 +37.8 +44.2 +- +- +- +- +- +- +- +- +- +Comic +30.4 +7.7 +12.2 +0.0 +1.3 +16.0 +8.5 +28.2 +38.4 +- +- +- +- +- +- +- +- +- +Objects365 +21.6 +10.3 +11.4 +3.0 +10.4 +20.4 +3.0 +15.4 +34.2 +- +- +- +- +- +- +- +- +- +OpenImages 17.3 +9.5 +9.7 +0.4 +2.3 +14.9 +6.5 +17.6 +29.6 +- +- +- +- +- +- +- +- +- +Table 13. Detailed zero-shot evaluation results on all benchmarks used in this work. +Precision +0 +0.2 +0.4 +0.6 +0.8 +1 +Recall +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Selective Search +Ours +Figure 7. Precision-recall curve for comparing selective search +and CutLER on VOC07 trainval. +better performing unsupervised method for region proposal +generation with real-time inference speed. +A.5. Training details for label-efficient and fully- +supervised learning +We train the detector on the COCO [32] dataset using the +bounding box, and instance mask labels. To evaluate label +efficiency, we subsample the training set to create subsets +with varying proportions of labeled image We train the de- +tector, initialized with CutLER, on each of these subsets. +As a baseline, we follow the settings from MoCo-v2 [9] +and train the same detection architecture initialized with a +MoCo-v2 ResNet50 model, given its strong performance on +object detection tasks. MoCo-v2 and our models use the +same training pipeline and hyper-parameters and are trained +for the 1× schedule using Detectron2 [51], except for ex- +tremely low-shot settings with 1% or 2% labels. Following +previous works [47], when training with 1% or 2% labels, +we train both MoCo-v2 and our model for 3,600 iterations +with a batch size of 16. +Our detector weights are initialized with ImageNet-1K +pre-trained CutLER, except for the weights of the final +bounding box prediction layer and the last layer of the mask +prediction head, which are randomly initialized with val- +ues taken from a normal distribution. For experiments on +COCO with labeling ratios below 50%, during model train- +ing, we use a batch size of 16, and learning rates of 0.04 and +0.08 for model weights loaded from the pre-trained CutLER +and randomly initialized, respectively. For experiments on +COCO with labeling ratios between 50% and 100%, the +learning rates of all layers decay by a factor of 2. +For a fair comparison, baselines and CutLER use the +13 + +Mask R-CNN +Cascade Mask R-CNN + Instance Segmentation: AP (%) +9 +11 +13 +15 +17 +19 +21 +23 +25 +27 +29 +31 +33 +35 +37 +39 +1 +2 +5 10 20 30 40 50 60 80100 +Object Detection: AP (%) +10 +12 +14 +16 +18 +20 +22 +24 +26 +28 +30 +32 +34 +36 +38 +40 +42 +1 +2 +5 10 20 30 40 50 60 80100 +% of Labeled Data +% of Labeled Data +CutLER +MoCo-v2 +CutLER +MoCo-v2 +8.8% +8.0% + Instance Segmentation: AP (%) +9 +11 +13 +15 +17 +19 +21 +23 +25 +27 +29 +31 +33 +35 +37 +39 +1 +2 +5 10 20 30 40 50 60 80100 +Object Detection: AP (%) +11 +13 +15 +17 +19 +21 +23 +25 +27 +29 +31 +33 +35 +37 +39 +41 +43 +45 +1 +2 +5 10 20 30 40 50 60 80100 +% of Labeled Data +% of Labeled Data +CutLER +MoCo-v2 +FreeSOLO +CutLER +MoCo-v2 +DETReg +7.3% +6.6% +Figure 8. Fine-tuning on MS-COCO with various annotation ratios. We report results using Mask R-CNN and Cascade Mask R-CNN +with a backbone of ResNet-50 as the detector. +same hyper-parameters and settings. +A.6. More visualizations +We provide more qualitative visualizations of CutLER’s +zero-shot predictions in Fig. 9. +14 + +Figure 9. More visualizations of CutLER’s predictions. +15 + +1tlegi,eCratn,DvsiVioge,war +color, fetttip,2009 +2,Frfiend'sportrait,Penei,2009CH54 +CAeskHhrESHCOm2 \ No newline at end of file diff --git a/eNFIT4oBgHgl3EQfoyuB/content/tmp_files/load_file.txt b/eNFIT4oBgHgl3EQfoyuB/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f902d0df9e9c58973e76d1a598ab9ec875bc4f4c --- /dev/null +++ b/eNFIT4oBgHgl3EQfoyuB/content/tmp_files/load_file.txt @@ -0,0 +1,1230 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf,len=1229 +page_content='Cut and Learn for Unsupervised Object Detection and Instance Segmentation Xudong Wang1,2 Rohit Girdhar1 Stella X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Yu2,3 Ishan Misra1 1FAIR, Meta AI 2UC Berkeley / ICSI 3University of Michigan Code: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='com/facebookresearch/CutLER natural images paintings sketches clip arts videos traffic images Domains Sample Results 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='7 Prev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' SOTA CutLER Watercolor 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='7 10 UVO 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='4 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='9 Comic 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='1 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='9 Clipart 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='4 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='7 KITTI 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='6 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='1 Objects365 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='9 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='9 VOC 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='4 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='7 COCO 20K 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='9 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='6 COCO 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='8 LVIS 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='3 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='9 OpenImages Datasets AP50 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Zero-shot unsupervised object detection and instance segmentation using our CutLER model, which is trained without human supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' We evaluate the model using the standard detection APbox 50 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' CutLER gives a strong performance on a variety of benchmarks spanning diverse image domains - video frames, paintings, clip arts, complex scenes, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Compared to the previous state- of-the-art method, FreeSOLO [47] with a backbone of ResNet101, CutLER with a backbone of ResNet50 provides strong gains on all benchmarks, increasing performance by more than 2× on 10 of the 11 benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' We evaluate [47] with its official code and checkpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Abstract We propose Cut-and-LEaRn (CutLER), a simple ap- proach for training unsupervised object detection and seg- mentation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' We leverage the property of self- supervised models to ‘discover’ objects without supervision and amplify it to train a state-of-the-art localization model without any human labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' CutLER first uses our proposed MaskCut approach to generate coarse masks for multiple objects in an image, and then learns a detector on these masks using our robust loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' We further improve performance by self-training the model on its predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Compared to prior work, CutLER is simpler, compatible with different detection architectures, and detects multiple objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' CutLER is also a zero-shot unsupervised detec- tor and improves detection performance AP50 by over 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='7× on 11 benchmarks across domains like video frames, paint- ings, sketches, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' With finetuning, CutLER serves as a low- shot detector surpassing MoCo-v2 by 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='3% APbox and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='6% APmask on COCO when training with 5% labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Introduction Object localization is a critical task in computer vision that enables AI systems to perceive, reason, plan and act in an object-centric manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Training models for localization require special annotations like object boxes, masks, local- ized points, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' which are both difficult and resource inten- sive to collect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Without accounting for overhead, annotating ∼164K images in the COCO dataset [32] with masks for just 80 classes took more than 28K human hours of annota- tion time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' In this work, we study unsupervised object detec- tion and instance segmentation models that can be trained without any human labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Our key insight is that simple probing and training mechanisms can amplify the innate lo- calization ability of self-supervised models [7], leading to state-of-the-art unsupervised zero-shot detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Our method Cut-and-LEaRn (CutLER) consists of three simple, architecture- and data-agnostic mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Con- sistent with prior self-supervised learning methods [7–9, 26], CutLER is trained exclusively on unlabeled ImageNet data without needing additional training data, but contrary to these methods, CutLER can be directly employed to per- form complex segmentation and detection tasks over a wide range of domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' First, we propose MaskCut that can au- tomatically produce multiple initial coarse masks for each image, using the pretrained self-supervised features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Sec- ond, we propose a simple loss dropping strategy to train detectors using the coarse masks while being robust to ob- jects missed by MaskCut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Finally, we observe that despite 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='11320v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='CV] 26 Jan 2023 Naranjd-OrangeLeaveitJUSTINEMARKOWSKlearning from these coarse masks, the detectors ‘clean’ the ground truth and produce masks (and boxes) that are bet- ter than the coarse masks used to train them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Therefore, we further show that multiple rounds of self-training on the models’ own predictions allow it to evolve from capturing the similarity of local pixels to capturing the global geome- try of the object, thus producing finer segmentation masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Prior work shows that a self-supervised vision trans- former (ViT) [15] can automatically learn patch-wise fea- tures that detect a single salient object in an image [7,38,43, 44,50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' However, unlike CutLER, such salient object detec- tion methods only locate a single, usually the most promi- nent, object and cannot be used for real world images con- taining multiple objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' While some recent methods, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=', FreeSOLO [47] and DETReg [3], also aim at unsupervised multi-object detection (or multi-object discovery), they rely on a particular detection architecture, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=', SOLO-v2 [48] or DDETR [5,54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Additionally, apart from self-supervised features trained on ImageNet [12], the current state-of-the- art methods FreeSOLO and MaskDistill [42] also require ‘in-domain’ unlabeled data for model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' In contrast, CutLER works with various detection archi- tectures and can be trained solely on ImageNet, without requiring in-domain unlabeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Thus, during model training, CutLER does not see any images from any target dataset and yields a zero-shot model capable of detecting and segmenting multiple objects in diverse domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Features of CutLER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' 1) Simplicity: CutLER is simple to train and agnostic to the choice of detection and backbone architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Thus, it can be integrated effortlessly into existing object detection and instance segmentation works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' 2) Zero-shot detector: CutLER trained solely on ImageNet shows strong zero-shot performance on 11 different bench- marks where it outperforms prior work trained with addi- tional in-domain data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' We double the APbox 50 performance on 10 of these benchmarks, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' 1, and even outperform supervised detectors on the UVO video instance segmentation benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' 3) Robustness: CutLER exhibits strong robustness against domain shifts when tested on im- ages from different domains such as video frames, sketches, paintings, clip arts, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' 4) Pretraining for supervised de- tection: CutLER can also serve as a pretrained model for training fully supervised object detection and instance seg- mentation models and improves performance on COCO, in- cluding on few-shot object detection benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Related Work Self-supervised feature learning involves inferring the patterns within the large-scale unlabeled data without us- ing human-annotated labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Contrastive learning based [8, 26, 34, 52] methods learn such representations that sim- ilar samples or various augmentations of the same instance are close to each other, while dissimilar instances are far DINO LOST TokenCut FreeSOLO Ours detect multiple objects \x17 \x13 \x17 \x13 \x13 zero-shot detector \x13 \x17 \x13 \x17 \x13 compatible with various detection architectures \x13 \x17 \x13 pretrained model for supervised detection \x13 \x17 \x17 \x13 \x13 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' We compare previous methods on unsupervised object detection, including DINO [7], LOST [38], TokenCut [50] and FreeSOLO [47], with our CutLER in term of key properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Our CutLER is the only method with all these desired properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' apart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Similarity-based self-supervised learning methods [10, 23] learn representations via minimizing the distance between different augmentations of the same instance and use only positive sample pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Clustering-based feature learning [1, 6, 46, 53, 55] automatically discovers the nat- ural grouping of data in the latent representation space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Re- cently, [2,25] have shown that masked autoencoders, which learn representations via masking out a large random subset of image patches and reconstructing the missing pixels or patches [2, 13, 14, 25], are scalable self-supervised learners for computer vision [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' In contrast to these unsupervised representation learn- ing efforts, our work aims to automatically discover natural pixel groupings and locate instances within each image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Unsupervised object detection and instance segmen- tation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' The main comparisons to previous works are listed in Table 1 and are elaborated as follows: DINO [7] observes that the underlying semantic segmen- tation of images can emerge from the self-supervised Vision Transformer (ViT) [15], which does not appear explicitly in either supervised ViT or ConvNets [7, 56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Based on this observation, LOST [38] and TokenCut [50] leverage self- supervised ViT features and propose to segment one single salient object [11,38,50] from each image based on a graph that is constructed with DINO’s patch features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' These previous works either can not detect more than one object from each image, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=', DINO and TokenCut, or can not improve the quality of features for better transfer to downstream detection and segmentation tasks, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=', To- kenCut and LOST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Unlike these works, CutLER can locate multiple objects and serve as a pretrained model for label- efficient and fully-supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' FreeSOLO [47] achieves unsupervised instance segmen- tation by extracting coarse object masks in an unsuper- vised manner, followed by mask refinement through a self- training procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' While FreeSOLO’s FreeMask stage can generate multiple coarse masks per image, the quality of these masks is often rather low [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' MaskDistill [42] dis- tills class-agnostic initial masks from the affinity graph pro- duced by a self-supervised DINO [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' However, it utilizes one single mask per image in the distillation stage, which 2 Lcls + Lbox + Lmask Lexp unlabeled data MaskCut ViT Detector self-training Ldrop Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Overview of CutLER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' We propose a simple yet effec- tive method to train an object detection and instance segmentation model without using any supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' We first propose MaskCut to extract initial coarse masks from the features of a self-supervised ViT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' We then learn a detector using our loss dropping strategy that is robust to objects missed by MaskCut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' We further improve the model using multiple rounds of self-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' greatly limits the model’s ability to detect multiple objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' By contrast, the initial masks generated by our Mask- Cut are usually better in quality and quantity than the ini- tial masks used by [42, 47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Therefore, CutLER achieves 2×∼4× higher APbox and APmask than FreeSOLO [47] and MaskDistill [42] on almost all experimented detection and segmentation benchmarks, even when FreeSOLO and MaskDistill are trained and tested on the same domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Method We tackle the problem of unsupervised object detection and segmentation with a simple cut-and-learn pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Our method builds upon insights from recent work [7,50], show- ing that self-supervised representations can discover ob- jects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' While these methods often find a single object per image, we propose a simple approach that can discover multiple objects and significantly improves segmentation and detection performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' The overview of our cut-and- learn pipeline is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' First, we propose MaskCut that generates multiple binary masks per image using self-supervised features from DINO [7] (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Second, we show a dynamic loss dropping strategy, called DropLoss, that can learn a detector from MaskCut’s ini- tial masks while encouraging the model to explore objects missed by MaskCut (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Third, we further improve the performance of our method through multiple rounds of self-training (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Preliminaries Normalized Cuts (NCut) treats the image segmentation problem as a graph partitioning task [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' We construct a fully connected undirected graph via representing each im- age as a node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Each pair of nodes is connected by edges with weights Wij that measure the similarity of the con- nected nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' NCut minimizes the cost of partitioning the graph into two sub-graphs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=', a bipartition, by solving a generalized eigenvalue system (D − W)x = λDx (1) for finding the eigenvector x that corresponds to the second smallest eigenvalue λ, where D is a N×N diagonal matrix with d(i) = � j Wij and W is a N×N symmetrical matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' DINO and TokenCut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' DINO [7] finds that the self- supervised ViT can automatically learn a certain degree of perceptual grouping of image patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' TokenCut [50] leverages the DINO features for NCut and obtaining fore- ground/background segments in an image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' The authors use the similarity of the patches in the DINO feature space as the similarity weight Wij in NCut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Specifically, follow- ing multiple recent methods [38, 42, 50], we use the cosine similarity of ‘key’ features from the last attention layer of DINO-pretrained model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=', Wij = KiKj ∥Ki∥2∥Kj∥2 where Ki is the ‘key’ feature of patch i, and solve Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' (1) for finding the second smallest eigenvector x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' A limitation of TokenCut is that it only computes a sin- gle binary mask for an image and thus only finds one object per image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Although we can use the other N −2 smallest eigenvectors to locate more than one instance, this signifi- cantly degrades the performance for multi-object discovery, as demonstrated in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' MaskCut for Discovering Multiple Objects As we discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='1, vanilla NCut is limited to discovering a single object in an image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' We propose Mask- Cut that extends NCut to discover multiple objects per im- age by iteratively applying NCut to a masked similarity ma- trix (illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' After getting the bipartition xt from NCut at stage t, we get two disjoint groups of patches and construct a binary mask M t, where M t ij = � 1, if M t ij ≥ mean(xt) 0, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' (2) To determine which group corresponds to the foreground, we make use of two criteria: 1) intuitively, the fore- ground patches should be more prominent than background patches [7, 43, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Therefore, the foreground mask should contain the patch corresponding to the maximum absolute value in the second smallest eigenvector M t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' 2) we in- corporate a simple but empirically effective object-centric prior [33]: the foreground set should contain less than two of the four corners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' We reverse the partitioning of the fore- ground and background, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=', M t ij = 1−M t ij, if the criteria 1 is not satisfied while the current foreground set contains two corners or the criteria 2 is not satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' In practice, we also set all Wij <τ ncut to 1e−5 and Wij ≥τ ncut to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' To get a mask for the (t+1)th object, we update the node similarity W t+1 ij via masking out these nodes corresponding to the foreground in previous stages: W t+1 ij = (Ki �t s=1 ˆ M s ij)(Kj �t s=1 ˆ M s ij) ∥Ki∥2∥Kj∥2 (3) 3 NCut patch-wise affinity matrix masked affinity matrix … mask 1 mask 2 patchified input pseudo masks masked affinity matrix n × n n2 × n2 n2 × n2 n2 × n2 1 4 7 2 5 8 3 6 9 ViT NCut Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' MaskCut can discover multiple object masks in an image without supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' We build upon [7, 50] and create a patch-wise similarity matrix for the image using a self-supervised DINO [7] model’s features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' We apply Normalized Cuts [37] to this matrix and obtain a single foreground object mask of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' We then mask out the affinity matrix values using the foreground mask and repeat the process, which allows MaskCut to discover multiple object masks in a single image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' In this pipeline illustration, we set n=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' where ˆ M s ij = 1−M s ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Using the updated W t+1 ij , we re- peat Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' (1) and (2) to get a mask M t+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' We repeat this process t times and set t=3 by default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' DropLoss for Exploring Image Regions A standard detection loss penalizes predicted regions ri that do not overlap with the ‘ground-truth’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Since the ‘ground-truth’ masks given by MaskCut may miss in- stances, the standard loss does not enable the detector to discover new instances not labeled in the ‘ground-truth’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Therefore, we propose to ignore the loss of predicted re- gions ri that have a small overlap with the ‘ground-truth’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' More specifically, during training, we drop the loss for each predicted region ri that has a maximum overlap of τ IoU with any of the ‘ground-truth’ instances: Ldrop(ri) = 1(IoUmax i > τ IoU)Lvanilla(ri) (4) where IoUmax i denotes the maximum IoU with all ‘ground- truth’ for ri and Lvanilla refers to the vanilla loss function of detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Ldrop does not penalize the model for detecting objects missed in the ‘ground-truth’ and thus encourages the exploration of different image regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' In practice, we use a low threshold τ IoU = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Multi-Round Self-Training Empirically, we find that despite learning from the coarse masks obtained by MaskCut, detection models ‘clean’ the ground truth and produce masks (and boxes) that are better than the initial coarse masks used for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' The detectors refine mask quality, and our DropLoss strategy encourages them to discover new object masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Thus, we leverage this property and use multiple rounds of self-training to improve the detector’s performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' We use the predicted masks and proposals with a confi- dence score over 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='75−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='5t from the tth-round as the addi- tional pseudo annotations for the (t + 1)th-round of self- training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' To de-duplicate the predictions and the ground truth from round t, we filter out ground-truth masks with an IoU > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='5 with the predicted masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' We found that three rounds of self-training are sufficient to obtain good performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Each round steadily increases the number of ‘ground-truth’ samples used to train the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Implementation Details Training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' We only use the images from the Ima- geNet [12] dataset (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='3 million images) for all parts of the CutLER model and do not use any type of annotations either for training or any supervised pretrained models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' MaskCut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' We use MaskCut with three stages on images resized to 480×480 pixels and compute a patch-wise affin- ity matrix using the ViT-B/8 [15] DINO [7] model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' We use Conditional Random Field (CRF) [30] to post-process the masks and compute their bounding boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' While CutLER is agnostic to the underlying detector, we use popular Mask R-CNN [27] and Cascade Mask R-CNN [4] for all experiments, and use Cascade Mask R-CNN by default, unless otherwise noted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' We train the detector on ImageNet with initial masks and bounding boxes for 160K iterations with a batch size of 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' When training the detectors with a ResNet-50 backbone [28], we initialize the model with the weights of a self-supervised pretrained DINO [7] model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' We explored other pre-trained models, including MoCo-v2 [9], SwAV [6], and CLD [46], and found that they gave similar detection performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' We also leverage the copy-paste augmentation [16, 19] during the model training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Rather than using the vanilla copy-paste augmentation, to improve the model’s ability to segment small objects, we randomly downsample the mask with a scalar uniformly sampled between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='3 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' We then optimize the detector for 160K iterations using SGD with a learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='005, which is decreased by 5 after 80K iterations, and a batch size of 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' We apply a weight decay of 5×10−5 and a momentum of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Self-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' We initialize the detection model in each stage using the weights from the previous stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' We op- timize the detector using SGD with a learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='01 for 80K iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Since the self-training stage can provide 4 Datasets → Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' COCO COCO20K VOC LVIS UVO Clipart Comic Watercolor KITTI Objects365 OpenImages Metrics → AP50 AR AP50 AR AP50 AR AP50 AR AP50 AR AP50 AR AP50 AR AP50 AR AP50 AR AP50 AR AP50 AR AP50 AR Prev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' SOTA [47] 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='0 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='4 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='6 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='6 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='7 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='6 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='9 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='8 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='4 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='0 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='9 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='1 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='9 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='3 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='7 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='7 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='1 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='1 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='2 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='9 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='9 CutLER 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='3 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='5 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='9 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='7 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='4 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='1 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='9 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='3 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='4 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='8 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='7 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='8 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='1 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='3 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='4 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='6 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='5 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='6 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='4 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='5 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='6 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='2 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='3 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='6 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' prev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' SOTA +15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='3 +22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='1 +12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='3 +20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='1 +12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='7 +20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='5 +21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='0 +23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='0 +4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='6 +15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='4 +21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='7 +28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='6 +13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='2 +26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='2 +20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='5 +22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='3 +30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='8 +28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='4 +10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='7 +20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='4 +13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='5 +24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='0 +7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='4 +14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='7 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' State-of-the-art zero-shot unsupervised object detection performance on 11 different datasets spanning a variety of domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' We report class-agnostic multi-object detection performance and the averaged results for 11 datasets using APbox 50 and ARbox 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Our CutLER is trained in an unsupervised manner solely on ImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' While the previous SOTA method [47] is typically fine-tuned on extra data, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=', ∼241k unlabeled COCO images, CutLER significantly outperforms it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Results of [47] are produced with official code and checkpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' a sufficient number of pseudo-masks for model training, we don’t use the DropLoss during the self-training stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' We provide more details on model implementation and training in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Experiments We evaluate CutLER on various detection and segmen- tation benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='1, we show that CutLER can discover objects without any supervision on completely un- seen images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Despite being evaluated in a zero-shot manner on eleven benchmarks, CutLER outperforms prior methods that use in-domain training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='2 shows that fine- tuning CutLER further improves detection performance, outperforming prior work like MoCo-V2 and FreeSOLO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Unsupervised Zero-shot Evaluations We conduct extensive experiments on eleven different datasets, covering various object categories, image styles, video frames, resolutions, camera angles, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' to verify the effectiveness of CutLER as a universal unsupervised object detection and segmentation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' We describe the differ- ent datasets used for zero-shot evaluation in detail in Ap- pendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' CutLER is trained solely using images from ImageNet and evaluated in a zero-shot manner on all down- stream datasets without finetuning on any labels or data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Evaluating unsupervised object detectors poses two unique challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' First, since the model is trained with- out any notion of semantic classes, it cannot be evaluated using the class-aware detection setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Thus, like prior work [3,38,48] we use the class-agnostic detection evalua- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Second, object detection datasets often only annotate a subset of the objects in the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' For example, while COCO and LVIS use the same images, COCO only labels 80 object classes, and LVIS labels 1203 object classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' In this partially labeled setup, Average Recall (AR) is a valu- able metric for unsupervised detection as it does not penal- ize the models for detecting novel objects unlabeled in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Thus, we additionally report AR for all datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Zero-shot detection on 11 benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' We evaluate Cut- LER on a variety of datasets and report the detection perfor- mance using APbox 50 and ARbox 100 metrics in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' 1 and Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' CutLER uses a smaller model size and less training data than prior work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Compared to the previous SOTA approach, ground truth CutLER (ours) FreeSOLO Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Compared to the previous state-of-the-art [47], our CutLER can better discriminate instances (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' person and skis in col.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' 1), discover more objects (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' apple and raisins in col.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' 2), and produce higher quality segmentation masks even for small ob- jects (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' kite in col.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' 3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' compared to human annotations, CutLER can locate novel instances that are overlooked by human annota- tors, such as the streetlight and clock tower in col.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Qualitative comparisons between previous SOTA methods (row 1) and our CutLER (row 2) on COCO, as well as ground truth annotations by human annotators (row 3), are visualized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' FreeSOLO [47] with a backbone of ResNet101, CutLER, with the smaller ResNet50 backbone, significantly outper- forms it in each of these benchmarks spanning various im- age distributions, more than doubling performance on 10 of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Also note that, FreeSOLO requires FreeMask pre- training using approximately 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='3M ImageNet images and model fine-tuning using additional data in test benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' We observe that on different domains, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' watercolor or frames from videos (UVO dataset), CutLER improves per- formance by over 4× and 2×, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' 1 shows some qualitative examples of CutLER’s predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Detailed comparisons on COCO20K and COCO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Table 3 presents detailed detection and segmentation evaluations (also referred to as ‘multi-object’ discovery) on two pop- ular benchmarks: COCO val2017 [32] and COCO 20K, which contains a subset of 20K images of COCO [38, 47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' CutLER consistently surpasses prior works by a large mar- 5 Methods Pretrain Detector Init.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' COCO 20K COCO val2017 APbox 50 APbox 75 APbox APmask 50 APmask 75 APmask APbox 50 APbox 75 APbox APmask 50 APmask 75 APmask non zero-shot methods LOST [38] IN+COCO FRCNN DINO 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='1 MaskDistill [42] IN+COCO MRCNN MoCo 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='9 FreeSOLO∗ [47] IN+COCO SOLOv2 DenseCL 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='1 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='3 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='2 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='3 zero-shot methods DETReg [3] IN DDETR SwAV 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='3 DINO [7] IN DINO 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='3 TokenCut [50] IN DINO 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='4 CutLER (ours) IN MRCNN DINO 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='8 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='1 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='1 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='6 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='0 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='3 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='1 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='2 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='9 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='9 CutLER (ours) IN Cascade DINO 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='4 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='5 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='9 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='6 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='2 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='9 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='8 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='3 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='9 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='7 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='2 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' prev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' SOTA +12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='7 +9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='3 +7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='8 +9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='9 +6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='6 +4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='9 +12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='3 +8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='7 +8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='1 +9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='5 +6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='4 +4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='9 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Unsupervised object detection and instance segmentation on COCO 20K and COCO val2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' We report the detection and segmentation metrics and note the pretraining data (Pretrain), detectors, and backbone initialization (Init.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Methods in the top half of the table train on extra unlabeled images from the downstream datasets, while zero-shot methods in the bottom half only train on ImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Despite using an older detector, CutLER outperforms all prior works on all evaluation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' ∗: results obtained with the official code and checkpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' IN, Cascade, MRCNN, and FRCNN denote ImageNet, Cascade Mask R-CNN, Mask R-CNN, and Faster R-CNN, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Methods AP50 AP75 AP APS APM APL rOSD [43] 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='3 LOD [44] 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='5 LOST [38] 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='8 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='7 FreeSOLO∗ [47] 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='3 CutLER (ours) 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='9 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='2 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='3 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='5 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='2 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' prev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' SOTA +17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='1 +15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='6 +13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='5 +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='3 +4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='5 +22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='9 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Zero-shot unsupervised object detection on VOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' ∗: re- produced results with official code and checkpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' gin (often gets 2∼3× higher AP) on both the segmentation and detection tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Although CutLER is not trained on any images from COCO, it surpasses existing methods trained on COCO by more than 10% in terms of APmask 50 and APbox 50 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' 4 shows the qualitative comparisons between [47] and our CutLER on COCO val2017, along with human annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Surprisingly, CutLER can often detect novel instances that human annotators miss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' We present detailed comparisons on COCO 20K, COCO val2017 and LVIS [24] benchmarks in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Detailed comparisons on UVO and VOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' For a com- prehensive comparison with existing unsupervised multi- object detection methods, we report the results for UVO val [45] and VOC trainval07 [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Table 4 shows that CutLER yields significant performance gains over previous SOTA, obtaining over 3× higher AP, with the most consid- erable improvement coming from APL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' On UVO, Table 5 shows that CutLER more than quadruples the AP of previ- ous SOTA and almost triples the APbox 50 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Our APmask 50 is even 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='8% higher than the fully-supervised SOLOv2 [48] trained on LVIS with 100% annotations, significantly narrowing the gap between supervised and unsupervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Methods APbox 50 APbox 75 APboxAPmask 50 APmask 75 APmask fully-supervised methods: SOLO-v2 (w/ COCO) [48] 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='0 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='9 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='4 Mask R-CNN (w/ COCO) [27] 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='0 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='2 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='9 SOLO-v2 (w/ LVIS) [48] 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='9 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='1 unsupervised methods: FreeSOLO∗ [47] 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='2 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='3 CutLER (ours) 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='7 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='1 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='1 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='8 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='1 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' prev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' SOTA +21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='7+12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='3+12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='9 +13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='3 +6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='0 +6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='8 Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Zero-shot unsupervised object detection and instance segmentation on the UVO val video benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' CutLER out- performs prior unsupervised methods and achieves better perfor- mance than the supervised SOLO-v2 model trained on the LVIS dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' ∗: reproduced results with official code and checkpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Label-Efficient and Fully-Supervised Learning We now evaluate CutLER as a pretraining method for training object detection and instance segmentation mod- els.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' While CutLER can discover objects without any su- pervision, finetuning it on a target dataset aligns the model output to the same set of objects labeled in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' We use CutLER to initialize a standard Cascade Mask R-CNN [4] detector with a ResNet50 [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Prior work uses more advanced detectors, SOLOv2 [48] used in [47] and DDETR [54] used in [3], that perform better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' However, we choose Cascade Mask R-CNN for its simplicity and show in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' 5 that CutLER’s performance improves with stronger detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' We train the detector on the COCO [32] dataset using the bounding box and instance mask labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' To evaluate label efficiency, we subsample the training set to create subsets with varying proportions of labeled images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' We train the detector, initialized with CutLER, on each of these subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' As a baseline,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' we follow the settings from MoCo-v2 [9] and train the same detection architecture ini- tialized with a MoCo-v2 ResNet50 model,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' given its strong 6 Instance Segmentation: AP (%) 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 1 2 5 10 20 30 40 50 60 80100 Object Detection: AP (%) 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 1 2 5 10 20 30 40 50 60 80100 % of Labeled Data % of Labeled Data CutLER MoCo-v2 FreeSOLO CutLER MoCo-v2 DETReg 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='3% 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='6% Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Finetuning CutLER for low-shot and fully super- vised detection and instance segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' We fine-tune a Cas- cade Mask R-CNN model initialized with CutLER or MoCo-v2 on varying amounts of labeled data on the COCO dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' We use the same schedule as the self-supervised pretrained MoCo-v2 coun- terpart and report the detection and instance segmentation perfor- mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' CutLER consistently outperforms the MoCo-v2 baseline: in the low-shot setting with 1% labels and the fully supervised set- ting using 100% labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' CutLER also outperforms FreeSOLO [47] and DETReg [3] on this benchmark despite using an older detec- tion architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Results with Mask R-CNN are in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' performance on object detection tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Both MoCo-v2 and our models are trained for the 1× schedule using Detec- tron2 [51], except for extremely low-shot settings with 1% or 2% labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Following previous works [47], when train- ing with 1% or 2% labels, we train both MoCo-v2 and our model for 3,600 iterations with a batch size of 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' 5 shows the results of fine-tuning the detec- tor on different subsets of COCO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' When tested with low- shot settings, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=', 2% and 5% labeled data, our approach achieves 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='4% and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='3% higher APbox than the MoCo-v2 baseline, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Even when training with full anno- tations, CutLER still consistently gives more than 2% im- provements, outperforming MoCo-v2 for both object detec- tion and segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' More impressively, CutLER out- performs prior SOTA methods - FreeSOLO [47] and DE- TReg [3] despite using an older detection architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Ablations We analyze the design decisions in CutLER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' We use sim- ilar settings to Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' 4 and train CutLER only on ImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' We use the Cascade Mask R-CNN detection architecture and evaluate our model primarily on the COCO and UVO unsupervised detection benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' All ablation studies are conducted without self-training unless otherwise noted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Importance of each component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' We analyze the main components of CutLER and report their relative contribu- tion in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' We report results on the popular COCO [32] dataset and a densely annotated video instance segmenta- Methods UVO COCO APmask 50 APmask APmask 50 APmask TokenCut [50] 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='0 Base 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='4 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='7 + MaskCut 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='3 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='1 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='8 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='7 + DropLoss 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='9 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='0 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='6 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='2 + copy-paste [16,19] 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='9 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='7 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='8 + self-train (CutLER) 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='8 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='1 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='9 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='7 Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Ablation study on the contribution of each component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Results reported on COCO and video segmentation dataset UVO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Methods APbox 50 APbox ARbox 100 APmask 50 APmask ARmask 100 TokenCut (1 eigenvec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=') 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='4 TokenCut (3 eigenvec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=') 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='7 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='9 MaskCut (t = 3) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='9 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='9 CutLER 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='9 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='3 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='7 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='9 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='7 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='1 Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' CutLER achieves much higher results even when com- pared to a modified TokenCut that can produce more than one mask per image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Compared to TokenCut, MaskCut gets a higher recall without reducing precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' We report results on COCO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' tion dataset UVO [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' We also report the performance of running TokenCut [50] on the COCO dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Next, we use TokenCut’s official codes to generate masks on ImageNet and use them for training a Cascade Mask R-CNN [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' This base model provides substantial gains over just using To- kenCut on COCO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' We add each of our proposed compo- nents to this strong base model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Using MaskCut increases APmask 50 and APmask by 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='7% and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='7%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Also, the improvements to APmask 50 is larger for densely annotated dataset UVO, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='7% vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='7%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' These results prove that MaskCut’s ability to segment multiple instances per image is vital for densely annotated datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Adding DropLoss brings another 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='6% and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='9% improvements to APmask 50 for UVO and COCO, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Multi-round of self-training increases the quantity and quality of pseudo-masks, leading to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='3% improvements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' These results show that each simple proposed component is critical for strong performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Comparison with TokenCut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' TokenCut [50] is also a zero-shot segmentation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' However, it only segments a single instance per image, as discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' In order to generate more than one segmentation mask per image, we use a modified TokenCut by using more of the smaller eigenvectors and combining all produced masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Table 7 shows the object detection performance on COCO’s validation set for vanilla TokenCut, our modified Token- Cut and CutLER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Although using more eigenvectors in- creases the recall ARbox 100, it significantly reduces the pre- cision APbox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' CutLER not only improves the average recall ARbox 100 by 4× but also surpasses TokenCut’s average preci- sion APbox by 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='8×, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' 480% relative improvements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' 7 Size → 240 360 480 640 APmask 50 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='1 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='6 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='7 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='9 (a) Image size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' τ ncut → 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='3 APmask 50 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='1 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='5 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='7 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='6 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='5 (b) τ ncut for MaskCut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' N → 2 3 4 APmask 50 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='9 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='7 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='7 (c) # masks per image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' τ IoU → 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='2 APmask 50 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='4 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='7 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='4 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='7 (d) τ IoU for DropLoss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Ablations for MaskCut and DropLoss used for training CutLER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' We report CutLER’s detection and instance segmentation performance on COCO val2017, without adding the self-training stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' (a) We vary the size of the image used for MaskCut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' (b) We vary the threshold τ ncut in MaskCut, which controls the sparsity of the affinity matrix used for Normalized Cuts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' (c) We vary the number of masks extracted using MaskCut and train different CutLER models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' (d) We vary τ IoU in DropLoss, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=', the maximum overlap between the predicted regions and the ground truth beyond which the loss for the predicted regions is ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Default settings are highlighted in gray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' UVO COCO APmask 50 APmask APmask 75 APmask 50 APmask APmask 75 1 round 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='6 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='7 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='8 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='0 2 rounds 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='2 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='6 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='5 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='8 3 rounds 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='8 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='1 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='0 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='9 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='7 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='2 4 rounds 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='8 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='2 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='2 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='9 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='8 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='3 Table 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Number of self-training rounds used in CutLER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' We find that 3 rounds of self-training are sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Self-training pro- vides larger gains for the densely labeled UVO dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' round 1 round 2 round 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='58 M 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='77 M 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='03 M # of masks Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Multiple rounds of self-training can improve the pseudo- masks in terms of quality and quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' We show qualitative visu- alizations and the number of pseudo-masks for all three rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Design choices in MaskCut and DropLoss and their im- pact on the final localization performance is presented in Ta- ble 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' We first study the effect of the image size used by MaskCut for generating the initial masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' As expected, Ta- ble 8a shows that MaskCut benefits from using higher reso- lution images presumably as it provides a higher resolution similarity between pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' We pick a resolution of 480px for a better trade-off between the speed of MaskCut and its performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' In Table 8b, we study the effect of the threshold used in MaskCut for producing a binary W ma- trix (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Overall, CutLER seems to be robust to the threshold values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' We understand the impact of the num- ber of masks per image generated by MaskCut in Table 8c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Increasing the number improves the performance of the re- sulting CutLER models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' This shows that MaskCut gener- ates high-quality masks that directly impact the overall per- formance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Finally, in Table 8d, we vary the IOU threshold used for DropLoss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' With a high threshold, we ignore the loss for a higher number of predicted regions while encour- aging the model to explore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='01 works best for the trade-off between exploration and detection performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Mask R-CNN Cascade Mask R-CNN ViTDet APbox 50 / APbox 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='3 / 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='6 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='8 / 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='5 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='5 / 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='8 APmask 50 / APmask 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='2 / 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='5 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='7 / 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='8 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='0 / 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='0 Table 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' CutLER with different detection architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' We report results on COCO and observe that CutLER is agnostic to the detection architecture and improves performance using stronger detection architectures such as ViTDet with a backbone of ViT-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Pre-train CutLER APbox 50 APbox 75 APbox APmask 50 APmask 75 APmask IN1K IN1K 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='8 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='8 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='5 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='7 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='8 YFCC1M YFCC1M 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='4 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='4 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='9 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='3 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='4 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='1 IN1K YFCC1M 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='9 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='6 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='2 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='4 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='9 YFCC1M IN1K 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='8 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='2 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='0 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='8 Table 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Impact of datasets used to pre-train DINO and train CutLER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' CutLER’s detection performance is similar when pre- training both DINO and CutLER with the same dataset: the object- centric ImageNet dataset or the non-object-centric YFCC dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Self-training and its impact on the final performance is an- alyzed in Table 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Self-training consistently improves per- formance across the UVO and COCO benchmarks and all metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' UVO, which has dense object annotations, benefits more from the multi-round of self-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' By default, Cut- LER uses 3 rounds of self-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' 6 shows qualitative examples of how self-training improves both the quality of predictions and the number of objects predicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Generalization to different detection architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' We use different detector architectures for training CutLER and measure their performance in Table 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' We observe that CutLER works with various architectures, and its perfor- mance is improved with stronger architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Impact of the pretraining dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' We now study the im- pact of the dataset used for 1) pretraining the self-supervised DINO model and 2) training the CutLER model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' The com- monly used ImageNet dataset has a well-known object- centric bias [12] which may affect the unsupervised detec- tion performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Thus, we also use YFCC [40], a non- object-centric dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' We control for the number of images in both ImageNet and YFCC for a fair comparison and use them for training DINO and CutLER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' As Table 11 shows, CutLER’s performance on COCO is robust to the choice of object-centric or non-object-centric datasets as long as the 8 4-2007same dataset is used to train DINO and CutLER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' This shows the generalization of CutLER to different data distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' However, training DINO and CutLER with different data leads to worse performance, suggesting the importance of using the same image distribution for learning both DINO and CutLER models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Summary Object localization is a fundamental task in computer vi- sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' In this paper, we have shown that a simple yet effec- tive cut-and-learn approach can achieve extraordinary per- formance on challenging object detection and instance seg- mentation tasks without needing to train with human an- notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' As a zero-shot unsupervised detector, CutLER, trained solely on ImageNet, outperforms the detection per- formance of previous works by over 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='7× on 11 bench- marks across various domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' References [1] YM Asano, C Rupprecht, and A Vedaldi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Self-labelling via simultaneous clustering and representation learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' In In- ternational Conference on Learning Representations, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' 2 [2] Hangbo Bao, Li Dong, Songhao Piao, and Furu Wei.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' 2 11 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Training details While CutLER is agnostic to the underlying detector, we use popular Mask R-CNN [27] and Cascade Mask R- CNN [4] for all experiments, and use Cascade Mask R- CNN by default, unless otherwise noted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' We train the de- tector on ImageNet with initial masks and bounding boxes for 160K iterations with a batch size of 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' When training the detectors with a ResNet-50 backbone [28], we initialize the model with the weights of a self-supervised pretrained DINO [7] model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' We explored other pre-trained models, in- cluding MoCo-v2 [9], SwAV [6], and CLD [46], and found that they give similar detection performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Therefore, we initialize model weights with DINO by default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' We also leverage the copy-paste augmentation [16, 19] during the model training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Rather than using the vanilla copy-paste augmentation to improve the model’s ability to segment small objects, we randomly downsam- ple the mask with a scalar uniformly sampled between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='3 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' We then optimize the detector for 160K iterations using SGD with a learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='005, which is decreased by 5 after 80K iterations and a batch size of 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' We apply a weight decay of 5×10−5 and a momentum of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' For the multi-round of self-training, in each stage, we initialize the detection model using the weights from the previous stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' We optimize the detector using SGD with a learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='01 for 80K iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Since the self- training stage can provide a sufficient number of pseudo- masks for model training, we don’t use the exploration loss during the self-training stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Datasets used for zero-shot evaluation COCO and COCO20K [32] is a large-scale object detec- tion and instance segmentation dataset, containing about 115K and 5K images in the training and validation split, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Additionally, COCO has an unannotated split of 123K images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' We test our model in a class-agnostic manner on COCO val2017 and COCO 20K, without fine-tuning on any images in COCO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' COCO 20K is a subset of the COCO trainval2014 [32], containing 19817 randomly sampled images, used as a benchmark in [38, 43, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' We report class-agnostic COCO style averaged precision and averaged recall for object detection and segmentation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Pascal VOC [17] is another popular benchmark for object dtetection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' We evaluate our model on its trainval07 split in COCO style evaluation matrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' UVO [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Unidentified Video Objects (UVO) is an exhaus- tively annotated dataset for video object detection and in- stance segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' We evaluate our model on UVO val by frame-by-frame inference and report results in COCO style evaluation matrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' LVIS [24] collected 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='2 million high-quality instance seg- mentation masks for over 1000 entry-level object cate- gories, which naturally constitutes the long-tailed data dis- tribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' We report class-agnostic object detection and in- stance segmentation results on LVIS val split, containing about 5K images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' CrossDomain [29] contains three subsets of watercolor, cli- part, and comics, in which objects are depicted in water- color, sketch and painting styles, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' We evaluate our model on all annotated images from these three datasets, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=', traintest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Objects365 V2 [36] presents a supervised object detection benchmark with a focus on diverse objects in the wild.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' We evaluate CutLER on the 80K images from its val split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' OpenImages V6 [31] unifies image classification, object de- tection, and instance segmentation, visual relationship de- tection, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' in one dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' We evaluate CutLER on its 42K images from the val split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' KITTI [18] presents a dataset captured from cameras mounted on mobile vehicles used for autonomous driv- ing research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' We evaluate CutLER on 7521 images from KITTI’s trainval split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' We provide the summary of these datasets used for zero- shot evaluation in Table 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Additional results for zero-shot detection & segmentation In this section, we use official COCO API and pro- vide more results with standard COCO metrics, including AP across various IoU thresholds - AP (averaged over IoU thresholds from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='5 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='95 with a step size of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='05), AP50 (IoU@0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='5) and AP75 (IoU@0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='75), and AP across scales - APS (small objects), APM (medium objects) and APL (large objects).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' We provide detailed results on all these bench- marks listed in Table 12 and report these results in Ta- ble 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' We report the performance of object detection for all datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' In addition, for those datasets that provide annota- tions for instance segmentation, we also present the perfor- mance of the instance segmentation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' It is worth noting that on these datasets without segmentation labels, CutLER can still predict instance segmentation masks, but since we do not have ground truth masks to be compared, we cannot evaluate the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' CutLER vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Selective Search Selective Search [41] is a popular unsupervised object discovery method, used in many early state-of-the-art de- tectors such as R-CNN [22] and Fast R-CNN [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' How- ever, generating possible object locations with sliding win- dows greatly reduces inference speed (please refer to [41] for more details on selective search).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' We compare Cut- LER’s performance to selective search in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' 7 and ob- serve that CutLER provides a significant improvement in both precision and recall,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' which indicates that CutLER is a 12 datasets domain testing data #images instance segmentation label COCO [32] natural images val2017 split 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='000 \x13 COCO20K [32] natural images a subset of COCO 20,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='000 \x13 UVO [45] video frames val split 7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='356 \x13 LVIS [24] natural images val split 19,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='809 \x13 KITTI [18] traffic images trainval split 7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='521 \x17 Pascal VOC [17] natural images trainval07 split 9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='963 \x17 Clipart [29] clip arts traintest split 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='000 \x17 Watercolor [29] paintings traintest split 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='000 \x17 Comic [29] sketches traintest split 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='000 \x17 Objects365-V2 [36] natural images val split 80,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='000 \x17 OpenImages-V6 [31] natural images val split 41,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='620 \x17 Table 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Summary of datasets used for zero-shot evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Datasets APbox 50 APbox 75 APbox APbox S APbox M APbox L ARbox 1 ARbox 10 ARbox 100 APmask 50 APmask 75 APmask APmask S APmask M APmask L ARmask 1 ARmask 10 ARmask 100 COCO 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='9 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='8 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='7 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='7 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='8 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='6 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='8 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='9 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='2 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='4 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='8 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='3 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='8 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='5 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='1 COCO20K 22.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='3 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='8 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='6 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='4 UVO 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='7 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='1 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='1 3.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='1 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='7 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='3 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='6 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='0 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='2 LVIS 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='7 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='1 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='1 2.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='3 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='9 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='5 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='6 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='6 Table 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Detailed zero-shot evaluation results on all benchmarks used in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Precision 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='8 1 Recall 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='0 Selective Search Ours Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Precision-recall curve for comparing selective search and CutLER on VOC07 trainval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' better performing unsupervised method for region proposal generation with real-time inference speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Training details for label-efficient and fully- supervised learning We train the detector on the COCO [32] dataset using the bounding box, and instance mask labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' To evaluate label efficiency, we subsample the training set to create subsets with varying proportions of labeled image We train the de- tector, initialized with CutLER, on each of these subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' As a baseline, we follow the settings from MoCo-v2 [9] and train the same detection architecture initialized with a MoCo-v2 ResNet50 model, given its strong performance on object detection tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' MoCo-v2 and our models use the same training pipeline and hyper-parameters and are trained for the 1× schedule using Detectron2 [51], except for ex- tremely low-shot settings with 1% or 2% labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Following previous works [47], when training with 1% or 2% labels, we train both MoCo-v2 and our model for 3,600 iterations with a batch size of 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Our detector weights are initialized with ImageNet-1K pre-trained CutLER, except for the weights of the final bounding box prediction layer and the last layer of the mask prediction head, which are randomly initialized with val- ues taken from a normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' For experiments on COCO with labeling ratios below 50%, during model train- ing, we use a batch size of 16, and learning rates of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='04 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='08 for model weights loaded from the pre-trained CutLER and randomly initialized, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' For experiments on COCO with labeling ratios between 50% and 100%, the learning rates of all layers decay by a factor of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' For a fair comparison, baselines and CutLER use the 13 Mask R-CNN Cascade Mask R-CNN Instance Segmentation: AP (%) 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 1 2 5 10 20 30 40 50 60 80100 Object Detection: AP (%) 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 1 2 5 10 20 30 40 50 60 80100 % of Labeled Data % of Labeled Data CutLER MoCo-v2 CutLER MoCo-v2 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='8% 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='0% Instance Segmentation: AP (%) 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 1 2 5 10 20 30 40 50 60 80100 Object Detection: AP (%) 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 1 2 5 10 20 30 40 50 60 80100 % of Labeled Data % of Labeled Data CutLER MoCo-v2 FreeSOLO CutLER MoCo-v2 DETReg 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='3% 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='6% Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' Fine-tuning on MS-COCO with various annotation ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' We report results using Mask R-CNN and Cascade Mask R-CNN with a backbone of ResNet-50 as the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' same hyper-parameters and settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' More visualizations We provide more qualitative visualizations of CutLER’s zero-shot predictions in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' 14 Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=' More visualizations of CutLER’s predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} +page_content=" 15 1tlegi,eCratn,DvsiVioge,war color, fetttip,2009 2,Frfiend'sportrait,Penei,2009CH54 CAeskHhrESHCOm2" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFIT4oBgHgl3EQfoyuB/content/2301.11320v1.pdf'} diff --git a/etAyT4oBgHgl3EQfw_lx/content/tmp_files/2301.00658v1.pdf.txt b/etAyT4oBgHgl3EQfw_lx/content/tmp_files/2301.00658v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..2c43c898f0a9b1a7968bd170411bc5ced5ea2c35 --- /dev/null +++ b/etAyT4oBgHgl3EQfw_lx/content/tmp_files/2301.00658v1.pdf.txt @@ -0,0 +1,2154 @@ +1 +Comparative Analysis of Terahertz Propagation +Under Dust Storm Conditions on Mars and Earth +Lasantha Thakshila Wedage, Bernard Butler, Sasitharan Balasubramaniam, Yevgeni Koucheryavy, +Mehmet C. Vuran +Abstract—Reliable Terahertz (THz) links are necessary for out- +door point-to-point communication with the exponential growth +of wireless data traffic. This study presents a modified Monte +Carlo simulation procedure for estimating THz link attenuation +due to multiple scattering by dust particles on the THz beam +propagation path. Scattering models are developed for beams +through dust, based on Mie and Rayleigh approximations for +corresponding frequencies for Earth (0.24 THz) and Mars +(1.64 THz). The simulation results are compared, considering +parameters such as the number of Monte-Carlo photon (MCP) +packets, visibility, dust particle placement density along the beam, +frequency, and distance between the transmitter and the receiver. +Moreover, a channel capacity model was proposed, considering +THz link attenuation due to dust storms, spreading loss and +molecular absorption loss for Earth and Mars outdoor environ- +ments. Simulation results for Earth show that link attenuation +increases with dust particle placement density, distance and +frequency, and attenuation decreases with visibility. On Mars, +similar results are obtained, except that the attenuation is variate +around a constant value with the frequency increase. Channel +capacity is estimated for Earth and Mars environments consid- +ering time and distance-dependent scenarios. Time windows that +show a sudden drop of dust particles along the beam provide +opportunities to communicate with high reliability. Moreover, +increasing the distance between the transmitter and receiver +severely reduces the channel capacity measurement in strong dust +storm conditions in both environments. Our study has found that +weak dust storms have relatively little effect on Mars, but much +larger effects on Earth. +Index Terms—THz Communication, Atmosphere, Attenuation, +Scattering, Dust. +I. INTRODUCTION +S +IXTH generation (6G) wireless networks aim to push +the frequency spectrum into the Terahertz (THz) band to +fulfill rising capacity demands and requirements, given the +opportunity for higher bandwidths [1]–[3]. The 0.1 to 10 +THz frequency range has the potential to (1) realize high +bandwidth transmissions that can allow hundreds of GB/s +data rates for communication [4]–[6], and (2) provide new +opportunities to create miniature THz-enabled antennas due +to the small wavelengths (30 µm – 3 mm), enabling us to +design arrays with a large number of antenna units [7]–[9]. +Numerous studies have shown that specific THz frequencies +suffer high molecular absorption due to atmospheric gases +(e.g., water vapor and oxygen). However, given the wavelength +and high energy photons of THz signals, other particles can +Lasantha Thakshila Wedage and Bernard Butler are with the Walton +Institute, South East Technological University, Ireland. +Sasitharan Balasubramaniam and Mehmet C. Vuran are with the University +of Nebraska-Lincoln, USA. +Yevgeni Koucheryavy is with the Tampere University of Technology, Finland. +also significantly impact the link budget, which can result in +scattering and absorption of signal power. +Recent studies have shown that solid particles such as dust, +sand and ice affect THz signals [10], in addition to molecular +absorption from atmospheric gases [11]. However, past studies +have paid little attention to signal attenuation caused by solid +particles such as dust and sand. Therefore, further investigation +is required to determine how dynamic environments composed +of solid particles, such as dust storms, affect THz links. +This requires further investigation, especially as we expand +connectivity in rural areas and other planets (e.g., Mars) to +interplanetary scale. In the case of Mars, the recent vision of +colonizing the planet will require high-bandwidth connectivity +to maximize chances for human survival. +A dust storm is a physical layer of dust and debris blown +into the atmosphere by winds with horizontal and vertical +velocity components. On Earth, the wall of dust can be miles +wide and several thousand feet high. Dust storms are more +frequently found in arid regions such as the Middle East +[12], North China [13], and North Africa [14] at specific +periods of the year. In more densely populated areas, human +activity creates dust when burning fossil fuels for heating, +cooking, or transport. Industrial and construction processes +also create dust. This study compares the effects of solid dust +particles on (sub)THz signals on both Earth and Mars, taking +account of varying environmental conditions. Considering the +differences in atmospheric conditions on Earth and Mars, with +or without dust, suggests the use of different frequencies +to enable relatively long-distance wireless communication on +both planets. Dust storms are one of the most remarkable +features on Mars. Even though wind speed on Mars is not +significantly higher than on Earth, the extremely dry, dusty +surface yields more dust storms. +Figure 1 provides an overview of selected wireless commu- +nications applications on Earth, and proposed wireless com- +munication applications on Mars. While the applications differ, +they are both affected by wireless channel losses, including +those caused by dust particles that can scatter the EM waves +used for communication. The rest of this paper considers the +similarities and differences in the channel conditions, and +includes models and simulations based on simulated dust +storms that result in beam scattering, as shown at the bottom +of Fig. 1. +In a dusty environment, the dust particle density is higher +than usual, and the effects of multiple scattering of EM waves +due to dust particles are non-negligible. Recent studies have +not considered this significant effect on the attenuation of +EM waves [15], [16]. The lack of consideration of multiple +arXiv:2301.00658v1 [cs.IT] 11 Dec 2022 + +2 +Immersive +XR +Wireless +Cognition +Mobile +Hologram +Smart ML/AI +Applications +1Tbps +Data rate +Wireless +Sensing +Smart space V2V +and Positioning +Smart +Spacesuit +comms +Wireless +Cognition +Smart +Rover +Smart +Habitats +Fig. 1: THz wireless communication applications and links through Earth and Mars atmospheric and environmental conditions. +scattering effects can result in significant gaps between theo- +retical and experimental results. This paper considers multiple +scattering of EM waves due to dust particles along the beam +propagation path. To this end, we model the EM wave as a +photon packet instead of a shower of photons. It is inaccurate +to consider the EM wave as a shower of photons characterized +by the position of a photon and its trajectory [17]. A photon +packet models a portion of the energy weight of the EM wave +rather than single photons (which have quantum behavior). +Therefore, we can consider an EM wave as a collection of +energy packets and model multiple scattering effects utilizing +the Monte Carlo algorithm, to infer the radiative transfer +equation. The THz link scattering loss measurement in this +study is inspired by [18], where the scattering loss due to +charged dust particles is calculated by considering the energy +of the transmitting signal as Monte Carlo Photon Packets. +Vertical THz attenuation is determined in [18], but this study +considers horizontal point-to-point communication for both +Earth and Mars in dusty atmospheric scenarios. +The contributions of this paper are: +1) A 3-D geometric scattering model for multiple photon- +dust particle interactions is presented, using both Mie +and Rayleigh approximations, to estimate the probability +that a photon packet arrives at the receiver. +2) The model is used in simulation to estimate the overall +channel capacity considering THz and sub-THz link +budget degradation due to the combination of scattering +by dust particles, molecular absorption loss due to the +atmosphere, and free-space spreading loss. +TABLE I: Atmospheric gas composition comparison between +Earth and Mars [16]; ppm is a concentration of parts per +million. +Gas +Composition on Earth +Composition on Mars +N2 +78.084% +2.7% +O2 +20.946% +0.13% +Ar +0.93% +1.6% +H2O +1-3% +100-400ppm +CO2 +0.003% +95.32% +CH4 +1.5ppm +- +SO2 +1ppm +- +O3 +0.05ppm +0.1ppm +N2O +0.02ppm +- +CO +0.01ppm +0.08% +NH3 +0.01ppm +- +NO +- +100ppm +3) Different communication channel conditions (on Earth +and Mars) and their effect on channel capacity, including +power loss caused by multiple scattering by dust, are +compared and analysed. +The rest of this paper is organized as follows: Section II +describes dust conditions and how they affect EM propagation +and contrasts the conditions that prevail on Earth and Mars; +Section III describes how 3D dust storm simulation is affected +by the number of dust particles on the EM wave propagation +path. Then Section IV explains the Monte-Carlo simulation +process for calculating the transmittance/attenuation when +photon packets are scattered by multiple dust particles. Section +V presents estimates of transmittance/attenuation obtained by +Monte-Carlo simulation, in various parameter settings. Section + +McPhoton3 +VI presents a channel capacity model that combines the effect +of spreading, molecular absorption and multiple scattering by +dust, with simulated results. Finally, Section VII presents our +conclusions. +II. BACKGROUND +A. THz link behaviour in Dust Storms +THz signal attenuation due to the scattering loss caused by +high dust particle density on the THz beam propagation path is +the main concern of this study. Dust particle density on Mars +is expected to be higher than on Earth because of the dusty +atmosphere with low water vapour concentration. Mars dust +consists of basalt and montmorillonite clay [19]. On the other +hand, Earth dust consists of pollen, bacteria, smoke, ash, salt +crystals from the ocean, and small amounts of dirt or various +rocks, including sand. Moreover, during dust storm conditions +on Mars, the effective radius of the dust particle varies from +1 to 4 microns with an effective variance of 0.2 – 0.4 [20]. +However, on Earth, the effective radius varies between 1 and +150 microns [18], [19]. +Many researchers investigated the THz [16], [18], [21], +[22] and lower frequency bands [23], [24] attenuation due to +the presence of dust particles on the beam propagation path. +In [18], Monte-Carlo simulation was used to calculate the +transmittance of EM waves when they propagate through dust, +considering multiple scattering effects for charged particles +in 20 and 75 GHz frequencies. Hongxia et al. [21] also +studied the attenuation characteristics of THz waves subject +to multiple scattering caused by dust storms in the Tengger +desert, using the Mie scattering approximation and Monte +Carlo simulation. In addition, considering the Mie theory, +Diao et al. [16] investigated THz wave attenuation due to +heavy dust in the Martian atmosphere in the 0.1-1 THz +frequency range and compared with Earth measurements. [22] +investigated attenuation at 0.625 THz caused by dust utilising +an experimental setup and found that degradation of the THz +link budget is minor due to dust, compared to that found using +IR beams with 1.5 µm wavelength, and average attenuation +of the THz link is proportional to the dust particle density. +Moreover, Elshaikh et al. [23] developed a mathematical +model to characterise the microwave attenuation due to dust, +considering parameters such as visibility, frequency, particle +size and complex permittivity. Li et al. [24] calculated the light +scattering properties of partially charged dust particles utilising +Mie scattering theory for various frequencies and found that +for higher THz frequency EM waves, the attenuation effect of +charge carried by sand particles can be ignored. Furthermore, +[25] presents the EM scattering properties of the small partially +charged sand/dust particles, using the Rayleigh approximation, +for microwave frequencies. +B. Atmospheric Condition Differences between Mars and +Earth +When THz radio waves pass through the atmosphere, the +signals experience attenuation due to many factors, which +differ in their impact between Earth and Mars. This study +focuses on point-to-point signal degradation in the lower +part of the atmosphere (the troposphere) on Earth and Mars, +when communicating antennas are placed 50 meters above +the ground. Apart from improved line-of-sight properties, [26] +shows that longer communication distances can be achieved +on Mars because dust particle density decreases with height. +The propagation medium in the troposphere of both planets +includes gases, water vapour, clouds, fog, ice, dust, and +assorted aerosols (haze), but the proportions vary. The impair- +ment mechanisms include absorption, scattering, refraction, +diffraction, multi-path, scintillation and Doppler shift. Impair- +ment phenomena include fading, attenuation, depolarization, +frequency broadening, and ray bending. However, this study +considers only Line-of-Sight (LoS) transmission under dust +storm scenarios through the troposphere of both Earth and +Mars. It considers signal attenuation based on three factors: (1) +free space path loss (which is the same for Earth and Mars), +(2) molecular absorption due to atmospheric gases (which are +different for Earth and Mars), and (3) scattering loss due to +dust particles along the propagation path (Mars and Earth +typically have different dust distributions). +Free space path loss occurs due to misalignment between the +transmitter and the receiver antennas. It is the same for both +environments because it only depends on carrier frequency and +distance. Molecular absorption loss plays a significant role on +both planets. It measures the fraction of power loss (of the +carrier wave) converted to kinetic energy due to molecular +vibration when EM waves propagate through molecules of +the atmosphere. Therefore, unlike spreading loss, molecular +absorption loss depends on local atmospheric gas composition +and density (see Table I), including carrier frequency and +distance between the transmitter and the receiver. According +to [27], certain frequencies of the THz spectrum, such as 183, +325, 380, 450, 550, and 760 GHz, suffer attenuation that is +significantly greater than the free space propagation loss, due +to water vapor absorption on Earth. However, the Martian +atmosphere contains only about 1/1,000 as much water as +Earth’s. Still, even this tiny amount can condense out, forming +clouds that ride high in the atmosphere or swirl around the +slopes of towering volcanoes [19]. This serious issue needs +to be considered for vertical communication of Mars surface +devices (Rovers, Habitats, etc.) and satellites. Since our study +focuses on horizontal point-to-point communication, we do +not need to consider upper atmospheric layer’s impact on THz +signal transmission. Therefore, at these frequencies, we expect +lower molecular absorption loss and higher channel capacity +on Mars compared to Earth. +To the best of our knowledge, this is the first study that +compares attenuation (at THz frequencies) due to dust storms +on Earth and Mars, applying Monte Carlo simulation to the +corresponding Mie and Rayleigh approximations. This paper +also presents a channel capacity model that includes the effect +of spreading, molecular absorption and dust scattering losses +for sub-THz and THz links on Earth and Mars, respectively. +III. THZ BEAM PROPAGATION THROUGH A SIMULATED 3D +DUST STORM +This section discusses THz wave propagation through a +randomly simulated 3D dust storm by simulating wind having + +4 +both vertical (up-draught) and horizontal velocity components. +This is used to calculate the number of dust particles on the +beam path. The simulated dust storm (see Fig. 2) consists +of a line source starting at X = 0 and a vertically upward +movement of dust based on the vertex motion due to wind +turbulence at (6000, 0, 0). The line source dust storm in this +study spreads for 8m along the Y-axis (−4 ≤ Y ≤ 4). Such a +line source is more realistic than a point source for dust storm +simulation on both Earth and Mars. +MATLAB’s wind package was used to simulate the storm +and considered an exponential movement of dust along the +positive X-axis coupled with strong wind in the same direction. +Also, dust particles gradually precipitate from the atmosphere, +when their weight exceeds upward forces. Dust particle move- +ment of the point source dust storm downstream of the line +source dust storm comprises both upward (point) wind and +horizontal (line) wind, resulting in a vortex flow (a simulated +whirlwind). +When counting the number of dust particles on the cone- +shaped THz beam, we followed the following method. First, +consider the THz beam starting at the point of (0, 0, h), +where h (50 m) is the transmitter antenna height and beam +propagation direction is aligned with the positive X-axis. In +this cone-shaped beam, the maximum radius of the impact area +is calculated to be approximately 15 cm for the corresponding +transmitting distance of 10 Km. Since it is difficult to calcu- +late the dust particle concentration on such a pencil-thin beam, +we sub-divided the cone-shaped beam into 1 × 106 disks with +1 cm distances between the disk centres (see Fig. 2 (a)). Then +we identified the position of each dust particle at 1 m below +the antenna height and recorded its position considering the +nearest two disks. By looping over the position of each dust +particle, and comparing the distances between the nearest disk +centres and the particle position to centre distance, we encoded +the position as being inside (1) or outside (0) of the THz beam +for each particle. From this, we calculated the number of dust +particles along the THz propagation path. +Considering the dust particle size on Mars to vary between +0.5 to 4 microns [20], [28], we simulate a scenario of sending +a THz signal with the transmitter position (5500,0,50) and +the receiver position (6500,0,50), which creates a 1000 m +distance between them while placing the point source vertex +movement at (6000,0,0). As a result, we found that the average +ratio of the number of generated dust particles along the line +of propagation of the THz transmission beam is 0.0022872. +Moreover, considering just the dust particles on the cone- +shaped beam path, their density averaged 10.1832 (say, 10) +particles per cm3. Hence we can conclude that assuming the +beam face area is 0.01 cm2, the number of dust particles along +the beam for a distance of 10 m between the transmitter and +the receiver is approximately 100 particles. However, in this +random walk simulation process, it is difficult to control the +effective radius of the dust particles. Therefore, depending on +the dust particle size, scattering effects (which depend on the +number and size of the particles) along the beam propagation +path can vary. +h +Si,1 +D +X +Y +x +y +z +φ +ϴ +X +Z +Tx +Rx +Si,2 +Si,3 +Si,j +Tx +Rx +Wi,0 +Wi,1 +Wi,j +Wi,2 +Wi,j-1 +MC Photon packet weight +Scatters +(a) +(b) +Inside (1) +Outside (0) +Center - Center +Distance +Center - Point +Distance +Radius +Fig. 2: Multiple scattering processes of EMWs in a sand/dust +storm with (a) the decision-making (in/out) method of dust +particles from the beam and (b) the local coordinate system. +IV. MODELLING MONTE CARLO PHOTON PACKETS +PROPAGATION THROUGH DUST PARTICLES +In this section, we calculate the transmittance and the corre- +sponding attenuation of the THz EM wave when it propagates +through suspended dust particles. The initial intention was +to consider the THz EM wave as a collection of photons. +However, photon position and trajectory are not meaningful +here [17] but collections of photons enable us to discretize the +beam in a physically meaningful way. Monte Carlo simulation +is used to estimate transmittance, where the incident plane +EM wave is discretized as Monte Carlo photon (MCP) pack- +ets/units. Such photon packets provide an appropriate physical +unit for discrete event simulation [18]. Each MCP packet is +considered to be an equally divided portion of the energy +weight of the EM wave field. In this simulation model, we +assume that the particle number concentration is uniformly +distributed throughout the THz beam area, and dust particles +are randomly positioned. +The intensity (I) of the incident THz EM wave can be +expressed as I0 = M.W, where M is the number of MCP +packets per unit area per unit time and W is the energy +weight of each MCP. Here we suppose that MCP packet +i (i = 1, 2, ..., M) is randomly scattered by dust particles j +before it either exits the beam cone or reaches the receiver +interface boundary at X = D. We assume that MCP packets +enter from the point (0, 0, h) (which is height h corresponding +to the transmitter antenna height) (see Fig.2) and are forward- +scattered by scattering particles Si,l (l = 1, 2, ..., j) whose +positions are denoted by (xi,l, yi,l, zi,l), which are assumed +random. Moreover, the algorithm will randomly select the +number of scattering particles (l) that collide with an MCP +packet from j dust particles on the EM wave propagation +path because each MCP packet will randomly change its + +150- +喜 +100.% +Y (m) +f00 +2000 +4000 +6000 +B000 +00001 +x(n)5 +propagation direction after every collision and dust particles +are uniformly distributed along the beam path. +We suppose the initial energy weight of each MCP packet is +W = Wi,0 = 1 and after scattering due to scattering particle +Si,l is Wi,l. We also define a local coordinate system xyz with +the origin located at the scattering particle. So, we can define +the propagation direction of the MCP packet i with respect to +the local coordinate system xyz, which is to the x direction +considering forward scattering. The propagation direction for +the scattering direction of the MCP packet i due to the +impact with scattering particle Si,l can be expressed using +the direction cosines (µx +i,l, µy +i,l, µz +i,l) in the global coordinate +system XY Z, which is defined with the help of scattering +(or deflection) angle θ and azimuth angle φ (see Fig. 2 (b)). +In each simulated collision, the scattering particle position +(xi,l, yi,l, zi,l) and the propagation angles (θ, φ) to calculate +the direction cosines were calculated randomly for the global +coordinate system, which will be explained in detail later. +Direction cosines can be calculated according to Fig. 2 (b) +as follows, +µx +i,l= −sin(θi,l)cos(φi,l) +� +1 − (µx +i,l−1)2 + µx +i,l−1cos(θi,l) +(1a) +µy +i,l= +sin(θi,l)(µy +i,l−1µx +i,l−1cos(φi,l) − µz +i,l−1sin(φi,l)) +� +1 − (µx +i,l−1)2 ++ µy +i,l−1cos(θi,l) +(1b) +µz +i,l= +sin(θi,l)(µz +i,l−1µx +i,l−1cos(φi,l) + µy +i,l−1sin(φi,l)) +� +1 − (µx +i,l−1)2 ++ µz +i,l−1cos(θi,l). +(1c) +If |µx +i,l−1| > 0.99999, then +µx +i,l = +µx +i,l−1 +|µx +i,l−1|cos(θi,l) +(2a) +µy +i,l = sin(θi,l)cos(φi,l) +(2b) +µz +i,l = sin(θi,l)sin(φi,l) +(2c) +In the initial stage, we suppose that the MCP packet i +enters the dust particle zone form X = 0 at point (0,0,h) and +propagates along the direction of (µx +i,0, µy +i,0, µz +i,0) = (1, 0, 0). +The simulation process depends on uniformly-distributed, ran- +domly generated numbers ϵi,l, νi,l, and χi,l ∼ Uniform(0, 1), +which are used to calculate the random variables ∆Si,l, θi,l +and φi,l. +The ∆Si,l is the travelling distance between the scattering +particles Si,l and Si,l−1 and defined as, +∆Si,l = −ln(ϵi,l) +Cext +(3) +where Cext is the total extinction cross-section efficiency of +spherical dust particles with radius r. The total extinction +cross-section efficiency is expressed [18] as, +Cext = +� rmax +rmin +N0P(r)cextd(r), +(4) +where P(r) is the log-normal size distribution of dust particles +for both Earth [29] and Mars [16] environments. Here, N0 +is the number of dust particles per unit volume, and it can +be expressed as a function of visibility (Vb) [16], which is +represented as, +N0 = +15 +0.034744Vb +� 2rmax +0 +πr2P(r)dr +. +(5) +On Earth, dust particle radius varies between 1 to 150 µm. +Therefore, when considering the approximate equality of the +effective diameter of the dust particles and the wavelength of +the THz frequency utilised in this study, we can use the Mie +approximation [30] to infer the total extinction cross-section +(cext) [19], which is the sum of the absorption cross-section +and scattering cross-section. The cext is expressed by the Mie +solution for spherical particles with dielectric constant ϵ [23] +as, +cext = k3rλ2 +2 +(c1 + c2(kr)2 + c3(kr)3) +(6) +where, +c1 = +6ϵ′′ +(ϵ′ + 2)2 + (ϵ′′)2 +(7a) +c2= ϵ′′�6 +5 +7(ϵ′)2 + 7(ϵ′′)2 + 4ϵ′ − 20 +[(ϵ′ + 2)2 + (ϵ′′)2]2 +� ++ 1 +15 + +5 +3[(2ϵ′ + 3)2 + 4(ϵ′′)2]2 +(7b) +c3= 4 +3 +�(ϵ′ − 1)2(ϵ′ + 2) + [2(ϵ′ − 1)(ϵ′ + 2) − 9] + (ϵ′′)4 +[(ϵ′ + 2)2 + (ϵ′′)2]2 +�. +(7c) +The complex refractive index of dry dust particles on Earth can +be expressed as +� +3 + ı. 18.256 +f +[21], where f is the frequency +and ı is the imaginary unit √−1. +On the other hand, the total extinction cross-section of the +dust particles on Mars can be evaluated utilising the Rayleigh +approximations [25] because the effective diameter of the dust +particles on Mars (1-8 µm) is less than one-tenth of the +wavelength of the frequency [31] used in this study. Therefore, +the total extinction cross-section can be expressed as, +cext = 8 +3πk4r6���ϵr − 1 +ϵr + 2 +��� +2 ++ 12πkϵ +′′ +r r3 +1 +|ϵr + 2|2 + π +6 +k4r6σ2 +E2 +0ϵ2 +0 +|ϵr − 1|2 +(8) +where ϵr is the relative permittivity [25]. Moreover, we can +take the complex refractive index of dust particles on Mars +as 1.52 + 0.01i [16], [26] corresponding to the radius range +(0.5-4 µm) used in this study. +The scattering angle, θi,l, due to the ith MCP packet impact +with scatter l is represented as, +θi,l = +� +cos−1� 1 +2g +� +(1 + g2) − +� +1 − g2 +1 − g + 2gνi,l +�2�� +for g ̸= 0 +cos−1(2νi,l − 1) for g = 0 +(9) +where φi,l is the azimuth angle due to the same impact and +φi,l = 2πχi,l +(10) +and g =< cos(θ) >∈ [0, 1] is the asymmetry factor (here, +g = 0 refers to the isotropic scattering and 1 refers to forward +direct scattering). We assume g varies uniformly between 0.5 +and 1 for our simulations which is the average value for the +direct forward scattering. +After calculating the random variables ϵi,l, νi,l, and χi,l, we +can determine the (random) position of the scattering particle +Si,j (Xi,l, Yi,l, Zi,l) in the global coordinate system utilising + +6 +the equations (1), (2) and (3) as below. The position of the +scattering particles can be expressed as, +Xi,l = Xi,l−1 + ∆Si,l µx +i,l−1 +Yi,l = Yi,l−1 + ∆Si,l µy +i,l−1 +Zi,l = Zi,l−1 + ∆Si,l µz +i,l−1 +(11) +Successful transmittance occurs only for MCP packets that +reached the receiver interface at a distance of D from the +transmitter. If Xi,l >= D, this means that Si,l−1 is the last +scattering particle encountered by the MCP packet i before +it leaves the region boundary (X = D) from the receiving +interface. Therefore, we can stop the simulation process and +go to the next MCP packet (i + 1) after calculating its current +energy weight (Wi,l). The energy weight follows the Beer- +Lambert law, which determines how Wi,l−1 is related to the +Wi,l [18], [32]. Thus, the energy weight of an MCP packet +after collision with a scattering particle can be expressed as, +Wi,l = Wi,l−1 exp +�−Cext(Xi,l − Xi,l−1) +µx +i,l−1 +� +. +(12) +If Xi,l < D, this means MCP packet i is unable to reach the +receiver interface after impacting with l scatters. From this +point, we focus our interest more on the energy weight of the +MCP packet i and calculate the energy weight of the MCP +packet Wi,l using eq. (12). In this instance, we assume that +the initial energy weight of the MCP packet is a unit (i.e., +Wi,0 = 1), where we define a threshold (ϵt) value of 1×10−5 +to consider as the minimum energy weight that an MCP packet +can take after l impacts with the scatters. If the energy weight +of an MCP packet does not exceed this minimum threshold +(i.e., Wi,l < ϵt), the packet is assumed not to reach the receiver +and is recorded as such. Therefore, we can set j = l and +Wi,l = 0. +Based on the calculation of energy weights of the MCP +packets that reached the receiver interface, we can calculate +the transmittance of the THz EM wave by, +TMS = +�M +i=1 Wi,l exp +�−Cext(D − Xi,l) +µx +i,l +� +I0 +. +(13) +From our calculations, we noticed that eq. (13) does not +converge to a finite value for every simulation. Therefore, we +assume the transmittance to be zero when it is divergent and +set the simulation to the next Monte-Carlo process. Based on +the transmittance measurements at the end of this procedure, +we can calculate the specific attenuation AMS in (dB/m) +according to [18], as, +AMS = −4.343 ln(TMS) +D +. +(14) +V. THZ CHANNEL CAPACITY +To evaluate the channel capacity in the THz band, we can +decompose the received signal as a sum of the sub-bands, +where each sub-band channel is narrow and has a flat-band +response [33]. +The ith frequency sub-band is defined as ∆fi = fi+1 − fi +with power Pi under the constraint �NB +i=1 Pi ≤ Pt, where NB +refers to the total number of sub-bands, and Pt stands for +the total transmit power. In the ith narrow-band, the sub-band +capacity, Ci, is expressed in [33] as, +Ci = ∆fi log +� +1 + |hLoS|2Pi +∆fiSD(fi) +� +, +(15) +where SD is the power spectral density of the additive white +Gaussian noise (AWGN) and hLoS is the frequency-dependent +channel response for attenuation due to dust particles including +spreading loss and molecular absorption loss due to gas +molecules on the LoS signal propagation path. According to +[33], hLoS can be expressed as, +hLoS(τ) = αLosδ(τ − τLoS). +(16) +where αLos refers to the attenuation and τLoS refers to the +propagation delay due to dust particles and gas molecules +on the signal propagation path and τLoS = dLoS +c +. Where +dLoS is the signal travelling distance through dust, which +is D in this study, because we calculate the transmittance +using the MCP packets that reached the fixed receiver in- +terface. Also, the power spectral density of AWGN can be +expressed as SD(τ) = n0 +2 δ(τ) and, in the frequency domain, +PSD(SD(f))= +n0 +2 . Utilizing the Wiener-Khinchin theorem, +the frequency-dependent channel response for LoS attenuation +can be expressed as [33], +hLoS(τ) = |HLoS(f)|δ(τ − τLoS). +(17) +The free space direct ray or LoS channel transfer function, +HLoS, consists of the spreading loss function (HSpr), the +molecular absorption loss function (HAbs), and scattering loss +function due to dust particles (HDust), which can be expressed +as, +HLoS(f) = HSpr.HAbs.HDuste−j2πfτLoS. +(18) +The free space path loss or the spreading loss (PLSpr) +measures the fraction of power lost by a beam with frequency +f over a distance D in a vacuum, and it can be expressed +according to [34] as, +PLspr = +�4πDf +c +�2 +, +(19) +where c is the speed of light in the medium. Thus, according +to [33] the corresponding channel transfer function for the +spreading loss can be expressed as, +Hspr = (PLspr)−1/2 = +� +c +4πDf +� +. +(20) +The molecular absorption loss measures the fraction of power +converted to kinetic energy due to molecular vibration when +EM waves propagate through molecules in the atmosphere. +Thus, when transmitting frequency f through a homogeneous +medium between a transmitter and receiver at a distance D, +the molecular absorption loss is obtained with the help of the +Beer-Lambert law [34], which is represented as, +PLabs = ek(f)D, +(21) +where k(f) = � +i,g ki +g(f) and ki +g(f) is the monochromatic +absorption coefficient of the ith isotopologue of gth gas at + +7 +frequency f. When calculating the absorption coefficient, we +consider water vapour and nine other gases for Earth and six +gases for Mars (see Table I), except for Argon. This allows +us to consider the vastly different gas concentrations between +the two planets. The monochromatic absorption coefficient for +each isotopologue of a particular gas in the Martian and Earth +atmosphere at frequency f is provided in [35], +ki +g(f) = Si +g(T)F i(f), +(22) +where Si +g(T) is the line intensity at temperature T (210K for +Mars) referenced to the temperature 296K of the ith isotopo- +logue of gth gas, which can be easily calculated using the high- +resolution transmission (HITRAN) molecular spectroscopic +data. Where, F i is the spectral line shape function at frequency +f. In the lower atmosphere on Earth, pressure broadening of +spectral lines dominates the line shape and a Lorentz profile +can be assumed as the line shape function and it is given by +[35], +F i +L(f) = 1 +π +γ(p, T) +γ(p, T)2 + [f − (f ig + δ(Pref)P)]2 +(23) +where f i +g is the resonant frequency for the isotopologue i of +gas g, γ(P, T) is the Lorentzian (pressure-broadened) HWHM +for a gas at pressure P (atm), temperature T (K), and δ(Pref) +is the pressure shift at reference pressure (Pref= 1 atm). +Since Doppler-broadening dominates the line shape in low- +pressure environments such as Martian environment, a Gaus- +sian profile can be assumed as the line shape function, and it +is given by, +F i +G(f) = +� +ln 2 +παi +D +2 exp +� +− (f − f i +g)2 ln 2 +αi +D +2 +� +, +(24) +where αi +D is the Doppler broadening half-width, +αi +D = f i +g +c +� +2NAkBT ln 2 +M i +, +(25) +where M i is the molar mass of isotopologues which can be +obtained from the HITRAN database [35], and NA and kB +are the Avogadro and Boltzmann constants. +Thus, according to [33] the corresponding channel transfer +function for the molecular absorption loss due to the gas +molecules on the atmosphere can be express as, +Habs = (PLAbs)−1/2 = e− 1 +2 k(f)D, +(26) +Finally, HDust is the transfer function for the attenuation +due to dust particles that can be expressed as following the +relationship between the transfer functions and the attenuation +functions for spreading loss and molecular absorption loss, +respectively, as well as utilising the dust attenuation function +in eq. (14) in dB, which is represented as, +Hdust = +1 +√ +10−0.4343 ln (TMS) . +(27) +Therefore, substituting the functions derived for HSpr, HAbs, +and HDust, to eq. (18), we can calculate the channel transfer +function. Moreover, in this study, we consider one narrow +frequency band for each environment, which is 0.22 - 0.24 +(a) Earth: Transmittance. +(b) Earth: Attenuation (dB/m). +(c) Martian: Transmittance. +(d) Martian: Attenuation (dB/m). +Fig. 3: Simulation measurements of a) the transmittance and +b) the attenuation for a THz beam of 0.24 THz and 1.64 THz +frequency for Earth and Mars by varying MCP packets from +10 to 10000 while fixing dust particle number (100/10000) +on the propagation path, visibility and the distance (10 m) +between transmitter and the receiver. +THz for Earth and 1.64 - 1.67 THz for Mars. Therefore, we +can suppose that Pi = Pt and the eq. (15) can be rewritten as, +C = ∆f log +� +1 + |hLoS|2Pt +∆fSD(f) +� +. +(28) +VI. RESULTS AND DISCUSSION +A. Transmittance and Attenuation measurements through +Monte-Carlo Simulations +This subsection presents the simulation results for the +transmittance and attenuation measurements of the THz link +and the generated estimation models for the THz attenuation +due to dust particles on the beam propagation path by varying +parameters such as the MCP packets, visibility, dust particle +number, the distance between transmitter and the receiver, and +EM frequency. When simulating data for a targeted parameter, +we have kept the other parameters constant to make the +interpretation easy (see Table II). Thus, we have chosen a +frequency of 1.64 THz as the constant frequency for Mars [26], +and 0.24 THz frequency for Earth [36], [37] corresponding +to low molecular absorption and high transmission distance. +However, it is crucial to consider molecular absorption on +Mars, even though it has a thin atmosphere with very low water +vapour concentration. Moreover, we have considered 100 dust +particles on the beam propagation path for a 10 m fixed +distance between the transmitter and receiver, as explained in +section III for simulations on Earth. It is unrealistic to consider +the same amount of dust particles for the Mars simulations be- +cause of the tiny particle sizes on Mars. Therefore, considering + +103 +Transmittance +10-4 +10-5 +10-6 +101 +102 +103 +104 +Number of MCP packetsSimulated data +Fitted curve +-- Prediction bounds +6 +3 +2 +101 +102 +103 +104 +Number of MCP packets100 +10-10 +10-20 +Transmittance +10~30 +10-40 +10~50 +101 +103 +102 +104 +Number of MCP packets100 +Simulated data +Fitted curve +.-- Prediction bounds +80 +Attenuation [dB/m] +60 +40 +20 +0 +-20 +101 +102 +103 +104 +Number of MCP packets8 +the blockage that this dust particle creates and the proportional +relationship between blockage area and dust particle radius, we +consider 10000 dust particles for Mars corresponding to 100 +dust particles on Earth. When selecting the number of MCP +packets, we considered 10000 packets in this study. +TABLE II: Channel conditions and simulation settings on +Earth and Mars. +Parameter +Earth +Mars +Frequency +0.24 THz +1.64 THz +MCP packets +104 +104 +Dust Density +102 per 10m +104 for 10 m +Dust radius +1–150 microns +0.5–4 microns +Dust size distribution +log-Normal +log-Normal +Antenna height +50 m +50 m +Approximation +Mie +Rayleigh +Distance +1 - 200 m +1 - 200 m +Temperature +288 K +210 K +Surface Pressure +1013 mb +6.1 mb +Surface density +1.29 Kg/m3 +0.02 Kg/m3 +Figure 3 illustrate the simulated data for the transmittance +and attenuation measurements using the simulation setup ex- +plained in section IV for Earth and Mars environments by +varying the number of MCP packets from 10 to 10000, while +keeping the other parameters constant. As we can see from +the figures, the transmittance measurements for both Earth +and Mars environments (see Fig. 3 (a) and (c)) are increasing +rapidly when the MCP packets increase at the beginning +up to 100. After that, it converges to a particular value +corresponding to each environment. Furthermore, attenuation +measurements decrease following a power function for both +Earth and Mars environments (see Fig. 3 (b) and (d)) and +converge approximately to a value of 3.6 dB/m and 2.8 +dB/m, respectively. In addition, the fitted power function +for attenuation against the MCP packets (NMCP ) can be +expressed as Attenuation (dB/m) = 2145N −2.721 +MCP ++ 2.839 +for Earth and Attenuation (dB/m) = 410N −0.8485 +MCP ++ 3.727 +for Mars. +Next, we investigated the effect of visibility on the trans- +mittance and attenuation measurement for Earth and Mars +environments by varying the parameter values from 10 to +10000 (see Fig. 4). Generally, when the visibility increases +between the transmitter and the receiver, we will see fewer +dust particles on the beam propagation path with a high +distance variance between the particles. Therefore, we can +expect high transmittance and low attenuation measurements +when the visibility increases. According to Fig. 4 (a), the +transmittance measurements of the Earth’s environment are +increasing dramatically, with the visibility and attenuation +measurements (see Fig. 4 (b)) decreasing following a power +function as expected. Moreover, the attenuation measurements +approximately converge to 2.1 dB/m value with an increase +of visibility near 10000 m, which we can consider as clear sky +condition. The fitted power function for the attenuation against +the visibility (V ) can be expressed as Attenuation (dB/m) = +63.41V −0.9694+2.105 for the Earth environment. On the other +hand, the transmittance measurements for the Mars environ- +ment (see Fig. 4 (c)) show an increasing trend with visibility. +However, transmittance measurements for some visibility pa- +rameter values diverge from the trend because, according to +our simulation process, each MCP packet will randomly select +the scatters. Therefore, even if we have low dust density on +the beam propagation path with high visibility, the transmit- +tance can be low due to high collision with dust particles. +Corresponding to the transmittance measurements, we can see +a slight linear decrease in attenuation on Mars (see Fig. 4 +(d)) with increased visibility. The fitted linear function can be +expressed as Attenuation (dB/m) = −4.184×10−5V +4.256. +(a) Earth: Transmittance. +(b) Earth: Attenuation (dB/m). +(c) Martian: Transmittance. +(d) Martian: Attenuation (dB/m). +Fig. 4: Simulation measurements of a) the transmittance and +b) the attenuation for a THz beam of 0.24 THz and 1.64 +THz frequency for Earth and Mars, respectively, by varying +the visibility from 10 to 10000 m while fixing MCP packets +(10000) and the distance (10 m) between transmitter and the +receiver. +The dust particle density can vary unpredictably with the +wind in a dust storm on Earth and Mars. There can be +time windows with very low and high dust particles on the +beam propagation path, which will be perfect for transmission. +To investigate the effect of dust particle count, we have +measured the transmittance and attenuation for Earth and +Mars environments by varying dust particle numbers from +10 (very low) to 10000 (very high). As shown in Fig. 5 (a) +and (c), the transmittance measurement drops dramatically +to near zero with the increase of dust particles for both +environments. This rapid transmittance drop happens due to +the high amount of scatters on the THz beam propagation path +that each MCP packet should randomly collide. Moreover, the +attenuation measurements (see Fig. 5 (b) and (d)) increased +rapidly following a power function for both environments. The +fitted power function for attenuation against the dust particle +number (DP N) can be expressed as Attenuation (dB/m) = +0.4423D0.2579 +P N ++ 1.213 for Earth and Attenuation (dB/m) = +7.534D0.04617 +P N +− 7.262 for Mars. Furthermore, as we can no- +tice, attenuation measurements do not converge to a particular +value as in previous cases when increasing the number of +dust particles on the beam propagation path. Therefore, we + +5 +4.8 +4.6 +4.4 +Attenuation [dB/m] +4.2 +4 +3.8 +3.6 +3.4 +Simulated data +3.2 +Fitted curve +Prediction bounds +.. +3 +101 +102 +103 +104 +visibility (m)X10-3 +9 +8 +7 +6 +Transmittance +5 +4 +3 +2 +1 +Oc +10l +102 +103 +104 +visibility (m)9 +Simulated data +Fitted curve + Prediction bounds +8 +7 +6 +5 +4 +3 +2 +102 +103 +101 +104 +visibility (m)X10-4 +1.8 +1.6 +1.4 +Transmittance +1.2 +0.8 +0.6 +0.4 +C +0.2 +101 +102 +103 +104 +visibility (m)9 +can expect a communication blackout in a regional/global dust +storm situation which will boost the dust particle density in +the communication area. +(a) Earth: Transmittance. +(b) Earth: Attenuation (dB/m). +(c) Martian: Transmittance. +(d) Martian: Attenuation (dB/m). +Fig. 5: Simulation measurements of a) the transmittance and +b) the attenuation for a THz beam of 0.24 THz and 1.64 THz +frequency for Earth and Mars, respectively, by varying the +dust particle number on the beam propagation path from 10 +to 10000 while fixing the number of MCP packets (10000) +and the distance (10 m) between transmitter and the receiver. +As mentioned above, dust density on the beam propaga- +tion path can vary significantly on Earth and Mars due to +the unpredictable wind, temperature and pressure behaviour. +Therefore, to conduct more realistic simulations, we have +investigated the effect of distance between the transmitter and +the receiver when we have various dust densities that vary with +the distance. Dust density usually measures the number of dust +particles per unit volume. However, in this study, we define it +as the number of dust particles per meter for simplicity because +the THz beam is assumed to be cone shape, and its face area is +very tiny. This means that if we consider 10 dust particles per +meter (10/m) for 100 m, we assume that 1000 dust particles +are uniformly distributed on the beam propagation path. As +we can see in Fig. 6 (a), the transmittance measurements on +Earth drop dramatically with the distance, and when increasing +the dust density, the transmittance measurements reaches near +zero rapidly beyond 100 m. However, on Mars (see Fig. 6 (c)), +the transmittance measurements decreases slightly compared +to Earth. Moreover, we can clearly see that the transmittance +measurements for 25/m dust density are significantly lower +than the dust density at 10/m, as expected. However, we can +not see much difference between the transmittance measure- +ments for 50/m and 100/m dust densities up to 150 m. On the +other hand, attenuation measurements (see Fig. 6 (b) and (d)) +advance with the increase of distance and dust density for both +environments corresponding to transmittance measurements. +Furthermore, we investigated the impact of frequency on +(a) Earth: Transmittance. +(b) Earth: Attenuation (dB). +(c) Martian: Transmittance. +(d) Martian: Attenuation (dB). +Fig. 6: Simulation measurements of a) the transmittance and +b) the attenuation for a THz beam of 0.24 THz and 1.64 THz +frequency for Earth and Mars, respectively, by varying the +distance between transmitter and the receiver from 1 to 200 +m for different particle number densities of 10/m, 25/m, 50/m +and 100/m on the beam propagation path while fixing MCP +packets (10000). +(a) Earth: Transmittance. +(b) Earth: Attenuation (dB/m). +(c) Martian: Transmittance. +(d) Martian: Attenuation (dB/m). +Fig. 7: Simulation for a THz beam by varying the frequency +from 0.1 to 10 THz while fixing dust particle number (100) +on the beam propagation path, MCP packets (10000), and the +distance (10 m) between the transmitter and the receiver. +the transmittance and attenuation measurements on Earth +and Mars. As illustrated in Fig. 7 (a), the transmittance +measurements for Earth’s environment decrease following an +exponential function with the frequency increase from 0.1 to + +0.014G +0.012 +0.01 +Transmittance +0.008 +0.006 +0.004 +0.002 +0 +11 +101 +102 +103 +104 +Particle Number6 +5 +3 +2 +Simulated data +Fitted curve +Prediction bounds +10' +102 +103 +104 +Particle Number0.09 +0.08Q +0.07 +0.06 +Transmittance +0.05 +0.04 +0.03 +0.02 +0.01 +0 +101 +102 +103 +104 +Particle Number4.5 +4 +3.5 +Attenuation [dB/m] +3 +2.5 +2 +Simulated data +Fitted curve +1.5 +Prediction bounds +1 +0.5 +101 +102 +103 +104 +Particle Number100 +PN = 10/m +PN = 25/m +- PN = 50/m +- PN = 100/m +10-5 +Transmittance +10-20 +50 +0 +100 +150 +200 +Distance (m)250 +200 +[dB +150 +Attenuation +PN = 10/m +100 +PN = 25/m +PN = 50/m +-..- PN = 100/m +50 +0 +0 +50 +100 +150 +200 +Distance (m)PN = 10/m +PN = 25/m +PN = 50/m +- PN = 100/m +102 +Transmittance +10~6 +50 +100 +150 +200 +0 +Distance (m)80 +70 +60 +Attenuation [dB] +50 +40 +30 +PN = 10/m +20 +PN = 25/m +PN = 50/m +PN = 100/m +10 +0 +50 +100 +150 +200 +Distance (m)10 +10-10 +Transmittance +10-30 +10-40 +0 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +4 +Frequency (THz)50 +Simulated data +Fitted curve +40 +- Prediction bounds +Attenuation [dB/m] +30 +20 +10 +0 +-10 +0 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +4 +Frequency (THz)X10-4 +2.5 +2 +Transmittance +1.5 +0.5 +0 +0 +2 +4 +6 +8 +10 +Frequency (THz)5.2 +5 +4.8 +Attenuation [dB/m] +4.6 +4.4 +4.2 +4 +3.8 +3.6 +0 +2 +4 +6 +8 +10 +Frequency (THz)10 +(a) Particle number count. +(b) Transmittance. +(c) Attenuation (dB/m). +Fig. 8: Comparison of simulations for time-dependent turbulence (turmoil after 5 seconds) corresponding to the measurements +of a) particle number count, b) the transmittance, and c) the attenuation (dB/m) considering Earth and Mars environments. +4 THz. We were unable to calculate transmittance measure- +ments following our simulation process beyond the 4 THz +frequency limit since we are considering the dust particles +with a radius of 1 to 150 microns for Earth environment, +wavelengths can be comparably low or approximately equal +to the dust particles’ size after some frequencies threshold, +creating more difficulties for data transmission. The corre- +sponding attenuation for the transmittance measurements on +Earth increases following a power function that can be fitted +as Attenuation (dB/m) = 2.277D2.054 +P N . On the other hand, +the transmittance and attenuation measurements for the Mars +environment do not show a particular increase or decrease +trend. However, we noticed that the attenuation measurements +vary around 4.2 dB/m with the frequency increase from 0.1 +to 10 THz. +Finally, we investigated the effect of time-dependent tur- +bulence on the transmittance and the attenuation of the THz +signal on Earth and Mars, which corresponds to the 0.24 +THz and 1.64 THz frequencies, respectively and the distance +between the transmitter and the receiver for 10 m. Here, we +compare Earth and Mars simulation scenarios in which we +assume that the dust particle number on the beam propagation +path will suddenly increase due to the unpredictable behaviour +of wind after 5 s. In the first 5 s, we assume that the dust +particle number on the beam will vary between 10-20 in +the Earth environment and 100-200 in the Mars environment +in clear sky conditions. After five seconds, the dust particle +number on the beam will increase between 100-200 on Earth +and 10000-20000 on Mars due to the sudden wind turbulence +in the communication area. As demonstrated in Fig. 8 (a), dust +particle number on the beam propagation path is low in the +first 5 s compared to the next 5 s for both environments. Also, +the dust particle number is higher for the Martian environment +than the Earth environment in the considered time interval. +When scrutinising the transmittance measurements (see Fig. +8 (b)), we can notice that transmittance drops suddenly after +5 s for both environments. However, average transmittance +measurement values are approximately similar within the +first 5 s. Also, the transmittance measurements on Mars +after the turbulence are significantly low compared to Earth. +Corresponding to the transmittance measurements, attenuation +measurements (see Fig. 8 (c)) increases dramatically after 5 +s and are high on Mars, concluding that the turbulence effect +on Mars should be investigated thoroughly. +Fig. 9: Measurements of Transmittance and Attenuation of +0.24 THz and 1.64 THz links for Earth and Mars due to the +sudden movement of dust particles on the beam propagation +path for a fixed transmitter and receiver distance of 1 m. +B. Channel Capacity simulation for Earth and Mars Under +Dust storm +This subsection investigates the channel capacity measure- +ment of the THz links considering two scenarios. In the +first scenario, we assume that there are time windows when +the number of dust particles on the THz beam propagation +path drops, creating opportunities to communicate with high +data rates for both Earth (0.24 THz) and Mars (1.64 THz) +environments. Here we analyse the channel capacity for Earth +and Mars environments considering spreading loss and Molec- +ular absorption loss with the THz attenuation due to dust +on the beam propagation path. Also, we investigate channel +capacity variations in this scenario for different transmitter +powers. In the second scenario, we investigate the channel + +105 +On Earth +On Mars +104 +Dust Particle Count +103 +102 +101 +2 +3 +4 +5 +6 +7 +8 +9 +10 +1 +Time (s)10' +On Earth +On Mars +Transmittance +10-3 +10-4 +10-5 +10~6 +2 +3 +4 +5 +6 +7 +8 +9 +10 +Time (s)6 +5.5 +5 +4.5 +Attenuation +A +3.5 +2.5 +2 +On Earth +-On Mars +1.5 +1 +2 +3 +4 +5 +6 +7 +8 +6 +10 +Time (s) number +10 +On Earth +- On Mars +Dust particle +10 +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +Transmittance +On Earth +On Mars +10 +2 +4 +8 +6 +10 +12 +14 +16 +18 +20 +[dB/m] +- On Earth +On Mars +Attenuation +40 +20 +0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +Time [s]11 +Fig. 10: Channel capacity variation comparison for the sce- +nario of a sudden drop in the number of dust particles on +the beam due to the wind behaviour on Earth and Mars +environments with considering spreading loss and molecular +absorption loss. +capacity variation with the distance in clear sky and dust storm +conditions. +In our first model, we assume that the dust density varies +randomly for the Earth environment from 100 and 200 and +Mars environment from 10000 to 20000 particles correspond- +ing to a 1 m distance between the transmitter and the receiver. +Here we know that considering a 1 m distance for simulation +is unrealistic. However, in this scenario, we need to infer +the effect of the sudden dust particle drop for the channel +capacity measurements. Therefore, it is adequate to consider +a 1 m distance for the experiment. As we mentioned above, +the significant variation of dust particles on Earth and Mars +is in its effective radius. On Earth, the average effective +radius of a dust particle varies between 1 and 150 microns +[18], and on Mars, it varies between 0.5 to 4 microns [20]. +Therefore, to measure transmittance/attenuation considering +approximately similar beam-blocking areas by dust particles +on both Earth and Mars, we should consider 100 times more +dust on Mars than on Earth, following the relationship between +dust effective radius and area. Moreover, we sampled the dust +particle number for each environment every second for 20 +s. In addition, we assume there are two-time intervals (t=[7 +s,9 s], t=[15 s,17 s]) when the dust particle number drops +to less than 30 and 300 particles (see Fig. 9) due to the +unpredictable behaviour of wind. Such time intervals might +represent occasions when the wind that causes dust particles +to be suspended in the atmosphere falls away to near zero. +Furthermore, as we can see from Fig. 9, the relative dust +particle density decrease at the interval centred on t=16 s is +much higher than at the interval centred on t=8 s for both +environments. We noticed that corresponding to the low dust +density, the transmittance measurement is higher at the interval +15-17 s than at the 7-9 s interval, and the attenuation shows +the opposite variation to transmittance. In principle, those time +intervals represent attractive time windows for communication +in both environments because of lower attenuation due to the +momentary absence of dust particles in the channel. +(a) On Earth +(b) On Mars +Fig. 11: Channel capacity measurement variation for the +scenario of a sudden drop in the number of dust particles on +the beam due to the wind behaviour for different Transmitted +powers (1, 5, 10 dBm) of antenna with considering spreading +and molecular absorption (a) on Earth and (b) on Mars. +Figure 10 illustrates the channel capacity variation in the +clear sky and dust storm conditions on Earth and Mars for the +same dusty scenario. When scrutinising Fig. 10, we noticed +that in clear sky conditions, channel capacity on Mars is +approximately 1.39 ×1012 bits/s, and on Earth, its nearly +0.85 ×1012 bits/s. This shows more than 550 GB/s differ- +ence between the channel capacity measurement in clear sky +conditions on Mars and Earth due to much higher molecular +absorption on Earth. Also, it is noticeable that dust appears +to have a more significant relative effect on channel capacity +on Mars than on Earth in a dust storm situation due to the +high number of tiny dust particles on the beam propagation +path. Moreover, channel capacity measurements in dust storm +situation on Earth is lower than in clear sky condition, but +the difference is minimal. However, the difference on Mars +is relatively enormous, and it is approximately two orders of +magnitude less than clear sky conditions. On the other hand, +channel capacity measurements in dust storm condition on +Mars is higher by approximately five orders of magnitude than +in dust storm condition on Earth. In addition, the free space +path loss is constant in this scenario because it only depends on +the carrier frequency and the distance between the transmitter +and the receiver, which are both constant in this simulation +process. +In Fig. 11, we investigated the channel capacity measure- +ment variations for different transmitter power in discrete +time windows considering the free space path loss and the +molecular absorption loss effect on the channel. This figure +shows that the channel capacity increases as antenna trans- +mitter power increases for both Earth and Mars environments +by approximately 5 to 20 GB/s. Also, we noticed that we +could have reliable communication links with high channel +capacities when communicating in the time windows with +low dust densities on Mars. Moreover, the channel capacity +measurements show a significant variation on Earth compared +to Mars for all transmitted powers. Therefore, these measure- +ments imply that we can reach high channel capacities on Mars +than on Earth using lower transmitter power antennas. + +×1012 +1.5 +1.4 +1.3 +1.2 +On Earth: Clear Sky +On Earth: Dust Storm +On Mars: Clear Sky +On Mars: Dust Storm +0.9 +0.8 +0.7 +2 +8 +10 +12 +4 +6 +14 +16 +18 +20 +Time[s]X101 +8.3 +On Earth: Tx power = 1 dBm +On Earth: Tx power= 5 dBm +8.25 +- On Earth: Tx power = 1O dBm +8.2 +Channel Capacity [bits/s] +8.15 +8.1 +8.05 +8 +7.95 +7.9 +7.85 +7.8 +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +Time「s]1.34 +1.32 +[bits/s] +1.3 +.28 +Channel Capacity [ +.26 +.24 +1.22 +1.2 +On Mars: Tx power= 1 dBm +1.18 +On Mars: Tx power = 5 dBm +On Mars: Tx power = 10 dBm +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +Time[s]12 +Fig. 12: Channel capacity measurements by varying the dis- +tance between the transmitter and the receiver with and without +dust storms situations on Mars and Earth. +In our second and final simulation scenario, we inves- +tigated the channel capacity for various distances between +the transmitter and the receiver for both Earth and Mars +environments, comparing clear sky and dust storm conditions +with different dust densities per meter (See Fig. 12). The +transmitter power was taken as 10 dBm in this simulation. +Here, we allowed the dust particle count density to vary as +a factor of distance to simulate more realistic dust storm +conditions. For the Earth’s environment, we have considered +dust storms that result in 10-20 (very low) and 100-200 (very +high) dust particles per meter dust densities on the THz beam +propagation path. Similar conditions for the mars environment +were considered by comparing the dust particle sizes with +Earth. Thus, we assumed that dust storms on Mars would +carry 100-200 and 1000-2000 dust particles per meter to the +beam propagation path. However, we should have considered +10000-20000 dust particles per meter on Mars. Nevertheless, +due to computational difficulties with high dust densities +with increasing the distance between the transmitter and the +receiver, we are considering above mentioned numbers for +channel capacity measurements. Moreover, we have taken +account of spreading loss and molecular absorption loss when +calculating the channel capacities for each distance. +As shown in Fig. 12, the channel capacity decreases grad- +ually for clear sky conditions for both environments showing +high channel capacities in the considering distance range. +However, the decrement is high on Earth due to high molecu- +lar absorption. Moreover, the channel capacity measurements +decrease dramatically on Earth with the distance for 100- +200/m dust storm conditions, showing communication black- +out beyond 70 m distance. Also, at 10-20/m dust storm, +channel capacity measurements decrease slowly, showing that +this dust particle number on the beam propagation path is +not a massive issue for achieving high channel capacities. +However, Earth’s channel capacity is significantly dropping +when compared with the channel capacity decrement on Mars +for the 100-200/m dust particle density. Again, 100-200/m +dust particles on the beam propagation path on Mars do +not significantly affect the channel capacity measurement. +Therefore, we can neglect the THz link budget degradation +due to the small amount of dust on Mars. In addition, when +investigating the high dust particle number density effect on +the channel capacity on Mars (1000-2000/m), the channel +capacity measurements are computationally difficult when the +distance is greater than 60 m. However, we can notice that +the channel capacity measurements for Mars are decreasing +rapidly with the increase in the distance. Also, we can see that +the channel capacities are equal at a distance of approximately +60 m for clear sky conditions on Earth and the high dust +density scenario on Mars. +VII. CONCLUSION +High-speed, reliable communication between devices on +Earth and Mars is needed to fulfil future communication +requirements. In this study, we investigated the impact of at- +mospheric dust and dust storms for communication using THz +links, utilising a modified Monte Carlo simulation algorithm. +The calculated transmittance and attenuation measurements +are based on Mie and Rayleigh approximations depending +on the dust particle sizes and carrier frequency utilised for +communication on the two planets. Moreover, we presented +a channel capacity model and analysed it for two different +time-dependent and distance-dependent scenarios. The Monte- +Carlo simulation results show that attenuation measurements +decrease for both Earth and Mars environments when the +MCP packets and visibility increase. In addition, for both +environments, the attenuation increases with higher dust par- +ticle number on the beam propagation path and distance +between the transmitter and the receiver. We noticed the exact +attenuation behaviour with the increased frequency for the +Earth’s environment. However, the attenuation measurements +vary around a constant value for the Mars environment. When +scrutinising the channel capacity measurements from the time- +dependent scenario, we can conclude that the time windows +showing sudden dust particle density drops create the best +communication opportunities with high data rates. Also, we +noticed that the channel capacity measurements dramatically +drop with the increase in distance between the transmitter and +the receiver in severe dust storm situations on both Earth (100- +200/m) and Mars (1000-2000/m) environments, even if we +use low molecular absorption frequencies and high transmitter +power antennas. However, the impact from the local dust +storm is negligible on Mars (100-200/m) but should be further +investigated on Earth (10-20/m). +ACKNOWLEDGMENT +This publication came from research conducted with the +financial support of Science Foundation Ireland (SFI) and +the Department of Agriculture, Food and Marine on behalf +of the Government of Ireland (Grant Number [16/RC/3835] +- VistaMilk), the support of YL Verkot, Finland, and US +National Science Foundation (NSF) ECCS-2030272 grant. + +Channel Capacity [bits/s] +On Mars: Clear Sky +- On Mars: Dust Storm (100-200/m) +On Mars: Dust Storm (1000-2000/m) +- On Earth: Clear Sky +On Earth: Dust Storm (10-20/m) +On Earth: Dust Storm (100-200/m) +20 +40 +60 +80 +100 +120 +140 +160 +180 +200 +Distance [m]13 +REFERENCES +[1] A. Shafie, G. N. Yang, C. Han, J. M. Jornet, M. Juntti, and T. Kurner, +“Terahertz communications for 6g and beyond wireless networks: Chal- +lenges, key advancements, and opportunities,” IEEE Network, 2022. +[2] I. 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He is currently pursuing +a PhD degree with the Department of Computing +and Mathematics, Walton Institute, South East tech- +nological University, Ireland. His current research +interests lie in Mathematical modelling and 5G/6G +Wireless communication and sensing. +BERNARD +BUTLER +[SM’22] +(bernard.butler@setu.ie) +received +his +PhD +from South East Technological University, Ireland. +He +was +a +Senior +Research +Scientist +in +the +U.K.’s National Physical Laboratory, focusing on +mathematics of measurement and sensing. He is +a Lecturer in SETU and is CONNECT Funded +Investigator and VistaMilk Academic Collaborator +with the Walton Institute, SETU. Research interests +include machine learning, wireless comms and edge +networking. +SASITHARAN BALASUBRAMANIAM [SM’14] +(sasi@unl.edu) received his Bachelors in Engineer- +ing and PhD degree from the University of Queens- +land, Australia in 1998 and 2005, respectively, and +Masters of Engineering Science from Queensland +University of Technology in 1999. He was a past +recipient of the Science Foundation Ireland Starter +Investigator Research Grant. He was also a past re- +cipient of the Academy of Finland Research Fellow +at Tampere University, Finland. He was previously +the Director of Research at the Walton Institute, +South East Technological University, Ireland. He is currently an Associate +Professor at the School of Computing, University of Nebraska-Lincoln. He is +currently the Editor-in-Chief of IEEE Transactions on Molecular, Biological +and Multi-scale Communications as well as an Associate Editor for IEEE +Transactions on Mobile Computing. He was an IEEE Distinguished Lecturer +for the IEEE Nanotechnology Council in 2018. His research interests lie in +molecular and nano communications, Internet of Bio-Nano Things, as well as +5G/6G networks. +YEVGENI +KOUCHERYAVY +[SM’08] +(yevgeni.koucheryavy@yl-verkot.com) +received +the Ph.D. degree from the Tampere University +of Technology, Finland, in 2004. He is currently +a +Full +Professor +with +the +Unit +of +Electrical +Engineering, +Tampere +University, +Finland. +He +has authored numerous publications in the field +of advanced wired and wireless networking and +communications. +His +current +research +interests +include various aspects in heterogeneous wireless +communication networks and systems, the Internet +of Things and its standardization, and nanocommunications. +Mehmet Can Vuran [M’07] (mcv@unl.edu) was +born in Istanbul, Turkey. He received his B.Sc. +degree in Electrical and Electronics Engineering +from Bilkent University, Ankara, Turkey, in 2002. +He received his M.S. and Ph.D. degrees in Electrical +and Computer Engineering from Georgia Institute +of Technology, Atlanta, GA., in 2004 and 2007, re- +spectively. Currently, he is the Dale M. Jensen Chair +Professor in Computing at the School of Computing +at the University of Nebraska-Lincoln. Dr. Vuran +has been recognized as a Highly Cited Researcher +three years in a row by Thomson Reuters ”in recognition of ranking among +the top 1% of researchers for most cited documents in Computer Science”. +Dr. Vuran was awarded an NSF CAREER award for the project “Bringing +Wireless Sensor Networks Underground”. He is a Daugherty Water of Food +Institute Fellow and a National Strategic Research Institute Fellow. He serves +on the editorial boards of IEEE Transactions on Wireless Communications, +IEEE Transactions on Mobile Computing, and IEEE Transactions on Network +Science and Engineering. His research interests are in 6G networks, the +Internet of Things (IoT), agricultural wireless networks, wireless underground +communications, and vehicular communications. + diff --git a/etAyT4oBgHgl3EQfw_lx/content/tmp_files/load_file.txt b/etAyT4oBgHgl3EQfw_lx/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..467fee23ae4c5389267ae82a284eabb2244280b0 --- /dev/null +++ b/etAyT4oBgHgl3EQfw_lx/content/tmp_files/load_file.txt @@ -0,0 +1,1003 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf,len=1002 +page_content='1 Comparative Analysis of Terahertz Propagation Under Dust Storm Conditions on Mars and Earth Lasantha Thakshila Wedage, Bernard Butler, Sasitharan Balasubramaniam, Yevgeni Koucheryavy, Mehmet C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Vuran Abstract—Reliable Terahertz (THz) links are necessary for out- door point-to-point communication with the exponential growth of wireless data traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' This study presents a modified Monte Carlo simulation procedure for estimating THz link attenuation due to multiple scattering by dust particles on the THz beam propagation path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Scattering models are developed for beams through dust, based on Mie and Rayleigh approximations for corresponding frequencies for Earth (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='24 THz) and Mars (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='64 THz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' The simulation results are compared, considering parameters such as the number of Monte-Carlo photon (MCP) packets, visibility, dust particle placement density along the beam, frequency, and distance between the transmitter and the receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Moreover, a channel capacity model was proposed, considering THz link attenuation due to dust storms, spreading loss and molecular absorption loss for Earth and Mars outdoor environ- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Simulation results for Earth show that link attenuation increases with dust particle placement density, distance and frequency, and attenuation decreases with visibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' On Mars, similar results are obtained, except that the attenuation is variate around a constant value with the frequency increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Channel capacity is estimated for Earth and Mars environments consid- ering time and distance-dependent scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Time windows that show a sudden drop of dust particles along the beam provide opportunities to communicate with high reliability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Moreover, increasing the distance between the transmitter and receiver severely reduces the channel capacity measurement in strong dust storm conditions in both environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Our study has found that weak dust storms have relatively little effect on Mars, but much larger effects on Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Index Terms—THz Communication, Atmosphere, Attenuation, Scattering, Dust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' INTRODUCTION S IXTH generation (6G) wireless networks aim to push the frequency spectrum into the Terahertz (THz) band to fulfill rising capacity demands and requirements, given the opportunity for higher bandwidths [1]–[3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' The 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='1 to 10 THz frequency range has the potential to (1) realize high bandwidth transmissions that can allow hundreds of GB/s data rates for communication [4]–[6], and (2) provide new opportunities to create miniature THz-enabled antennas due to the small wavelengths (30 µm – 3 mm), enabling us to design arrays with a large number of antenna units [7]–[9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Numerous studies have shown that specific THz frequencies suffer high molecular absorption due to atmospheric gases (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=', water vapor and oxygen).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' However, given the wavelength and high energy photons of THz signals, other particles can Lasantha Thakshila Wedage and Bernard Butler are with the Walton Institute, South East Technological University, Ireland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Sasitharan Balasubramaniam and Mehmet C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Vuran are with the University of Nebraska-Lincoln, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Yevgeni Koucheryavy is with the Tampere University of Technology, Finland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' also significantly impact the link budget, which can result in scattering and absorption of signal power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Recent studies have shown that solid particles such as dust, sand and ice affect THz signals [10], in addition to molecular absorption from atmospheric gases [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' However, past studies have paid little attention to signal attenuation caused by solid particles such as dust and sand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Therefore, further investigation is required to determine how dynamic environments composed of solid particles, such as dust storms, affect THz links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' This requires further investigation, especially as we expand connectivity in rural areas and other planets (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=', Mars) to interplanetary scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' In the case of Mars, the recent vision of colonizing the planet will require high-bandwidth connectivity to maximize chances for human survival.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' A dust storm is a physical layer of dust and debris blown into the atmosphere by winds with horizontal and vertical velocity components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' On Earth, the wall of dust can be miles wide and several thousand feet high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Dust storms are more frequently found in arid regions such as the Middle East [12], North China [13], and North Africa [14] at specific periods of the year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' In more densely populated areas, human activity creates dust when burning fossil fuels for heating, cooking, or transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Industrial and construction processes also create dust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' This study compares the effects of solid dust particles on (sub)THz signals on both Earth and Mars, taking account of varying environmental conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Considering the differences in atmospheric conditions on Earth and Mars, with or without dust, suggests the use of different frequencies to enable relatively long-distance wireless communication on both planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Dust storms are one of the most remarkable features on Mars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Even though wind speed on Mars is not significantly higher than on Earth, the extremely dry, dusty surface yields more dust storms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Figure 1 provides an overview of selected wireless commu- nications applications on Earth, and proposed wireless com- munication applications on Mars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' While the applications differ, they are both affected by wireless channel losses, including those caused by dust particles that can scatter the EM waves used for communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' The rest of this paper considers the similarities and differences in the channel conditions, and includes models and simulations based on simulated dust storms that result in beam scattering, as shown at the bottom of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' In a dusty environment, the dust particle density is higher than usual, and the effects of multiple scattering of EM waves due to dust particles are non-negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Recent studies have not considered this significant effect on the attenuation of EM waves [15], [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' The lack of consideration of multiple arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='00658v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='IT] 11 Dec 2022 2 Immersive XR Wireless Cognition Mobile Hologram Smart ML/AI Applications 1Tbps Data rate Wireless Sensing Smart space V2V and Positioning Smart Spacesuit comms Wireless Cognition Smart Rover Smart Habitats Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' 1: THz wireless communication applications and links through Earth and Mars atmospheric and environmental conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' scattering effects can result in significant gaps between theo- retical and experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' This paper considers multiple scattering of EM waves due to dust particles along the beam propagation path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' To this end, we model the EM wave as a photon packet instead of a shower of photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' It is inaccurate to consider the EM wave as a shower of photons characterized by the position of a photon and its trajectory [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' A photon packet models a portion of the energy weight of the EM wave rather than single photons (which have quantum behavior).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Therefore, we can consider an EM wave as a collection of energy packets and model multiple scattering effects utilizing the Monte Carlo algorithm, to infer the radiative transfer equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' The THz link scattering loss measurement in this study is inspired by [18], where the scattering loss due to charged dust particles is calculated by considering the energy of the transmitting signal as Monte Carlo Photon Packets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Vertical THz attenuation is determined in [18], but this study considers horizontal point-to-point communication for both Earth and Mars in dusty atmospheric scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' The contributions of this paper are: 1) A 3-D geometric scattering model for multiple photon- dust particle interactions is presented, using both Mie and Rayleigh approximations, to estimate the probability that a photon packet arrives at the receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' 2) The model is used in simulation to estimate the overall channel capacity considering THz and sub-THz link budget degradation due to the combination of scattering by dust particles, molecular absorption loss due to the atmosphere, and free-space spreading loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' TABLE I: Atmospheric gas composition comparison between Earth and Mars [16];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' ppm is a concentration of parts per million.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Gas Composition on Earth Composition on Mars N2 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='084% 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='7% O2 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='946% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='13% Ar 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='93% 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='6% H2O 1-3% 100-400ppm CO2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='003% 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='32% CH4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='5ppm SO2 1ppm O3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='05ppm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='1ppm N2O 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='02ppm CO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='01ppm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='08% NH3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='01ppm NO 100ppm 3) Different communication channel conditions (on Earth and Mars) and their effect on channel capacity, including power loss caused by multiple scattering by dust, are compared and analysed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' The rest of this paper is organized as follows: Section II describes dust conditions and how they affect EM propagation and contrasts the conditions that prevail on Earth and Mars;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Section III describes how 3D dust storm simulation is affected by the number of dust particles on the EM wave propagation path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Then Section IV explains the Monte-Carlo simulation process for calculating the transmittance/attenuation when photon packets are scattered by multiple dust particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Section V presents estimates of transmittance/attenuation obtained by Monte-Carlo simulation, in various parameter settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Section McPhoton3 VI presents a channel capacity model that combines the effect of spreading, molecular absorption and multiple scattering by dust, with simulated results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Finally, Section VII presents our conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' BACKGROUND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' THz link behaviour in Dust Storms THz signal attenuation due to the scattering loss caused by high dust particle density on the THz beam propagation path is the main concern of this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Dust particle density on Mars is expected to be higher than on Earth because of the dusty atmosphere with low water vapour concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Mars dust consists of basalt and montmorillonite clay [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' On the other hand, Earth dust consists of pollen, bacteria, smoke, ash, salt crystals from the ocean, and small amounts of dirt or various rocks, including sand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Moreover, during dust storm conditions on Mars, the effective radius of the dust particle varies from 1 to 4 microns with an effective variance of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='2 – 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='4 [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' However, on Earth, the effective radius varies between 1 and 150 microns [18], [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Many researchers investigated the THz [16], [18], [21], [22] and lower frequency bands [23], [24] attenuation due to the presence of dust particles on the beam propagation path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' In [18], Monte-Carlo simulation was used to calculate the transmittance of EM waves when they propagate through dust, considering multiple scattering effects for charged particles in 20 and 75 GHz frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Hongxia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' [21] also studied the attenuation characteristics of THz waves subject to multiple scattering caused by dust storms in the Tengger desert, using the Mie scattering approximation and Monte Carlo simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' In addition, considering the Mie theory, Diao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' [16] investigated THz wave attenuation due to heavy dust in the Martian atmosphere in the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='1-1 THz frequency range and compared with Earth measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' [22] investigated attenuation at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='625 THz caused by dust utilising an experimental setup and found that degradation of the THz link budget is minor due to dust, compared to that found using IR beams with 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='5 µm wavelength, and average attenuation of the THz link is proportional to the dust particle density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Moreover, Elshaikh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' [23] developed a mathematical model to characterise the microwave attenuation due to dust, considering parameters such as visibility, frequency, particle size and complex permittivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' [24] calculated the light scattering properties of partially charged dust particles utilising Mie scattering theory for various frequencies and found that for higher THz frequency EM waves, the attenuation effect of charge carried by sand particles can be ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Furthermore, [25] presents the EM scattering properties of the small partially charged sand/dust particles, using the Rayleigh approximation, for microwave frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Atmospheric Condition Differences between Mars and Earth When THz radio waves pass through the atmosphere, the signals experience attenuation due to many factors, which differ in their impact between Earth and Mars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' This study focuses on point-to-point signal degradation in the lower part of the atmosphere (the troposphere) on Earth and Mars, when communicating antennas are placed 50 meters above the ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Apart from improved line-of-sight properties, [26] shows that longer communication distances can be achieved on Mars because dust particle density decreases with height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' The propagation medium in the troposphere of both planets includes gases, water vapour, clouds, fog, ice, dust, and assorted aerosols (haze), but the proportions vary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' The impair- ment mechanisms include absorption, scattering, refraction, diffraction, multi-path, scintillation and Doppler shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Impair- ment phenomena include fading, attenuation, depolarization, frequency broadening, and ray bending.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' However, this study considers only Line-of-Sight (LoS) transmission under dust storm scenarios through the troposphere of both Earth and Mars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' It considers signal attenuation based on three factors: (1) free space path loss (which is the same for Earth and Mars), (2) molecular absorption due to atmospheric gases (which are different for Earth and Mars), and (3) scattering loss due to dust particles along the propagation path (Mars and Earth typically have different dust distributions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Free space path loss occurs due to misalignment between the transmitter and the receiver antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' It is the same for both environments because it only depends on carrier frequency and distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Molecular absorption loss plays a significant role on both planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' It measures the fraction of power loss (of the carrier wave) converted to kinetic energy due to molecular vibration when EM waves propagate through molecules of the atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Therefore, unlike spreading loss, molecular absorption loss depends on local atmospheric gas composition and density (see Table I), including carrier frequency and distance between the transmitter and the receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' According to [27], certain frequencies of the THz spectrum, such as 183, 325, 380, 450, 550, and 760 GHz, suffer attenuation that is significantly greater than the free space propagation loss, due to water vapor absorption on Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' However, the Martian atmosphere contains only about 1/1,000 as much water as Earth’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Still, even this tiny amount can condense out, forming clouds that ride high in the atmosphere or swirl around the slopes of towering volcanoes [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' This serious issue needs to be considered for vertical communication of Mars surface devices (Rovers, Habitats, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=') and satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Since our study focuses on horizontal point-to-point communication, we do not need to consider upper atmospheric layer’s impact on THz signal transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Therefore, at these frequencies, we expect lower molecular absorption loss and higher channel capacity on Mars compared to Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' To the best of our knowledge, this is the first study that compares attenuation (at THz frequencies) due to dust storms on Earth and Mars, applying Monte Carlo simulation to the corresponding Mie and Rayleigh approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' This paper also presents a channel capacity model that includes the effect of spreading, molecular absorption and dust scattering losses for sub-THz and THz links on Earth and Mars, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' THZ BEAM PROPAGATION THROUGH A SIMULATED 3D DUST STORM This section discusses THz wave propagation through a randomly simulated 3D dust storm by simulating wind having 4 both vertical (up-draught) and horizontal velocity components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' This is used to calculate the number of dust particles on the beam path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' The simulated dust storm (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' 2) consists of a line source starting at X = 0 and a vertically upward movement of dust based on the vertex motion due to wind turbulence at (6000, 0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' The line source dust storm in this study spreads for 8m along the Y-axis (−4 ≤ Y ≤ 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Such a line source is more realistic than a point source for dust storm simulation on both Earth and Mars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' MATLAB’s wind package was used to simulate the storm and considered an exponential movement of dust along the positive X-axis coupled with strong wind in the same direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Also, dust particles gradually precipitate from the atmosphere, when their weight exceeds upward forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Dust particle move- ment of the point source dust storm downstream of the line source dust storm comprises both upward (point) wind and horizontal (line) wind, resulting in a vortex flow (a simulated whirlwind).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' When counting the number of dust particles on the cone- shaped THz beam, we followed the following method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' First, consider the THz beam starting at the point of (0, 0, h), where h (50 m) is the transmitter antenna height and beam propagation direction is aligned with the positive X-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' In this cone-shaped beam, the maximum radius of the impact area is calculated to be approximately 15 cm for the corresponding transmitting distance of 10 Km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Since it is difficult to calcu- late the dust particle concentration on such a pencil-thin beam, we sub-divided the cone-shaped beam into 1 × 106 disks with 1 cm distances between the disk centres (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' 2 (a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Then we identified the position of each dust particle at 1 m below the antenna height and recorded its position considering the nearest two disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' By looping over the position of each dust particle, and comparing the distances between the nearest disk centres and the particle position to centre distance, we encoded the position as being inside (1) or outside (0) of the THz beam for each particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' From this, we calculated the number of dust particles along the THz propagation path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Considering the dust particle size on Mars to vary between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='5 to 4 microns [20], [28], we simulate a scenario of sending a THz signal with the transmitter position (5500,0,50) and the receiver position (6500,0,50), which creates a 1000 m distance between them while placing the point source vertex movement at (6000,0,0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' As a result, we found that the average ratio of the number of generated dust particles along the line of propagation of the THz transmission beam is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='0022872.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Moreover, considering just the dust particles on the cone- shaped beam path, their density averaged 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='1832 (say, 10) particles per cm3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Hence we can conclude that assuming the beam face area is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='01 cm2, the number of dust particles along the beam for a distance of 10 m between the transmitter and the receiver is approximately 100 particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' However, in this random walk simulation process, it is difficult to control the effective radius of the dust particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Therefore, depending on the dust particle size, scattering effects (which depend on the number and size of the particles) along the beam propagation path can vary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' h Si,1 D X Y x y z φ ϴ X Z Tx Rx Si,2 Si,3 Si,j Tx Rx Wi,0 Wi,1 Wi,j Wi,2 Wi,j-1 MC Photon packet weight Scatters (a) (b) Inside (1) Outside (0) Center - Center Distance Center - Point Distance Radius Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' 2: Multiple scattering processes of EMWs in a sand/dust storm with (a) the decision-making (in/out) method of dust particles from the beam and (b) the local coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' MODELLING MONTE CARLO PHOTON PACKETS PROPAGATION THROUGH DUST PARTICLES In this section, we calculate the transmittance and the corre- sponding attenuation of the THz EM wave when it propagates through suspended dust particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' The initial intention was to consider the THz EM wave as a collection of photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' However, photon position and trajectory are not meaningful here [17] but collections of photons enable us to discretize the beam in a physically meaningful way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Monte Carlo simulation is used to estimate transmittance, where the incident plane EM wave is discretized as Monte Carlo photon (MCP) pack- ets/units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Such photon packets provide an appropriate physical unit for discrete event simulation [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Each MCP packet is considered to be an equally divided portion of the energy weight of the EM wave field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' In this simulation model, we assume that the particle number concentration is uniformly distributed throughout the THz beam area, and dust particles are randomly positioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' The intensity (I) of the incident THz EM wave can be expressed as I0 = M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='W, where M is the number of MCP packets per unit area per unit time and W is the energy weight of each MCP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Here we suppose that MCP packet i (i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=', M) is randomly scattered by dust particles j before it either exits the beam cone or reaches the receiver interface boundary at X = D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' We assume that MCP packets enter from the point (0, 0, h) (which is height h corresponding to the transmitter antenna height) (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='2) and are forward- scattered by scattering particles Si,l (l = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=', j) whose positions are denoted by (xi,l, yi,l, zi,l), which are assumed random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Moreover, the algorithm will randomly select the number of scattering particles (l) that collide with an MCP packet from j dust particles on the EM wave propagation path because each MCP packet will randomly change its 150- 喜 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='% Y (m) f00 2000 4000 6000 B000 00001 x(n)5 propagation direction after every collision and dust particles are uniformly distributed along the beam path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' We suppose the initial energy weight of each MCP packet is W = Wi,0 = 1 and after scattering due to scattering particle Si,l is Wi,l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' We also define a local coordinate system xyz with the origin located at the scattering particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' So, we can define the propagation direction of the MCP packet i with respect to the local coordinate system xyz, which is to the x direction considering forward scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' The propagation direction for the scattering direction of the MCP packet i due to the impact with scattering particle Si,l can be expressed using the direction cosines (µx i,l, µy i,l, µz i,l) in the global coordinate system XY Z, which is defined with the help of scattering (or deflection) angle θ and azimuth angle φ (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' 2 (b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' In each simulated collision, the scattering particle position (xi,l, yi,l, zi,l) and the propagation angles (θ, φ) to calculate the direction cosines were calculated randomly for the global coordinate system, which will be explained in detail later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Direction cosines can be calculated according to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' 2 (b) as follows, µx i,l= −sin(θi,l)cos(φi,l) � 1 − (µx i,l−1)2 + µx i,l−1cos(θi,l) (1a) µy i,l= sin(θi,l)(µy i,l−1µx i,l−1cos(φi,l) − µz i,l−1sin(φi,l)) � 1 − (µx i,l−1)2 + µy i,l−1cos(θi,l) (1b) µz i,l= sin(θi,l)(µz i,l−1µx i,l−1cos(φi,l) + µy i,l−1sin(φi,l)) � 1 − (µx i,l−1)2 + µz i,l−1cos(θi,l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' (1c) If |µx i,l−1| > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='99999, then µx i,l = µx i,l−1 |µx i,l−1|cos(θi,l) (2a) µy i,l = sin(θi,l)cos(φi,l) (2b) µz i,l = sin(θi,l)sin(φi,l) (2c) In the initial stage, we suppose that the MCP packet i enters the dust particle zone form X = 0 at point (0,0,h) and propagates along the direction of (µx i,0, µy i,0, µz i,0) = (1, 0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' The simulation process depends on uniformly-distributed, ran- domly generated numbers ϵi,l, νi,l, and χi,l ∼ Uniform(0, 1), which are used to calculate the random variables ∆Si,l, θi,l and φi,l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' The ∆Si,l is the travelling distance between the scattering particles Si,l and Si,l−1 and defined as, ∆Si,l = −ln(ϵi,l) Cext (3) where Cext is the total extinction cross-section efficiency of spherical dust particles with radius r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' The total extinction cross-section efficiency is expressed [18] as, Cext = � rmax rmin N0P(r)cextd(r), (4) where P(r) is the log-normal size distribution of dust particles for both Earth [29] and Mars [16] environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Here, N0 is the number of dust particles per unit volume, and it can be expressed as a function of visibility (Vb) [16], which is represented as, N0 = 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='034744Vb � 2rmax 0 πr2P(r)dr .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' (5) On Earth, dust particle radius varies between 1 to 150 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Therefore, when considering the approximate equality of the effective diameter of the dust particles and the wavelength of the THz frequency utilised in this study, we can use the Mie approximation [30] to infer the total extinction cross-section (cext) [19], which is the sum of the absorption cross-section and scattering cross-section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' The cext is expressed by the Mie solution for spherical particles with dielectric constant ϵ [23] as, cext = k3rλ2 2 (c1 + c2(kr)2 + c3(kr)3) (6) where, c1 = 6ϵ′′ (ϵ′ + 2)2 + (ϵ′′)2 (7a) c2= ϵ′′�6 5 7(ϵ′)2 + 7(ϵ′′)2 + 4ϵ′ − 20 [(ϵ′ + 2)2 + (ϵ′′)2]2 � + 1 15 + 5 3[(2ϵ′ + 3)2 + 4(ϵ′′)2]2 (7b) c3= 4 3 �(ϵ′ − 1)2(ϵ′ + 2) + [2(ϵ′ − 1)(ϵ′ + 2) − 9] + (ϵ′′)4 [(ϵ′ + 2)2 + (ϵ′′)2]2 �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' (7c) The complex refractive index of dry dust particles on Earth can be expressed as � 3 + ı.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='256 f [21], where f is the frequency and ı is the imaginary unit √−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' On the other hand, the total extinction cross-section of the dust particles on Mars can be evaluated utilising the Rayleigh approximations [25] because the effective diameter of the dust particles on Mars (1-8 µm) is less than one-tenth of the wavelength of the frequency [31] used in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Therefore, the total extinction cross-section can be expressed as, cext = 8 3πk4r6���ϵr − 1 ϵr + 2 ��� 2 + 12πkϵ ′′ r r3 1 |ϵr + 2|2 + π 6 k4r6σ2 E2 0ϵ2 0 |ϵr − 1|2 (8) where ϵr is the relative permittivity [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Moreover, we can take the complex refractive index of dust particles on Mars as 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='52 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='01i [16], [26] corresponding to the radius range (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='5-4 µm) used in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' The scattering angle, θi,l, due to the ith MCP packet impact with scatter l is represented as, θi,l = � cos−1� 1 2g � (1 + g2) − � 1 − g2 1 − g + 2gνi,l �2�� for g ̸= 0 cos−1(2νi,l − 1) for g = 0 (9) where φi,l is the azimuth angle due to the same impact and φi,l = 2πχi,l (10) and g =< cos(θ) >∈ [0, 1] is the asymmetry factor (here, g = 0 refers to the isotropic scattering and 1 refers to forward direct scattering).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' We assume g varies uniformly between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='5 and 1 for our simulations which is the average value for the direct forward scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' After calculating the random variables ϵi,l, νi,l, and χi,l, we can determine the (random) position of the scattering particle Si,j (Xi,l, Yi,l, Zi,l) in the global coordinate system utilising 6 the equations (1), (2) and (3) as below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' The position of the scattering particles can be expressed as, Xi,l = Xi,l−1 + ∆Si,l µx i,l−1 Yi,l = Yi,l−1 + ∆Si,l µy i,l−1 Zi,l = Zi,l−1 + ∆Si,l µz i,l−1 (11) Successful transmittance occurs only for MCP packets that reached the receiver interface at a distance of D from the transmitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' If Xi,l >= D, this means that Si,l−1 is the last scattering particle encountered by the MCP packet i before it leaves the region boundary (X = D) from the receiving interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Therefore, we can stop the simulation process and go to the next MCP packet (i + 1) after calculating its current energy weight (Wi,l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' The energy weight follows the Beer- Lambert law, which determines how Wi,l−1 is related to the Wi,l [18], [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Thus, the energy weight of an MCP packet after collision with a scattering particle can be expressed as, Wi,l = Wi,l−1 exp �−Cext(Xi,l − Xi,l−1) µx i,l−1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' (12) If Xi,l < D, this means MCP packet i is unable to reach the receiver interface after impacting with l scatters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' From this point, we focus our interest more on the energy weight of the MCP packet i and calculate the energy weight of the MCP packet Wi,l using eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' In this instance, we assume that the initial energy weight of the MCP packet is a unit (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=', Wi,0 = 1), where we define a threshold (ϵt) value of 1×10−5 to consider as the minimum energy weight that an MCP packet can take after l impacts with the scatters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' If the energy weight of an MCP packet does not exceed this minimum threshold (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=', Wi,l < ϵt), the packet is assumed not to reach the receiver and is recorded as such.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Therefore, we can set j = l and Wi,l = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Based on the calculation of energy weights of the MCP packets that reached the receiver interface, we can calculate the transmittance of the THz EM wave by, TMS = �M i=1 Wi,l exp �−Cext(D − Xi,l) µx i,l � I0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' (13) From our calculations, we noticed that eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' (13) does not converge to a finite value for every simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Therefore, we assume the transmittance to be zero when it is divergent and set the simulation to the next Monte-Carlo process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Based on the transmittance measurements at the end of this procedure, we can calculate the specific attenuation AMS in (dB/m) according to [18], as, AMS = −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='343 ln(TMS) D .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' (14) V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' THZ CHANNEL CAPACITY To evaluate the channel capacity in the THz band, we can decompose the received signal as a sum of the sub-bands, where each sub-band channel is narrow and has a flat-band response [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' The ith frequency sub-band is defined as ∆fi = fi+1 − fi with power Pi under the constraint �NB i=1 Pi ≤ Pt, where NB refers to the total number of sub-bands, and Pt stands for the total transmit power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' In the ith narrow-band, the sub-band capacity, Ci, is expressed in [33] as, Ci = ∆fi log � 1 + |hLoS|2Pi ∆fiSD(fi) � , (15) where SD is the power spectral density of the additive white Gaussian noise (AWGN) and hLoS is the frequency-dependent channel response for attenuation due to dust particles including spreading loss and molecular absorption loss due to gas molecules on the LoS signal propagation path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' According to [33], hLoS can be expressed as, hLoS(τ) = αLosδ(τ − τLoS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' (16) where αLos refers to the attenuation and τLoS refers to the propagation delay due to dust particles and gas molecules on the signal propagation path and τLoS = dLoS c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Where dLoS is the signal travelling distance through dust, which is D in this study, because we calculate the transmittance using the MCP packets that reached the fixed receiver in- terface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Also, the power spectral density of AWGN can be expressed as SD(τ) = n0 2 δ(τ) and, in the frequency domain, PSD(SD(f))= n0 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Utilizing the Wiener-Khinchin theorem, the frequency-dependent channel response for LoS attenuation can be expressed as [33], hLoS(τ) = |HLoS(f)|δ(τ − τLoS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' (17) The free space direct ray or LoS channel transfer function, HLoS, consists of the spreading loss function (HSpr), the molecular absorption loss function (HAbs), and scattering loss function due to dust particles (HDust), which can be expressed as, HLoS(f) = HSpr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='HAbs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='HDuste−j2πfτLoS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' (18) The free space path loss or the spreading loss (PLSpr) measures the fraction of power lost by a beam with frequency f over a distance D in a vacuum, and it can be expressed according to [34] as, PLspr = �4πDf c �2 , (19) where c is the speed of light in the medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Thus, according to [33] the corresponding channel transfer function for the spreading loss can be expressed as, Hspr = (PLspr)−1/2 = � c 4πDf � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' (20) The molecular absorption loss measures the fraction of power converted to kinetic energy due to molecular vibration when EM waves propagate through molecules in the atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Thus, when transmitting frequency f through a homogeneous medium between a transmitter and receiver at a distance D, the molecular absorption loss is obtained with the help of the Beer-Lambert law [34], which is represented as, PLabs = ek(f)D, (21) where k(f) = � i,g ki g(f) and ki g(f) is the monochromatic absorption coefficient of the ith isotopologue of gth gas at 7 frequency f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' When calculating the absorption coefficient, we consider water vapour and nine other gases for Earth and six gases for Mars (see Table I), except for Argon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' This allows us to consider the vastly different gas concentrations between the two planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' The monochromatic absorption coefficient for each isotopologue of a particular gas in the Martian and Earth atmosphere at frequency f is provided in [35], ki g(f) = Si g(T)F i(f), (22) where Si g(T) is the line intensity at temperature T (210K for Mars) referenced to the temperature 296K of the ith isotopo- logue of gth gas, which can be easily calculated using the high- resolution transmission (HITRAN) molecular spectroscopic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Where, F i is the spectral line shape function at frequency f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' In the lower atmosphere on Earth,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' pressure broadening of spectral lines dominates the line shape and a Lorentz profile can be assumed as the line shape function and it is given by [35],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' F i L(f) = 1 π γ(p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' T) γ(p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' T)2 + [f − (f ig + δ(Pref)P)]2 (23) where f i g is the resonant frequency for the isotopologue i of gas g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' γ(P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' T) is the Lorentzian (pressure-broadened) HWHM for a gas at pressure P (atm),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' temperature T (K),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' and δ(Pref) is the pressure shift at reference pressure (Pref= 1 atm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Since Doppler-broadening dominates the line shape in low- pressure environments such as Martian environment,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' a Gaus- sian profile can be assumed as the line shape function,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' and it is given by,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' F i G(f) = � ln 2 παi D 2 exp � − (f − f i g)2 ln 2 αi D 2 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' (24) where αi D is the Doppler broadening half-width,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' αi D = f i g c � 2NAkBT ln 2 M i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' (25) where M i is the molar mass of isotopologues which can be obtained from the HITRAN database [35],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' and NA and kB are the Avogadro and Boltzmann constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Thus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' according to [33] the corresponding channel transfer function for the molecular absorption loss due to the gas molecules on the atmosphere can be express as,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Habs = (PLAbs)−1/2 = e− 1 2 k(f)D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' (26) Finally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' HDust is the transfer function for the attenuation due to dust particles that can be expressed as following the relationship between the transfer functions and the attenuation functions for spreading loss and molecular absorption loss,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' as well as utilising the dust attenuation function in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' (14) in dB, which is represented as, Hdust = 1 √ 10−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='4343 ln (TMS) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' (27) Therefore, substituting the functions derived for HSpr, HAbs, and HDust, to eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' (18), we can calculate the channel transfer function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Moreover, in this study, we consider one narrow frequency band for each environment, which is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='22 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='24 (a) Earth: Transmittance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' (b) Earth: Attenuation (dB/m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' (c) Martian: Transmittance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' (d) Martian: Attenuation (dB/m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' 3: Simulation measurements of a) the transmittance and b) the attenuation for a THz beam of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='24 THz and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='64 THz frequency for Earth and Mars by varying MCP packets from 10 to 10000 while fixing dust particle number (100/10000) on the propagation path, visibility and the distance (10 m) between transmitter and the receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' THz for Earth and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='64 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='67 THz for Mars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Therefore, we can suppose that Pi = Pt and the eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' (15) can be rewritten as, C = ∆f log � 1 + |hLoS|2Pt ∆fSD(f) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' (28) VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' RESULTS AND DISCUSSION A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Transmittance and Attenuation measurements through Monte-Carlo Simulations This subsection presents the simulation results for the transmittance and attenuation measurements of the THz link and the generated estimation models for the THz attenuation due to dust particles on the beam propagation path by varying parameters such as the MCP packets, visibility, dust particle number, the distance between transmitter and the receiver, and EM frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' When simulating data for a targeted parameter, we have kept the other parameters constant to make the interpretation easy (see Table II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Thus, we have chosen a frequency of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='64 THz as the constant frequency for Mars [26], and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='24 THz frequency for Earth [36], [37] corresponding to low molecular absorption and high transmission distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' However, it is crucial to consider molecular absorption on Mars, even though it has a thin atmosphere with very low water vapour concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Moreover, we have considered 100 dust particles on the beam propagation path for a 10 m fixed distance between the transmitter and receiver, as explained in section III for simulations on Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' It is unrealistic to consider the same amount of dust particles for the Mars simulations be- cause of the tiny particle sizes on Mars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Therefore, considering 103 Transmittance 10-4 10-5 10-6 101 102 103 104 Number of MCP packetsSimulated data Fitted curve -- Prediction bounds 6 3 2 101 102 103 104 Number of MCP packets100 10-10 10-20 Transmittance 10~30 10-40 10~50 101 103 102 104 Number of MCP packets100 Simulated data Fitted curve .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='-- Prediction bounds 80 Attenuation [dB/m] 60 40 20 0 20 101 102 103 104 Number of MCP packets8 the blockage that this dust particle creates and the proportional relationship between blockage area and dust particle radius, we consider 10000 dust particles for Mars corresponding to 100 dust particles on Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' When selecting the number of MCP packets, we considered 10000 packets in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' TABLE II: Channel conditions and simulation settings on Earth and Mars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Parameter Earth Mars Frequency 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='24 THz 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='64 THz MCP packets 104 104 Dust Density 102 per 10m 104 for 10 m Dust radius 1–150 microns 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='5–4 microns Dust size distribution log-Normal log-Normal Antenna height 50 m 50 m Approximation Mie Rayleigh Distance 1 - 200 m 1 - 200 m Temperature 288 K 210 K Surface Pressure 1013 mb 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='1 mb Surface density 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='29 Kg/m3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='02 Kg/m3 Figure 3 illustrate the simulated data for the transmittance and attenuation measurements using the simulation setup ex- plained in section IV for Earth and Mars environments by varying the number of MCP packets from 10 to 10000, while keeping the other parameters constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' As we can see from the figures, the transmittance measurements for both Earth and Mars environments (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' 3 (a) and (c)) are increasing rapidly when the MCP packets increase at the beginning up to 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' After that, it converges to a particular value corresponding to each environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Furthermore, attenuation measurements decrease following a power function for both Earth and Mars environments (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' 3 (b) and (d)) and converge approximately to a value of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='6 dB/m and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='8 dB/m, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' In addition, the fitted power function for attenuation against the MCP packets (NMCP ) can be expressed as Attenuation (dB/m) = 2145N −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='721 MCP + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='839 for Earth and Attenuation (dB/m) = 410N −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='8485 MCP + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='727 for Mars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Next, we investigated the effect of visibility on the trans- mittance and attenuation measurement for Earth and Mars environments by varying the parameter values from 10 to 10000 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Generally, when the visibility increases between the transmitter and the receiver, we will see fewer dust particles on the beam propagation path with a high distance variance between the particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Therefore, we can expect high transmittance and low attenuation measurements when the visibility increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' According to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' 4 (a), the transmittance measurements of the Earth’s environment are increasing dramatically, with the visibility and attenuation measurements (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' 4 (b)) decreasing following a power function as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Moreover, the attenuation measurements approximately converge to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='1 dB/m value with an increase of visibility near 10000 m, which we can consider as clear sky condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' The fitted power function for the attenuation against the visibility (V ) can be expressed as Attenuation (dB/m) = 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='41V −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='9694+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='105 for the Earth environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' On the other hand, the transmittance measurements for the Mars environ- ment (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' 4 (c)) show an increasing trend with visibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' However, transmittance measurements for some visibility pa- rameter values diverge from the trend because, according to our simulation process, each MCP packet will randomly select the scatters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Therefore, even if we have low dust density on the beam propagation path with high visibility, the transmit- tance can be low due to high collision with dust particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Corresponding to the transmittance measurements, we can see a slight linear decrease in attenuation on Mars (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' 4 (d)) with increased visibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' The fitted linear function can be expressed as Attenuation (dB/m) = −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='184×10−5V +4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' (a) Earth: Transmittance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' (b) Earth: Attenuation (dB/m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' (c) Martian: Transmittance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' (d) Martian: Attenuation (dB/m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' 4: Simulation measurements of a) the transmittance and b) the attenuation for a THz beam of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='24 THz and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='64 THz frequency for Earth and Mars, respectively, by varying the visibility from 10 to 10000 m while fixing MCP packets (10000) and the distance (10 m) between transmitter and the receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' The dust particle density can vary unpredictably with the wind in a dust storm on Earth and Mars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' There can be time windows with very low and high dust particles on the beam propagation path, which will be perfect for transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' To investigate the effect of dust particle count, we have measured the transmittance and attenuation for Earth and Mars environments by varying dust particle numbers from 10 (very low) to 10000 (very high).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' 5 (a) and (c), the transmittance measurement drops dramatically to near zero with the increase of dust particles for both environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' This rapid transmittance drop happens due to the high amount of scatters on the THz beam propagation path that each MCP packet should randomly collide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Moreover, the attenuation measurements (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' 5 (b) and (d)) increased rapidly following a power function for both environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' The fitted power function for attenuation against the dust particle number (DP N) can be expressed as Attenuation (dB/m) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='4423D0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='2579 P N + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='213 for Earth and Attenuation (dB/m) = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='534D0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='04617 P N − 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='262 for Mars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Furthermore, as we can no- tice, attenuation measurements do not converge to a particular value as in previous cases when increasing the number of dust particles on the beam propagation path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Therefore, we 5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='4 Attenuation [dB/m] 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='2 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='4 Simulated data 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='2 Fitted curve Prediction bounds .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='. 3 101 102 103 104 visibility (m)X10-3 9 8 7 6 Transmittance 5 4 3 2 1 Oc 10l 102 103 104 visibility (m)9 Simulated data Fitted curve Prediction bounds 8 7 6 5 4 3 2 102 103 101 104 visibility (m)X10-4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='4 Transmittance 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='4 C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='2 101 102 103 104 visibility (m)9 can expect a communication blackout in a regional/global dust storm situation which will boost the dust particle density in the communication area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' (a) Earth: Transmittance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' (b) Earth: Attenuation (dB/m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' (c) Martian: Transmittance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' (d) Martian: Attenuation (dB/m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' 5: Simulation measurements of a) the transmittance and b) the attenuation for a THz beam of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='24 THz and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='64 THz frequency for Earth and Mars, respectively, by varying the dust particle number on the beam propagation path from 10 to 10000 while fixing the number of MCP packets (10000) and the distance (10 m) between transmitter and the receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' As mentioned above, dust density on the beam propaga- tion path can vary significantly on Earth and Mars due to the unpredictable wind, temperature and pressure behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Therefore, to conduct more realistic simulations, we have investigated the effect of distance between the transmitter and the receiver when we have various dust densities that vary with the distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Dust density usually measures the number of dust particles per unit volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' However, in this study, we define it as the number of dust particles per meter for simplicity because the THz beam is assumed to be cone shape, and its face area is very tiny.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' This means that if we consider 10 dust particles per meter (10/m) for 100 m, we assume that 1000 dust particles are uniformly distributed on the beam propagation path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' As we can see in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' 6 (a), the transmittance measurements on Earth drop dramatically with the distance, and when increasing the dust density, the transmittance measurements reaches near zero rapidly beyond 100 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' However, on Mars (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' 6 (c)), the transmittance measurements decreases slightly compared to Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Moreover, we can clearly see that the transmittance measurements for 25/m dust density are significantly lower than the dust density at 10/m, as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' However, we can not see much difference between the transmittance measure- ments for 50/m and 100/m dust densities up to 150 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' On the other hand, attenuation measurements (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' 6 (b) and (d)) advance with the increase of distance and dust density for both environments corresponding to transmittance measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Furthermore, we investigated the impact of frequency on (a) Earth: Transmittance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' (b) Earth: Attenuation (dB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' (c) Martian: Transmittance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' (d) Martian: Attenuation (dB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' 6: Simulation measurements of a) the transmittance and b) the attenuation for a THz beam of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='24 THz and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='64 THz frequency for Earth and Mars, respectively, by varying the distance between transmitter and the receiver from 1 to 200 m for different particle number densities of 10/m, 25/m, 50/m and 100/m on the beam propagation path while fixing MCP packets (10000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' (a) Earth: Transmittance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' (b) Earth: Attenuation (dB/m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' (c) Martian: Transmittance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' (d) Martian: Attenuation (dB/m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' 7: Simulation for a THz beam by varying the frequency from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='1 to 10 THz while fixing dust particle number (100) on the beam propagation path, MCP packets (10000), and the distance (10 m) between the transmitter and the receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' the transmittance and attenuation measurements on Earth and Mars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' 7 (a), the transmittance measurements for Earth’s environment decrease following an exponential function with the frequency increase from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='1 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='014G 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='01 Transmittance 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content="002 0 11 101 102 103 104 Particle Number6 5 3 2 Simulated data Fitted curve Prediction bounds 10' 102 103 104 Particle Number0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='08Q 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='06 Transmittance 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='01 0 101 102 103 104 Particle Number4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='5 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='5 Attenuation [dB/m] 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='5 2 Simulated data Fitted curve 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='5 Prediction bounds 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='5 101 102 103 104 Particle Number100 PN = 10/m PN = 25/m PN = 50/m PN = 100/m 10-5 Transmittance 10-20 50 0 100 150 200 Distance (m)250 200 [dB 150 Attenuation PN = 10/m 100 PN = 25/m PN = 50/m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='.- PN = 100/m 50 0 0 50 100 150 200 Distance (m)PN = 10/m PN = 25/m PN = 50/m PN = 100/m 102 Transmittance 10~6 50 100 150 200 0 Distance (m)80 70 60 Attenuation [dB] 50 40 30 PN = 10/m 20 PN = 25/m PN = 50/m PN = 100/m 10 0 50 100 150 200 Distance (m)10 10-10 Transmittance 10-30 10-40 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='5 4 Frequency (THz)50 Simulated data Fitted curve 40 Prediction bounds Attenuation [dB/m] 30 20 10 0 10 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='5 4 Frequency (THz)X10-4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='5 2 Transmittance 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='5 0 0 2 4 6 8 10 Frequency (THz)5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='2 5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='8 Attenuation [dB/m] 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='2 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='6 0 2 4 6 8 10 Frequency (THz)10 (a) Particle number count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' (b) Transmittance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' (c) Attenuation (dB/m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' 8: Comparison of simulations for time-dependent turbulence (turmoil after 5 seconds) corresponding to the measurements of a) particle number count, b) the transmittance, and c) the attenuation (dB/m) considering Earth and Mars environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' 4 THz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' We were unable to calculate transmittance measure- ments following our simulation process beyond the 4 THz frequency limit since we are considering the dust particles with a radius of 1 to 150 microns for Earth environment, wavelengths can be comparably low or approximately equal to the dust particles’ size after some frequencies threshold, creating more difficulties for data transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' The corre- sponding attenuation for the transmittance measurements on Earth increases following a power function that can be fitted as Attenuation (dB/m) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='277D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='054 P N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' On the other hand, the transmittance and attenuation measurements for the Mars environment do not show a particular increase or decrease trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' However, we noticed that the attenuation measurements vary around 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='2 dB/m with the frequency increase from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='1 to 10 THz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Finally, we investigated the effect of time-dependent tur- bulence on the transmittance and the attenuation of the THz signal on Earth and Mars, which corresponds to the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='24 THz and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='64 THz frequencies, respectively and the distance between the transmitter and the receiver for 10 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Here, we compare Earth and Mars simulation scenarios in which we assume that the dust particle number on the beam propagation path will suddenly increase due to the unpredictable behaviour of wind after 5 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' In the first 5 s, we assume that the dust particle number on the beam will vary between 10-20 in the Earth environment and 100-200 in the Mars environment in clear sky conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' After five seconds, the dust particle number on the beam will increase between 100-200 on Earth and 10000-20000 on Mars due to the sudden wind turbulence in the communication area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' As demonstrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' 8 (a), dust particle number on the beam propagation path is low in the first 5 s compared to the next 5 s for both environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Also, the dust particle number is higher for the Martian environment than the Earth environment in the considered time interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' When scrutinising the transmittance measurements (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' 8 (b)), we can notice that transmittance drops suddenly after 5 s for both environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' However, average transmittance measurement values are approximately similar within the first 5 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Also, the transmittance measurements on Mars after the turbulence are significantly low compared to Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Corresponding to the transmittance measurements, attenuation measurements (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' 8 (c)) increases dramatically after 5 s and are high on Mars, concluding that the turbulence effect on Mars should be investigated thoroughly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' 9: Measurements of Transmittance and Attenuation of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='24 THz and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='64 THz links for Earth and Mars due to the sudden movement of dust particles on the beam propagation path for a fixed transmitter and receiver distance of 1 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Channel Capacity simulation for Earth and Mars Under Dust storm This subsection investigates the channel capacity measure- ment of the THz links considering two scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' In the first scenario, we assume that there are time windows when the number of dust particles on the THz beam propagation path drops, creating opportunities to communicate with high data rates for both Earth (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='24 THz) and Mars (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='64 THz) environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Here we analyse the channel capacity for Earth and Mars environments considering spreading loss and Molec- ular absorption loss with the THz attenuation due to dust on the beam propagation path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Also, we investigate channel capacity variations in this scenario for different transmitter powers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=" In the second scenario, we investigate the channel 105 On Earth On Mars 104 Dust Particle Count 103 102 101 2 3 4 5 6 7 8 9 10 1 Time (s)10' On Earth On Mars Transmittance 10-3 10-4 10-5 10~6 2 3 4 5 6 7 8 9 10 Time (s)6 5." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='5 5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='5 Attenuation A 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='5 2 On Earth On Mars 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='5 1 2 3 4 5 6 7 8 6 10 Time (s) number 10 On Earth On Mars Dust particle 10 2 4 6 8 10 12 14 16 18 20 Transmittance On Earth On Mars 10 2 4 8 6 10 12 14 16 18 20 [dB/m] On Earth On Mars Attenuation 40 20 0 2 4 6 8 10 12 14 16 18 20 Time [s]11 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' 10: Channel capacity variation comparison for the sce- nario of a sudden drop in the number of dust particles on the beam due to the wind behaviour on Earth and Mars environments with considering spreading loss and molecular absorption loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' capacity variation with the distance in clear sky and dust storm conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' In our first model, we assume that the dust density varies randomly for the Earth environment from 100 and 200 and Mars environment from 10000 to 20000 particles correspond- ing to a 1 m distance between the transmitter and the receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Here we know that considering a 1 m distance for simulation is unrealistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' However, in this scenario, we need to infer the effect of the sudden dust particle drop for the channel capacity measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Therefore, it is adequate to consider a 1 m distance for the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' As we mentioned above, the significant variation of dust particles on Earth and Mars is in its effective radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' On Earth, the average effective radius of a dust particle varies between 1 and 150 microns [18], and on Mars, it varies between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='5 to 4 microns [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Therefore, to measure transmittance/attenuation considering approximately similar beam-blocking areas by dust particles on both Earth and Mars, we should consider 100 times more dust on Mars than on Earth, following the relationship between dust effective radius and area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Moreover, we sampled the dust particle number for each environment every second for 20 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' In addition, we assume there are two-time intervals (t=[7 s,9 s], t=[15 s,17 s]) when the dust particle number drops to less than 30 and 300 particles (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' 9) due to the unpredictable behaviour of wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Such time intervals might represent occasions when the wind that causes dust particles to be suspended in the atmosphere falls away to near zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Furthermore, as we can see from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' 9, the relative dust particle density decrease at the interval centred on t=16 s is much higher than at the interval centred on t=8 s for both environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' We noticed that corresponding to the low dust density, the transmittance measurement is higher at the interval 15-17 s than at the 7-9 s interval, and the attenuation shows the opposite variation to transmittance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' In principle, those time intervals represent attractive time windows for communication in both environments because of lower attenuation due to the momentary absence of dust particles in the channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' (a) On Earth (b) On Mars Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' 11: Channel capacity measurement variation for the scenario of a sudden drop in the number of dust particles on the beam due to the wind behaviour for different Transmitted powers (1, 5, 10 dBm) of antenna with considering spreading and molecular absorption (a) on Earth and (b) on Mars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Figure 10 illustrates the channel capacity variation in the clear sky and dust storm conditions on Earth and Mars for the same dusty scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' When scrutinising Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' 10, we noticed that in clear sky conditions, channel capacity on Mars is approximately 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='39 ×1012 bits/s, and on Earth, its nearly 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='85 ×1012 bits/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' This shows more than 550 GB/s differ- ence between the channel capacity measurement in clear sky conditions on Mars and Earth due to much higher molecular absorption on Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Also, it is noticeable that dust appears to have a more significant relative effect on channel capacity on Mars than on Earth in a dust storm situation due to the high number of tiny dust particles on the beam propagation path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Moreover, channel capacity measurements in dust storm situation on Earth is lower than in clear sky condition, but the difference is minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' However, the difference on Mars is relatively enormous, and it is approximately two orders of magnitude less than clear sky conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' On the other hand, channel capacity measurements in dust storm condition on Mars is higher by approximately five orders of magnitude than in dust storm condition on Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' In addition, the free space path loss is constant in this scenario because it only depends on the carrier frequency and the distance between the transmitter and the receiver, which are both constant in this simulation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' 11, we investigated the channel capacity measure- ment variations for different transmitter power in discrete time windows considering the free space path loss and the molecular absorption loss effect on the channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' This figure shows that the channel capacity increases as antenna trans- mitter power increases for both Earth and Mars environments by approximately 5 to 20 GB/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Also, we noticed that we could have reliable communication links with high channel capacities when communicating in the time windows with low dust densities on Mars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Moreover, the channel capacity measurements show a significant variation on Earth compared to Mars for all transmitted powers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Therefore, these measure- ments imply that we can reach high channel capacities on Mars than on Earth using lower transmitter power antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' ×1012 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='2 On Earth: Clear Sky On Earth: Dust Storm On Mars: Clear Sky On Mars: Dust Storm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='7 2 8 10 12 4 6 14 16 18 20 Time[s]X101 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='3 On Earth: Tx power = 1 dBm On Earth: Tx power= 5 dBm 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='25 On Earth: Tx power = 1O dBm 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='2 Channel Capacity [bits/s] 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='15 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='1 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='05 8 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='95 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='9 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='85 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='8 2 4 6 8 10 12 14 16 18 20 Time「s]1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='34 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='32 [bits/s] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='28 Channel Capacity [ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='26 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='24 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='22 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='2 On Mars: Tx power= 1 dBm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='18 On Mars: Tx power = 5 dBm On Mars: Tx power = 10 dBm 2 4 6 8 10 12 14 16 18 20 Time[s]12 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' 12: Channel capacity measurements by varying the dis- tance between the transmitter and the receiver with and without dust storms situations on Mars and Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' In our second and final simulation scenario, we inves- tigated the channel capacity for various distances between the transmitter and the receiver for both Earth and Mars environments, comparing clear sky and dust storm conditions with different dust densities per meter (See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' The transmitter power was taken as 10 dBm in this simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Here, we allowed the dust particle count density to vary as a factor of distance to simulate more realistic dust storm conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' For the Earth’s environment, we have considered dust storms that result in 10-20 (very low) and 100-200 (very high) dust particles per meter dust densities on the THz beam propagation path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Similar conditions for the mars environment were considered by comparing the dust particle sizes with Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Thus, we assumed that dust storms on Mars would carry 100-200 and 1000-2000 dust particles per meter to the beam propagation path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' However, we should have considered 10000-20000 dust particles per meter on Mars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Nevertheless, due to computational difficulties with high dust densities with increasing the distance between the transmitter and the receiver, we are considering above mentioned numbers for channel capacity measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Moreover, we have taken account of spreading loss and molecular absorption loss when calculating the channel capacities for each distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' 12, the channel capacity decreases grad- ually for clear sky conditions for both environments showing high channel capacities in the considering distance range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' However, the decrement is high on Earth due to high molecu- lar absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Moreover, the channel capacity measurements decrease dramatically on Earth with the distance for 100- 200/m dust storm conditions, showing communication black- out beyond 70 m distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Also, at 10-20/m dust storm, channel capacity measurements decrease slowly, showing that this dust particle number on the beam propagation path is not a massive issue for achieving high channel capacities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' However, Earth’s channel capacity is significantly dropping when compared with the channel capacity decrement on Mars for the 100-200/m dust particle density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Again, 100-200/m dust particles on the beam propagation path on Mars do not significantly affect the channel capacity measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Therefore, we can neglect the THz link budget degradation due to the small amount of dust on Mars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' In addition, when investigating the high dust particle number density effect on the channel capacity on Mars (1000-2000/m), the channel capacity measurements are computationally difficult when the distance is greater than 60 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' However, we can notice that the channel capacity measurements for Mars are decreasing rapidly with the increase in the distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Also, we can see that the channel capacities are equal at a distance of approximately 60 m for clear sky conditions on Earth and the high dust density scenario on Mars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' CONCLUSION High-speed, reliable communication between devices on Earth and Mars is needed to fulfil future communication requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' In this study, we investigated the impact of at- mospheric dust and dust storms for communication using THz links, utilising a modified Monte Carlo simulation algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' The calculated transmittance and attenuation measurements are based on Mie and Rayleigh approximations depending on the dust particle sizes and carrier frequency utilised for communication on the two planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Moreover, we presented a channel capacity model and analysed it for two different time-dependent and distance-dependent scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' The Monte- Carlo simulation results show that attenuation measurements decrease for both Earth and Mars environments when the MCP packets and visibility increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' In addition, for both environments, the attenuation increases with higher dust par- ticle number on the beam propagation path and distance between the transmitter and the receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' We noticed the exact attenuation behaviour with the increased frequency for the Earth’s environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' However, the attenuation measurements vary around a constant value for the Mars environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' When scrutinising the channel capacity measurements from the time- dependent scenario, we can conclude that the time windows showing sudden dust particle density drops create the best communication opportunities with high data rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Also, we noticed that the channel capacity measurements dramatically drop with the increase in distance between the transmitter and the receiver in severe dust storm situations on both Earth (100- 200/m) and Mars (1000-2000/m) environments, even if we use low molecular absorption frequencies and high transmitter power antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' However, the impact from the local dust storm is negligible on Mars (100-200/m) but should be further investigated on Earth (10-20/m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' ACKNOWLEDGMENT This publication came from research conducted with the financial support of Science Foundation Ireland (SFI) and the Department of Agriculture, Food and Marine on behalf of the Government of Ireland (Grant Number [16/RC/3835] VistaMilk), the support of YL Verkot, Finland, and US National Science Foundation (NSF) ECCS-2030272 grant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Channel Capacity [bits/s] On Mars: Clear Sky On Mars: Dust Storm (100-200/m) On Mars: Dust Storm (1000-2000/m) On Earth: Clear Sky On Earth: Dust Storm (10-20/m) On Earth: Dust Storm (100-200/m) 20 40 60 80 100 120 140 160 180 200 Distance [m]13 REFERENCES [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' 16, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' 1–35, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' 14 LASANTHA THAKSHILA WEDAGE [S’22] (thakshila.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='wedage@waltoninstitute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='ie) received his B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' degree in Mathematics from University of Ruhuna, Sri Lanka, in 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' He is currently pursuing a PhD degree with the Department of Computing and Mathematics, Walton Institute, South East tech- nological University, Ireland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' His current research interests lie in Mathematical modelling and 5G/6G Wireless communication and sensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' BERNARD BUTLER [SM’22] (bernard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='butler@setu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='ie) received his PhD from South East Technological University, Ireland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' He was a Senior Research Scientist in the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='K.’s National Physical Laboratory, focusing on mathematics of measurement and sensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' He is a Lecturer in SETU and is CONNECT Funded Investigator and VistaMilk Academic Collaborator with the Walton Institute, SETU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Research interests include machine learning, wireless comms and edge networking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' SASITHARAN BALASUBRAMANIAM [SM’14] (sasi@unl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='edu) received his Bachelors in Engineer- ing and PhD degree from the University of Queens- land, Australia in 1998 and 2005, respectively, and Masters of Engineering Science from Queensland University of Technology in 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' He was a past recipient of the Science Foundation Ireland Starter Investigator Research Grant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' He was also a past re- cipient of the Academy of Finland Research Fellow at Tampere University, Finland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' He was previously the Director of Research at the Walton Institute, South East Technological University, Ireland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' He is currently an Associate Professor at the School of Computing, University of Nebraska-Lincoln.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' He is currently the Editor-in-Chief of IEEE Transactions on Molecular, Biological and Multi-scale Communications as well as an Associate Editor for IEEE Transactions on Mobile Computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' He was an IEEE Distinguished Lecturer for the IEEE Nanotechnology Council in 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' His research interests lie in molecular and nano communications, Internet of Bio-Nano Things, as well as 5G/6G networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' YEVGENI KOUCHERYAVY [SM’08] (yevgeni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='koucheryavy@yl-verkot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='com) received the Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' degree from the Tampere University of Technology, Finland, in 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' He is currently a Full Professor with the Unit of Electrical Engineering, Tampere University, Finland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' He has authored numerous publications in the field of advanced wired and wireless networking and communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' His current research interests include various aspects in heterogeneous wireless communication networks and systems, the Internet of Things and its standardization, and nanocommunications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Mehmet Can Vuran [M’07] (mcv@unl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='edu) was born in Istanbul, Turkey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' He received his B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='Sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' degree in Electrical and Electronics Engineering from Bilkent University, Ankara, Turkey, in 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' He received his M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' and Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' degrees in Electrical and Computer Engineering from Georgia Institute of Technology, Atlanta, GA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=', in 2004 and 2007, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Currently, he is the Dale M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Jensen Chair Professor in Computing at the School of Computing at the University of Nebraska-Lincoln.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Vuran has been recognized as a Highly Cited Researcher three years in a row by Thomson Reuters ”in recognition of ranking among the top 1% of researchers for most cited documents in Computer Science”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' Vuran was awarded an NSF CAREER award for the project “Bringing Wireless Sensor Networks Underground”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' He is a Daugherty Water of Food Institute Fellow and a National Strategic Research Institute Fellow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' He serves on the editorial boards of IEEE Transactions on Wireless Communications, IEEE Transactions on Mobile Computing, and IEEE Transactions on Network Science and Engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} +page_content=' His research interests are in 6G networks, the Internet of Things (IoT), agricultural wireless networks, wireless underground communications, and vehicular communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfw_lx/content/2301.00658v1.pdf'} diff --git a/etE0T4oBgHgl3EQfXAB2/vector_store/index.faiss b/etE0T4oBgHgl3EQfXAB2/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..d21c92397d033ba0a19b825719fcbf66b892efec --- /dev/null +++ b/etE0T4oBgHgl3EQfXAB2/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:b336283f816421e542ab03dd0b35bb096b8028024fa6a82ecca3a22cd3a9d592 +size 3342381 diff --git a/etE0T4oBgHgl3EQfXAB2/vector_store/index.pkl b/etE0T4oBgHgl3EQfXAB2/vector_store/index.pkl new file mode 100644 index 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[math.DG] 4 Jan 2023 +Some notes on the impact of Lagrange’s memoir +“On the construction of geographical maps" +Athanase Papadopoulos∗ +Abstract +These are notes on the impact of Lagrange’s memoir on the con- +struction of geographical maps. We mention the relations of some ideas +and questions introduced in this memoir with other notions that ap- +peared later in the works of several mathematicians, including in par- +ticular Chebyshev (19th c.) and Darboux (19th-20th c.), two math- +ematicians who were particularly interested in geography. The final +version of this paper appears in the book: Mathematical Geography in +the Eighteenth Century: Euler, Lagrange and Lambert (ed. R. Caddeo +and A. Papadopoulos), Springer, 2022. +1 +Introduction +In this chapter, I shall mention the impact of some ideas and questions +introduced by Lagrange in his memoir Sur la construction des cartes géo- +graphiques (On the construction of geographical maps) [10] on the works of +some later authors including Darboux (§2), Chebyshev (§3) and others (§3 +and 4). The last section (§5) contains some notes on the use of Lagrange’s +ideas in the actual construction of geographical maps. +2 +Lagrange’s memoir in the work of Darboux +Lagrange, in his memoir Sur la construction des cartes géographiques, moti- +vated by works of Lambert and Euler, addressed the following problem: +∗Institut +de +Recherche +Mathématique +Avancée +(Université +de +Strasbourg +et +CNRS) +7 +rue +René +Descartes +67084 +Strasbourg +Cedex +France +email +: +athanase.papadopoulos@math.unistra.fr +1 + +To find all the conformal projections from the sphere (and more gen- +erally from a surface of revolution) onto the Euclidean plane by which the +meridians and parallels are sent to circles. +This problem of Lagrange is mentioned several times by Darboux, in his +Leçons sur la théorie générale des surfaces et les applications géométriques +du calcul infinitésimal [5], and in particular in Chapter IV of Book II of +Part I, which concerns conformal maps between surfaces. In that chapter, +Darboux, after having studied in the preceding chapters the general meth- +ods of isothermal coordinates, applies them to the resolution of problems +posed by the construction of geographical maps. He considers Lagrange’s +problem (referring to the latter as the first mathematician who addressed +it) in the setting of the spheroid. Let us recall that in this setting, a merid- +ian is an ellipse (or half-ellipse) whose rotation about an axis generates the +spheroid, and a parallel is the orbit of a point on this ellipse by this rota- +tion. The meridians are geodesics of the resulting surface but the parallels +are not. (This holds in the general case of a surface of revolution in Eu- +clidean space). Darboux makes the remark that if one of the two families +(the meridians and the parallels) is represented by circles, then the same +holds for the second family. Thus, Lagrange’s conditions (that both fam- +ilies are represented by circles) are redundant. He discusses at length the +various solutions to this problem using geometric transformations, see [5, p. +236-241]. In the same chapter, Darboux reviews a method of Gauss on the +conformal representation on a sphere of a region of the Earth whose figure +is assumed to be spheroidal such that the similarity ratio is constant along a +meridian. Applying this method to the map of France, Darboux finds that +the variation of the similarity ratio, on the whole extent of the map, is only +of the order of +1 +400.000 of its actual value. +Darboux gives a modern formulation and a detailed modern proof of the +main problem stated by Lagrange in his memoir and which we recalled at +the beginning of this section. In Darboux’ formulation, the solution consists +of the following steps: +1. Perform a stereographic projection on the equator plane, with the +North pole as center. This gives a first map, call it A. +2. Transform the map A by a mapping defined by the equations +ρc = ρ′, +cω = ω′ +where (ρ, ω) and (ρ′, ω′) are polar coordinates and where the pole +is taken to be center of the sphere (Darboux calls this a Lambert +2 + +transformation). The constant c is what Lagrange calls the exponent +of the projection. This gives a second map, call it B. +3. Apply an arbitrary inversion, with pole in the plane of the equator. +This gives the needed map. +Let me mention now another problem of Lagrange to which he was led in +the same memoir, and of which Darboux gave a solution. This is a problem +in plane geometry, whose statement is the following: +Given three points R, R′, R′′, to construct on a fixed basis AB three +triangles whose vertices are the given points and such that: +1. the differences of the angles at the vertices BRA, BR′A, BR′′A are +given; +2. the ratios of the sides containing these angles, that is, RB +RA, R′B +R′A, R′′B +R′′A, +are given. +Lagrange says that this problem seems to him quite difficult to solve by +geometry, and that he did not try the algebraic solution because this seemed +to him useless, unless one can reduce it to an easy construction. He then +sketches a solution of the problem using the analytical methods he developed +in the first part of the memoir. +Darboux, some 130 years after Lagrange, published an article titled Sur +un problème posé par Lagrange (On a problem posed by Lagrange) [6] in +which he gave a solution of this problem using the new geometrical methods +that were available at his time, after transforming it into a problem of finding +an inversion in the plane which transforms one triangle into another one. +Darboux writes that the advances made in geometrical methods allow now +to solve easily and elegantly the problem that Lagrange indicated, and that +this problem is reduced to the following: +Given a triangle ABC in the plane, to find an inversion that turns it +into a triangle equal to another given triangle A0B0C0. +Darboux then gives a concise geometric and complete solution to this +problem. (The same solution by Darboux is contained in Part I, §133, Chap- +ter IV of his Leçons sur la théorie générale des surfaces et les applications +géométriques du calcul infinitésimal, p. 241-243.) +3 +Chebyshev’s theorem and developments +At the end of §2 of his memoir [10], Lagrange gives a formula for the infinites- +imal dilatation at a point of an angle-preserving map between (a subset of) +3 + +the sphere and the plane. The notation is the following: s, t are the curvi- +linear coordinates of a point on the surface of the Earth (assimilated to a +sphere of radius 1), and x, y are the rectangular coordinates of its image +in the plane. The differential ds denotes the difference of the two arcs of +meridian passing through the infinitesimally close points on the sphere and +qdt denotes the arc of parallel contained by these two meridians. +Then, +(s + ds, t + dt) and (x + dx, y + dy) are respectively the curvilinear coor- +dinates of a point on the sphere which is infinitesimally close to (s, t) and +those of its image by the map. With this, Lagrange writes the following +formula for the infinitesimal dilatation at the given point on the sphere: +m = +� +dx2 + dy2 +ds2 + q2dt2 . +This formula has been used by P. L. Chebyshev as a starting point for +his investigations on the construction of geographical maps that minimize +distortion. More precisely, using the above notation, in two papers, carrying +the same titles as those of Lagrange, Chebyshev addressed the question +of determining the maps that minimize the deviation of the infinitesimal +dilatation m from its integral over the region considered on the sphere. He +used other formulae established by Lagrange in his paper and he obtained a +new result, namely, that among all the conformal representations of a subset +of the sphere onto the plane, the representation that minimizes distortion is +the one for which this distortion is constant on the boundary of the region, +see the memoirs [3] and [4]. Chebyshev also made a relation between this +study and the Laplace equation. +Since in the previous section we talked about Darboux’ work on geogra- +phy, let us mention here that in 1911, Darboux published a memoir, whose +title is the same as those of Lagrange and Chebyshev (On the construction of +geographical maps), in which he presented a proof of Chebyshev’s theorem, +see [7]. As a matter of fact, in his paper, Darboux attributes to Chebyshev +a more general result, valid for any surface of revolution. +Let us also mention the work of D. A. Gravé, a student of Chebyshev. +Whereas Lagrange, in his paper [10], considered the problem of finding all +the angle-preserving maps for which the meridians and the parallels are sent +to circles or straight lines, Gravé worked on the problem of characterizing +the area-preserving maps for which the meridians and the parallels are sent +to circles or straight lines. He gave a complete solution of this problem in a +paper which is also titled Sur la construction des cartes géographiques, see [8]. +Furthermore, in 1894, Gravé presented an outline of a proof of Chebyshev’s +4 + +theorem (which the latter has only sketched), at the annual meeting of the +Association française pour l’avancement des sciences which took place in +Caen.1 In 1896, Gravé published in Russian a complete proof of Chebyshev’s +theorem, and in 1911 he published a paper in French titled Sur un théorème +de Tchébychef généralisé (on a generalized theorem of Chebyshev), in which +he gave a detailed proof of a slightly more general result, valid not only for +the sphere, but for an arbitrary surface whose curvature does not change +sign (see [9]). +Finally, we note that a modern proof of Chebyshev’s theorem, which is +based upon his ideas, was given about a century after Chebyshev found it, by +John Milnor [14], who also pointed out further developments, highlighting +the case where the region of the sphere which is mapped is geodesically +convex. Milnor writes in his paper: “This result has been available for more +than a hundred years, but to my knowledge it has never been used by actual +map makers." +4 +Lagrange’s memoir in the modern literature +We start this section by recalling the idea of the Schwarzian derivative. This +is a function associated with a smooth function of a real variable, or a holo- +morphic function of a complex variable, which measures the deviation of +such a function from being a Möbius transformation. A more precise def- +inition considers the Schwarzian derivative as a 1-cocycle on the group of +diffeomorphisms of the projective space RP1 with coefficients in the space of +1There is a written note by Gravé titled Sur une question de Tchébychef (On a question +of Chebyshev) in the 1895 publications of the Association. Regarding the many possible +transliteration of Chebyshev’s name from the cyrillic into Latin characters, Gravé, uses +Tchebichef, the one that was used by Chebyshev himself when he signed the (numerous) +papers he wrote in French. In fact, this is the unique way in which Chebyshev wanted his +name to be transliterated. M. d’Ocagne published a note on this matter in the Bulletin +des sciences mathématiques, titled “On the spelling of the name of Tchebichef" [15], in +which he writes: “Because of the many variants of the spelling of the name Tchebichef +which one can find in the contemporary mathematical publications, let me recall the +following fact which I have already pointed out. During his last stay in Paris, in May +1893, the illustrious Russian geometer accepted to entrust me with the task of writing +a complete description (which had never been given even in Russian) of his intriguing +arithmetic machine, a description which he reviewed himself with the greatest care. On +this occasion, he personally prescribed the French spelling of his name under the form +‘Tchebichef’, formulating the wish that it is exclusively adopted in all his writings written +with our characters. It is because he recommended me to ensure his desideratum regarding +this point, from which common usage deviates, that the preceding lines have been written." +Obviously, Chebyshev’s wish was not fulfilled. +5 + +quadratic differentials (holomorphic quadratic differential forms) and kernel +in PSL(2, R); the Schwarzian derivative is the only such cocycle, see [17]. +The clearest geometric definition is probably that given by Thurston in [18], +in which a Schwarzian derivative becomes simply a quadratic differential. +His aim in his paper is to study a topology on the space of conformal maps +from the unit disc to simply connected domains of the complex plane; such +a topology is induced by the topology of uniform convergence of Schwarzian +derivatives. In the introduction of his paper, Thurston makes an analogy +between the notion of Schwarzian derivative and that of curvature in differ- +ential geometry. +V. Ovsienko and S. Tabachnikov notice in [16, 17] that the Schwarzian +derivative was already introduced by Lagrange in his paper on the construc- +tion of geographical maps. It is also reported in [17] that Hermann Schwarz, +who introduced Schwarzian derivatives, declared that this notion is present +at least implicitly in Lagrange’s memoir. According to an explanation given +to the author of this chapter by Ovsienko, it is in §10 of Lagrange’s memoir, +starting with the definition of ϕ and Φ, that this notion occurs. Indeed, +Lagrange works with the quotients ϕ′′/ϕ and Φ′′/Φ and he compares them, +and two functions, f and F appear in this context. A small computation +of ϕ′′/ϕ in terms of f and Φ′′/Φ in terms of F shows that they are equal +precisely to 1/2S(f) and 1/2S(F), where S denotes the Schwarzian deriva- +tive. The result is that Lagrange expresses the property of sending circles +to circles in terms of having equal Schwarzian derivatives. +Another relation of Lagrange’s memoir with modern works concerns the +point of view of metric geometry. The problem of finding maps from the +sphere to the plane in which circles are sent to straight lines, which is a +particular case of the general problem Lagrange addressed in his memoir on +geography, has been considered by V. S. Matveev and others as an early +formulation of the general problem of finding maps between surfaces that +send geodesics to geodesics, and of the theory of geodesically equivalent +metrics, see the papers [13, 2], and the older paper by Beltrami [1] in which +the latter says that his work is motivated by the question of constructing +geographical maps. +5 +On Lagrange’s work and practical cartography +It is not clear to what extent Lagrange’s work has been used in practical +cartography, but there are indications for the fact that some attempts were +made for that. In a letter to Lagrange dated February 14, 1782, Laplace +6 + +writes [12, Vol. XIV, p. 111]: “[. . . ] Your two memoirs on the construction +of geographical maps gave me as much pleasure. Above all, I admired the +elegant manner with which you extracted from the general solution of the +problem the case where the meridian[s] and the parallels are represented by +circles. Besides, your analysis has the merit of being useful in the practice +of constructing particular maps, and I have engaged one of my friends, who +just announced a big atlas, to use it." +Around the beginning of the 20th century, Jules de Schokalsky, a colonel +of the Imperial Russian Navy and secretary of the Physics Section of the +Imperial Russian Geographical Society, was entrusted with the publication +of an atlas of universal geography, containing, among other things, a series +of maps of European Russia on a scale of 1: 1,640,000. From N. de Zinger’s +article La projection de Lagrange appliquée à la carte de la Russie d’Europe +(The Lagrange projection applied to the map of European Europe) [19], we +learn that it was a so-called “Lagrange projection" that was chosen, with +some convenient constants. +Even though it is not clear to what part of +Lagrange’s memoir the author refers, this indicates at least that Lagrange’s +work was used in geography. The article contains details on the choice of +the constants. +6 +In guise of a conclusion +Let us conclude this chapter by recalling that Lagrange, besides being a +mathematician, was also an astronomer,2 and that he was aware of the +fact that astronomical observations, together with spherical trigonometry, +were at the foundations of geodesy. Let me quote from §9 of his memoir +Solution de quelques problèmes relatifs aux triangles sphériques, avec une +analyse complète de ces triangles (Solution of some problems relative to +spherical triangles, with a complete analysis of these triangles) [11], men- +tioning geographical applications of his trigonometric formula. He writes: +“[. . . ] the formula we just gave will also be useful for measuring spherical +surfaces terminated by arcs of great circles. Thus it can be employed with +great advantage to determine the extent of a country, when we know the +latitudes and differences in longitude of several points placed at the circum- +ference; for, by linking these points by arcs of great circles, we shall have +a spherical polygon, whose area we shall easily find by decomposing it into +2For instance, Lagrange’s work on the two body problem (applied to the Eath-moon, to +the Earth-sun, and to other pairs), in which he introduced the equilibrium points known +now under the name Lagrange points, is a major contribution to celestial mechanics. +7 + +quadrilaterals formed by the circles of latitude and by the arcs of the equator +intercepted between these circles." +References +[1] E. Beltrami, Risoluzione del problema: riportare i punti di una superfi- +cie sopra un piano in modo che le linee geodetiche vengano rappresen- +tate da linee rette, Ann. Mat., 1(1865), no. 7, 185-204. +[2] A. V. Bolsinov and V. S. Matveev: Local normal forms for geodesically +equivalent pseudo-Riemannian metrics, J. of Geometry and Physics, 44 +(2003), 489-506. +[3] P. L. Chebyshev, Sur la construction des cartes géographiques. Bulletin +de la classe physico-mathématique de l’Académie Impériale des Sciences +de Saint-Pétersbourg, Tome VIV, 1856, p. 257-261. Reprinted in P. L. +Tchebycheff, Œuvres, Vol. 1, p. 233–236, Saint Petersburg, 1899, +[4] P. L. Chebyshev, Sur la construction des cartes géographiques. Dis- +cours prononcé le 8 février 1856 dans la séance solennelle de l’Université +Impériale de Saint-Pétersbourg, transl. A. Gravé, reprinted in P. L. +Tchebycheff, Œuvres, Vol. 1, p. 239-247, Saint Petersburg, 1899. +[5] G. Darboux, Leçons sur la théorie générale des surfaces et les appli- +cations géométriques du calcul infinitésimal, Gauthier-Villars, Paris, 4 +volumes, 1st edition, 1889. +[6] G. Darboux, Sur un problème posé par Lagrange, Bulletin des sciences +mathématiques, 35 (1911), p. 28-30. +[7] G. Darboux. Sur la construction des cartes géographiques, Bulletin des +Sciences Mathématiques 35 (1911), 23-28. +[8] D. A. Gravé, Sur la construction des cartes géographiques, Journal de +mathématiques pures et appliquées 5e série, 2 (1896), 317-362. +[9] D. A. Gravé, Démonstration d’un théorème de Tchébychef généralisé, +Journal für die reine und angewandte Mathematik 40 (1911), No. 4, p. +247-251. +[10] J.-L. Lagrange, Sur la construction des cartes géographiques, Nouveaux +Mémoires de l’Académie royale des Sciences et des Belles-Lettres de +Berlin, 1779, 161–210. English translation, this volume, Chapter 15. +8 + +[11] J.-L. Lagrange, Solution de quelques problèmes relatifs aux triangles +sphériques, avec une analyse complète de ces triangles, Journal de +l’Ecole Polytechnique, 2 (1800), p. 270-297. +[12] J.-L. Lagrange, Œuvres, ed. J.-A. Serret and G. Darboux, Paris, +Gauthier-Villars, 1867-1892. +[13] S. Matveev, +Riemannian metrics having common geodesics with +Berwald metrics, Publ. Math. Debrecen, 74/3-4 (2009), 405-416. +[14] J. Milnor, A problem in cartography, The American Mathematical +Monthly Vol. 76, No. 10 (Dec. 1969), 1101-1112. +[15] M. d’Ocagne, Sur l’orthographe du nom de Tchebichef, Bull. Sc. Math. +55 (1931), p. 98. +[16] V. Ovsienko, Schwarzian derivative and symplectic Sturm theory, An- +nales de la Faculté des sciences de Toulouse, Mathématiques (1993), +Vol. 2, Issue 1, p. 73-96. +[17] V. Ovsienko and S. Tabachnikov, What is . . . the Schwarzian derivative? +Notices of the AMS 56 (2009) (1), p. 34-36. +[18] W. P. Thurston, Zippers and univalent functions. The Bieberbach +conjecture (West Lafayette, Ind., 1985), p. 185-197, Math. Surveys +Monogr., 21, Amer. Math. Soc., Providence, RI, 1986. +[19] N. de Zinger, La projection de Lagrange appliquée à la carte de la Russie +d’Europe, Comptes Rendus Ac. Sci. Paris, 1906, p. 211-213. +9 + diff --git a/jNAzT4oBgHgl3EQfpP3S/content/tmp_files/load_file.txt b/jNAzT4oBgHgl3EQfpP3S/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..4787648315817bcb93a2b9df78bee15038bc5bb8 --- /dev/null +++ b/jNAzT4oBgHgl3EQfpP3S/content/tmp_files/load_file.txt @@ -0,0 +1,232 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf,len=231 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content='01611v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content='DG] 4 Jan 2023 Some notes on the impact of Lagrange’s memoir “On the construction of geographical maps" Athanase Papadopoulos∗ Abstract These are notes on the impact of Lagrange’s memoir on the con- struction of geographical maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' We mention the relations of some ideas and questions introduced in this memoir with other notions that ap- peared later in the works of several mathematicians, including in par- ticular Chebyshev (19th c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=') and Darboux (19th-20th c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content='), two math- ematicians who were particularly interested in geography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' The final version of this paper appears in the book: Mathematical Geography in the Eighteenth Century: Euler, Lagrange and Lambert (ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' Caddeo and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' Papadopoulos), Springer, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' 1 Introduction In this chapter, I shall mention the impact of some ideas and questions introduced by Lagrange in his memoir Sur la construction des cartes géo- graphiques (On the construction of geographical maps) [10] on the works of some later authors including Darboux (§2), Chebyshev (§3) and others (§3 and 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' The last section (§5) contains some notes on the use of Lagrange’s ideas in the actual construction of geographical maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' 2 Lagrange’s memoir in the work of Darboux Lagrange, in his memoir Sur la construction des cartes géographiques, moti- vated by works of Lambert and Euler, addressed the following problem: ∗Institut de Recherche Mathématique Avancée (Université de Strasbourg et CNRS) 7 rue René Descartes 67084 Strasbourg Cedex France email : athanase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content='papadopoulos@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content='unistra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content='fr 1 To find all the conformal projections from the sphere (and more gen- erally from a surface of revolution) onto the Euclidean plane by which the meridians and parallels are sent to circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' This problem of Lagrange is mentioned several times by Darboux, in his Leçons sur la théorie générale des surfaces et les applications géométriques du calcul infinitésimal [5], and in particular in Chapter IV of Book II of Part I, which concerns conformal maps between surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' In that chapter, Darboux, after having studied in the preceding chapters the general meth- ods of isothermal coordinates, applies them to the resolution of problems posed by the construction of geographical maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' He considers Lagrange’s problem (referring to the latter as the first mathematician who addressed it) in the setting of the spheroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' Let us recall that in this setting, a merid- ian is an ellipse (or half-ellipse) whose rotation about an axis generates the spheroid, and a parallel is the orbit of a point on this ellipse by this rota- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' The meridians are geodesics of the resulting surface but the parallels are not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' (This holds in the general case of a surface of revolution in Eu- clidean space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' Darboux makes the remark that if one of the two families (the meridians and the parallels) is represented by circles, then the same holds for the second family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' Thus, Lagrange’s conditions (that both fam- ilies are represented by circles) are redundant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' He discusses at length the various solutions to this problem using geometric transformations, see [5, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' 236-241].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' In the same chapter, Darboux reviews a method of Gauss on the conformal representation on a sphere of a region of the Earth whose figure is assumed to be spheroidal such that the similarity ratio is constant along a meridian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' Applying this method to the map of France, Darboux finds that the variation of the similarity ratio, on the whole extent of the map, is only of the order of 1 400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content='000 of its actual value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' Darboux gives a modern formulation and a detailed modern proof of the main problem stated by Lagrange in his memoir and which we recalled at the beginning of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' In Darboux’ formulation, the solution consists of the following steps: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' Perform a stereographic projection on the equator plane, with the North pole as center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' This gives a first map, call it A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' Transform the map A by a mapping defined by the equations ρc = ρ′, cω = ω′ where (ρ, ω) and (ρ′, ω′) are polar coordinates and where the pole is taken to be center of the sphere (Darboux calls this a Lambert 2 transformation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' The constant c is what Lagrange calls the exponent of the projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' This gives a second map, call it B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' Apply an arbitrary inversion, with pole in the plane of the equator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' This gives the needed map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' Let me mention now another problem of Lagrange to which he was led in the same memoir, and of which Darboux gave a solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' This is a problem in plane geometry, whose statement is the following: Given three points R, R′, R′′, to construct on a fixed basis AB three triangles whose vertices are the given points and such that: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' the differences of the angles at the vertices BRA, BR′A, BR′′A are given;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' the ratios of the sides containing these angles, that is, RB RA, R′B R′A, R′′B R′′A, are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' Lagrange says that this problem seems to him quite difficult to solve by geometry, and that he did not try the algebraic solution because this seemed to him useless, unless one can reduce it to an easy construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' He then sketches a solution of the problem using the analytical methods he developed in the first part of the memoir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' Darboux, some 130 years after Lagrange, published an article titled Sur un problème posé par Lagrange (On a problem posed by Lagrange) [6] in which he gave a solution of this problem using the new geometrical methods that were available at his time, after transforming it into a problem of finding an inversion in the plane which transforms one triangle into another one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' Darboux writes that the advances made in geometrical methods allow now to solve easily and elegantly the problem that Lagrange indicated, and that this problem is reduced to the following: Given a triangle ABC in the plane, to find an inversion that turns it into a triangle equal to another given triangle A0B0C0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' Darboux then gives a concise geometric and complete solution to this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' (The same solution by Darboux is contained in Part I, §133, Chap- ter IV of his Leçons sur la théorie générale des surfaces et les applications géométriques du calcul infinitésimal, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' 241-243.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=') 3 Chebyshev’s theorem and developments At the end of §2 of his memoir [10], Lagrange gives a formula for the infinites- imal dilatation at a point of an angle-preserving map between (a subset of) 3 the sphere and the plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' The notation is the following: s, t are the curvi- linear coordinates of a point on the surface of the Earth (assimilated to a sphere of radius 1), and x, y are the rectangular coordinates of its image in the plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' The differential ds denotes the difference of the two arcs of meridian passing through the infinitesimally close points on the sphere and qdt denotes the arc of parallel contained by these two meridians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' Then, (s + ds, t + dt) and (x + dx, y + dy) are respectively the curvilinear coor- dinates of a point on the sphere which is infinitesimally close to (s, t) and those of its image by the map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' With this, Lagrange writes the following formula for the infinitesimal dilatation at the given point on the sphere: m = � dx2 + dy2 ds2 + q2dt2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' This formula has been used by P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' Chebyshev as a starting point for his investigations on the construction of geographical maps that minimize distortion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' More precisely, using the above notation, in two papers, carrying the same titles as those of Lagrange, Chebyshev addressed the question of determining the maps that minimize the deviation of the infinitesimal dilatation m from its integral over the region considered on the sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' He used other formulae established by Lagrange in his paper and he obtained a new result, namely, that among all the conformal representations of a subset of the sphere onto the plane, the representation that minimizes distortion is the one for which this distortion is constant on the boundary of the region, see the memoirs [3] and [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' Chebyshev also made a relation between this study and the Laplace equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' Since in the previous section we talked about Darboux’ work on geogra- phy, let us mention here that in 1911, Darboux published a memoir, whose title is the same as those of Lagrange and Chebyshev (On the construction of geographical maps), in which he presented a proof of Chebyshev’s theorem, see [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' As a matter of fact, in his paper, Darboux attributes to Chebyshev a more general result, valid for any surface of revolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' Let us also mention the work of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' Gravé, a student of Chebyshev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' Whereas Lagrange, in his paper [10], considered the problem of finding all the angle-preserving maps for which the meridians and the parallels are sent to circles or straight lines, Gravé worked on the problem of characterizing the area-preserving maps for which the meridians and the parallels are sent to circles or straight lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' He gave a complete solution of this problem in a paper which is also titled Sur la construction des cartes géographiques, see [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' Furthermore, in 1894, Gravé presented an outline of a proof of Chebyshev’s 4 theorem (which the latter has only sketched), at the annual meeting of the Association française pour l’avancement des sciences which took place in Caen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content='1 In 1896, Gravé published in Russian a complete proof of Chebyshev’s theorem, and in 1911 he published a paper in French titled Sur un théorème de Tchébychef généralisé (on a generalized theorem of Chebyshev), in which he gave a detailed proof of a slightly more general result, valid not only for the sphere, but for an arbitrary surface whose curvature does not change sign (see [9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' Finally, we note that a modern proof of Chebyshev’s theorem, which is based upon his ideas, was given about a century after Chebyshev found it, by John Milnor [14], who also pointed out further developments, highlighting the case where the region of the sphere which is mapped is geodesically convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' Milnor writes in his paper: “This result has been available for more than a hundred years, but to my knowledge it has never been used by actual map makers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content='" 4 Lagrange’s memoir in the modern literature We start this section by recalling the idea of the Schwarzian derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' This is a function associated with a smooth function of a real variable, or a holo- morphic function of a complex variable, which measures the deviation of such a function from being a Möbius transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' A more precise def- inition considers the Schwarzian derivative as a 1-cocycle on the group of diffeomorphisms of the projective space RP1 with coefficients in the space of 1There is a written note by Gravé titled Sur une question de Tchébychef (On a question of Chebyshev) in the 1895 publications of the Association.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' Regarding the many possible transliteration of Chebyshev’s name from the cyrillic into Latin characters, Gravé, uses Tchebichef, the one that was used by Chebyshev himself when he signed the (numerous) papers he wrote in French.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' In fact, this is the unique way in which Chebyshev wanted his name to be transliterated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' d’Ocagne published a note on this matter in the Bulletin des sciences mathématiques, titled “On the spelling of the name of Tchebichef" [15], in which he writes: “Because of the many variants of the spelling of the name Tchebichef which one can find in the contemporary mathematical publications, let me recall the following fact which I have already pointed out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' During his last stay in Paris, in May 1893, the illustrious Russian geometer accepted to entrust me with the task of writing a complete description (which had never been given even in Russian) of his intriguing arithmetic machine, a description which he reviewed himself with the greatest care.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' On this occasion, he personally prescribed the French spelling of his name under the form ‘Tchebichef’, formulating the wish that it is exclusively adopted in all his writings written with our characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' It is because he recommended me to ensure his desideratum regarding this point, from which common usage deviates, that the preceding lines have been written.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content='" Obviously, Chebyshev’s wish was not fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' 5 quadratic differentials (holomorphic quadratic differential forms) and kernel in PSL(2, R);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' the Schwarzian derivative is the only such cocycle, see [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' The clearest geometric definition is probably that given by Thurston in [18], in which a Schwarzian derivative becomes simply a quadratic differential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' His aim in his paper is to study a topology on the space of conformal maps from the unit disc to simply connected domains of the complex plane;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' such a topology is induced by the topology of uniform convergence of Schwarzian derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' In the introduction of his paper, Thurston makes an analogy between the notion of Schwarzian derivative and that of curvature in differ- ential geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' Ovsienko and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' Tabachnikov notice in [16, 17] that the Schwarzian derivative was already introduced by Lagrange in his paper on the construc- tion of geographical maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' It is also reported in [17] that Hermann Schwarz, who introduced Schwarzian derivatives, declared that this notion is present at least implicitly in Lagrange’s memoir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' According to an explanation given to the author of this chapter by Ovsienko, it is in §10 of Lagrange’s memoir, starting with the definition of ϕ and Φ, that this notion occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' Indeed, Lagrange works with the quotients ϕ′′/ϕ and Φ′′/Φ and he compares them, and two functions, f and F appear in this context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' A small computation of ϕ′′/ϕ in terms of f and Φ′′/Φ in terms of F shows that they are equal precisely to 1/2S(f) and 1/2S(F), where S denotes the Schwarzian deriva- tive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' The result is that Lagrange expresses the property of sending circles to circles in terms of having equal Schwarzian derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' Another relation of Lagrange’s memoir with modern works concerns the point of view of metric geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' The problem of finding maps from the sphere to the plane in which circles are sent to straight lines, which is a particular case of the general problem Lagrange addressed in his memoir on geography, has been considered by V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' Matveev and others as an early formulation of the general problem of finding maps between surfaces that send geodesics to geodesics, and of the theory of geodesically equivalent metrics, see the papers [13, 2], and the older paper by Beltrami [1] in which the latter says that his work is motivated by the question of constructing geographical maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' 5 On Lagrange’s work and practical cartography It is not clear to what extent Lagrange’s work has been used in practical cartography, but there are indications for the fact that some attempts were made for that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' In a letter to Lagrange dated February 14, 1782, Laplace 6 writes [12, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' XIV, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' 111]: “[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' ] Your two memoirs on the construction of geographical maps gave me as much pleasure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' Above all, I admired the elegant manner with which you extracted from the general solution of the problem the case where the meridian[s] and the parallels are represented by circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' Besides, your analysis has the merit of being useful in the practice of constructing particular maps, and I have engaged one of my friends, who just announced a big atlas, to use it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content='" Around the beginning of the 20th century, Jules de Schokalsky, a colonel of the Imperial Russian Navy and secretary of the Physics Section of the Imperial Russian Geographical Society, was entrusted with the publication of an atlas of universal geography, containing, among other things, a series of maps of European Russia on a scale of 1: 1,640,000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' From N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' de Zinger’s article La projection de Lagrange appliquée à la carte de la Russie d’Europe (The Lagrange projection applied to the map of European Europe) [19], we learn that it was a so-called “Lagrange projection" that was chosen, with some convenient constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' Even though it is not clear to what part of Lagrange’s memoir the author refers, this indicates at least that Lagrange’s work was used in geography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' The article contains details on the choice of the constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' 6 In guise of a conclusion Let us conclude this chapter by recalling that Lagrange, besides being a mathematician, was also an astronomer,2 and that he was aware of the fact that astronomical observations, together with spherical trigonometry, were at the foundations of geodesy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' Let me quote from §9 of his memoir Solution de quelques problèmes relatifs aux triangles sphériques, avec une analyse complète de ces triangles (Solution of some problems relative to spherical triangles, with a complete analysis of these triangles) [11], men- tioning geographical applications of his trigonometric formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' He writes: “[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' ] the formula we just gave will also be useful for measuring spherical surfaces terminated by arcs of great circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' Thus it can be employed with great advantage to determine the extent of a country, when we know the latitudes and differences in longitude of several points placed at the circum- ference;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' for, by linking these points by arcs of great circles, we shall have a spherical polygon, whose area we shall easily find by decomposing it into 2For instance, Lagrange’s work on the two body problem (applied to the Eath-moon, to the Earth-sun, and to other pairs), in which he introduced the equilibrium points known now under the name Lagrange points, is a major contribution to celestial mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' 7 quadrilaterals formed by the circles of latitude and by the arcs of the equator intercepted between these circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content='" References [1] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' Beltrami, Risoluzione del problema: riportare i punti di una superfi- cie sopra un piano in modo che le linee geodetiche vengano rappresen- tate da linee rette, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=', 1(1865), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' 7, 185-204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' [2] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' Bolsinov and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' Matveev: Local normal forms for geodesically equivalent pseudo-Riemannian metrics, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' of Geometry and Physics, 44 (2003), 489-506.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' [3] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' Chebyshev, Sur la construction des cartes géographiques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' Bulletin de la classe physico-mathématique de l’Académie Impériale des Sciences de Saint-Pétersbourg, Tome VIV, 1856, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' 257-261.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' Reprinted in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' Tchebycheff, Œuvres, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' 1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' 233–236, Saint Petersburg, 1899, [4] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' Chebyshev, Sur la construction des cartes géographiques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' Dis- cours prononcé le 8 février 1856 dans la séance solennelle de l’Université Impériale de Saint-Pétersbourg, transl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' Gravé, reprinted in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' Tchebycheff, Œuvres, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' 1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' 239-247, Saint Petersburg, 1899.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' [5] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} +page_content=' Darboux, Leçons sur la théorie générale des surfaces et les appli- cations géométriques du calcul infinitésimal, Gauthier-Villars, Paris, 4 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf'} diff --git a/jdE1T4oBgHgl3EQfgASs/content/tmp_files/2301.03225v1.pdf.txt b/jdE1T4oBgHgl3EQfgASs/content/tmp_files/2301.03225v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..0691e717f7a28b5029f58640d8a49a2256c84266 --- /dev/null +++ b/jdE1T4oBgHgl3EQfgASs/content/tmp_files/2301.03225v1.pdf.txt @@ -0,0 +1,668 @@ +ONLINE FAKE REVIEW DETECTION USING +SUPERVISED MACHINE LEARNING AND BERT MODEL +1Abrar Qadir Mir, 2Furqan Yaqub Khan, Mohammad Ahsan Chishti +1,3Dept. of IT, CU Kashmir, +J&K, India, 191131 +2Dept. of CSE, IIT Patna, Bihar, India 801106 +Corresponding Author: furkaan309@gmail.com + +Abstract +Online shopping stores have grown steadily over the past few years. Due to the massive growth +of these businesses, the detection of fake reviews has attracted attention. Fake reviews are +seriously trying to mislead customers and thereby undermine the honesty and authenticity of +online shopping environments. So far, various of fake review classifiers have been proposed +that take into account the actual content of the review. To improve the accuracies of existing +fake review classification or detection approaches, we propose to use BERT (Bidirectional +Encoder Representation from Transformers) model to extract word embeddings from texts (i.e. +reviews). Word embeddings are obtained in various basic methods such as SVM (Support +vector machine), Random Forests, Naive Bayes and others. The confusion matrix method was +also taken into account to evaluate and graphically represent the results. The results indicate +that the SVM classifiers outperforms the others in terms of accuracy and f1-score with an +accuracy of 87.81%, which is 7.6% higher than the classifier used in the previous study [5]. +Keywords: BERT Model ; Machine Learning; Fake reviews; inserting words; Control detection +1. Introduction +Online reviewing play vital role in various online shopping environments such as online hotel +reservations, restaurant reservations and more importantly in the case of e-commerce sites as +they make it possible for customers to judge the quality/price of different products in order. +evaluate the seller's services and also make purchasing decisions. The users using online +shopping platforms has been increasing significantly for quite some time now. Let’s take an +example of TripAdvisor, globally the most popular travel website, has approximately 455.5 +million users and more than 600 million user reviews covering more than 7.5 million hotel, +restaurant, and airline reservations [1]. However, some unscrupulous businesses, motivated by +self-interest, mislead some less cautions customers by writing fake reviews. In order to over +promote products of their choice while disparaging their alternate choices. Fake reviewing +drastically affect and threaten the authenticity of online shopping environments. Therefore, +detecting and removing such fake reviews has become vital and unconditionally important area +of research for the present and if not dealt properly possibly for future as well. The Fake review +detection can be manual or automatic but manual execution of the process is considered +expensive, time-consuming and incorrect compared to automated detection methods [2]. Over + +the past decade, many significant advances have been made in the automatic identification of +fake reviews. Many ML based tools and techniques have been effectively detecting fake +reviews, some of the tools and techniques are SVM and Naïve Bayes[3]. This study mainly +focuses on review content to detect fake reviews. NLP is used to create some review features +which are not directly related to data or text provided. We used BERT (Bidirectional Encoder +Representation from Transformers) to perform NLP tasks and other text processing such as +feature selection. +2. Related work +The online shopping environment has started to gain massive interest in recent past and plays +a important role in people's lives, saving time or saving money. People seem to rely quite +heavily on such online shopping. From this point of view, finding out online reviews is equally +important and is one of the hot research topics today. The problem of detecting fake reviewing +had been addressed since 2007 [4]. The basic categories for fake review detection research are +categorized as; textual and behavioural features. Text characters means characterization of text +of reviews or to analyse and characterise the content of review text. Behavioural traits +characterise the non-verbal characteristics of review data corpus. It mainly depends on the +behaviour of reviewers, like style of writing, expressions which express emotions, and +frequency of reviewing. Liu [6] proposed the first review detection model using review +similarity and products features as criteria. The quality of fake reviews, also called "spammers", +to post fake reviews of their products has been exploited. Ott [7] proposed a classification +model based on text polarity of review data corpus; created a dataset that contained four- +hundred reviews divided into 2 groups real and fake or artificial and calculated the +effectiveness of his model using different classifiers and on different system configurations. +Savage [8] improved the efficiency of Ott’s model by updating it with some new syntactic +features. Farris [9] proposed another model for classification of fake reviews using Genetic +Algorithm with Random Weight Network. This model specifically addressed spam detection. +This model generated excellent results and it was concluded with this model that spam reviews +can be classified with good precision, recall and accuracy using automated classifiers. Fake +reviewer identification: [10] Although the above methods worked well in detecting fake +comments and reviews, it was recognized that fake reviewer detection is an additional part and +needs to be invented. Huang [11] came up with another theory that inauthentic reviewers +usually write comments frequently in particular time periods, so he defined time frame for +comment upload time and estimated upload frequencies of fake comments which where +unusually different from real reviews, +i.e.; If, uploads > threshold; The reviewer is said to be fake. +Lim [12] examined a large number of reviewers and similar data and pointed out two features +of fake reviewers: 1) fake reviewers target review a “particular” product or a specific online +purchase. 2) The pattern and style of writing and the frequency of posting reviews is generally +different for a real and a fake reviewer. Based on these two complimentary traits of reviewing, +it modelled the reviewer's behaviour and calculated the reviewer's score using another +algorithm. In [13], Supervised ML were used to classify reveiws. The supervised classification + +techniques used are Naïve-Bayes, SVM, K-Star, KNN, and Decision trees. The +experimentation was performed on three different movie review labelled datasets.[14] having +1400, 2000, 10662 reviews per dataset. In [15], the Naïve-Bayes, SVM, Decision Tree +techniques were used along with classifiers like Random Forest and Maximum Entropy +Classifier. The dataset forming of Samsung products and services reviews with 10000 fake +reviews. In [16], Chicago hotel review dataset of 1600 reviews was used. In [17], The RNN, +Average GRNN, GRNN, CNN, and Bi-directional Average GRNN’s were used to classify +data corpuse. The dataset used was from [18] having both fake and real reviews based on +hotels, restaurants and doctors. Only textual features were taken into consideration discarding +behavioural features. + +3. Background +To parse the high volume data with velocity which can’t be dealt through routine algorithms +for time critical tasks with accuracy machine learning is tool of choice. Machine Learning has +an advantage of learning and developing algorithms on its own based on data patterns and +improvise them while in use with further data[19]. The classification of machine learning +algorithms is based on labels and action-reward theory[20]; Based on Labels the classes are +supervised, semi-supervised and unsupervised ML techniques while based on action-reward +theory reinforced machine learning techniques are there. Labels and data are both needed in +supervised machine learning techniques[21] while in unsupervised ML only data is provided +and relation is found between different data points based on desired attributes or functions. The +semi-supervised techniques are hybrid of supervised and unsupervised ML techniques. Finally, +Reinforced learning deals with reacting to particular scenario or action if the reaction is in the +right direction the priority is added to step taken that is the step is rewarded if not the priority +is decreased for that particular step that means the step is punished. Mainly Supervised ML +classifiers are used in this article. Few prominent supervised classifiers used are SVM which +separates the two needed classes via a hyperplane[22] Another one is naïve bayes classifier +which uses bayes theorm to get the probability of particular review to be of one of the given +classes. The equation for bayes theorm is P(A—B) = P(B—A)*P(A) P(B) [23]. +The K-Nearest Neighbour’s (KNN) algorithm [24] mostly used in statistical estimations and +as well in pattern recognitions. The distance function is used to classify the least distant +attributes into one cluster thus trying to minimize the intra-cluster distance and maximise the +inter-cluster distances in order to classify more accurately[25] Decision Tree [26] classifies +data based values of gini-ratio, entropy, information gain etc. Thus each decision point is +represented as an internal node of tree while each class is represented by leaf node of tree giving +the algorithm’s representation a tree like shape hence called decision tree. Random Forest [27] +is an ensemble of more than one decision trees usually all unique thus eliminating the problem +of overfitting which usually occurs in decision tree algorithm. Logistic regression [28] also +develops a hyperplane between two different datatypes based on logistic function or log +function. + + +4. Data files +Primarily four datasets viz hotel, doctor, restaurant and amazon datasets are used in this article +for experimentation(Table 1). Cornell Universities Positive review[29] and negative +review[30] datasets are merged to form hotel dataset as described in[29-30].These datasets are +considered the gold standard mock review data corpuses[30]. Turks or assigned Fake reviewers +were the ones responsible for creating fake reviews, one review per Turk. The reviews were +filtered as short and plagiarized review were removed. Turkers were instructed to write a +review which seems realistic in nature. Correspondingly geniune reviews were taken from +various online reviewing sites like Expedia, TripAdvisor and Hotels.com. The dataset contains +reviews of 20 hotels with 80 reviews of each one divided equally into 2 classes thus forming +1600 reviews half of class true reviews and half of class false reviews. Corresponding actual +true and false labels were also included for each review. Thus each review in the dataset has a +true/false label, hotel information, travel agency name, polarity (positive/negative), and review +content with average word count of 152 per review. Similarly two more datasets were created +[31] namely restaurant review dataset and doctor review dataset. Twenty fake reviews were +created for 10 most famous restaurants in Chicago and 356 positive fake reviews for doctor +review dataset were created by well rated American Turkers and true reviews of same number +were taken from customer reviews of those hotels and doctors. Amazon's dataset consists of +21,000 reviews, of which 10,500 were identified as fake by Amazon. Some additional +information was also provided about the reviews like class label, ratings, verified purchase (yes +or no), product category, and product ID. With average review rating as 4.13/5 as well with +55.7% of the data from verified purchases. The reviews were form 30 product categories 700 +reviews per category. These categories are identified as non-compliant with Amazon's policies. + + +Table 1 Fake review datasets used in this study + +5. The proposed system +The proposed approach shown in Figure 1. The approach is divided into 3 stages and provides +us with the best model for fake review classification given as follows: +Dataset +Fake/truthful reviews +Polarity +Aver. review length (words) +Hotel [29, 30] + +400/400 +Positive and negative +151.9 +Restaurant [31] +200/200 +Positive +137.1 +Doctor[32] +356/200 +Positive +102.4 + + + + + + + + + + + + + + + + + + + Fig 1: Working of the proposed model +1) The dataset is loaded into a BERT (Bidirectional Encoder Representation from +Transformers) to generate word embeddings, which are large vector representations of the text +words in the dataset. 2) The input is then loaded into the classification models for their training. +Training and testing data are divided in the ratio: 80:20, i.e.: 80% for training and the rest for +testing the model. 3) Results are evaluated using a confusion matrix representation for +Precision, Recall, F1score and Accuracy. 4) The best performing classifier is then saved and +later used to detect user reviews as fake or real. + + +6. Operation of the BERT model +BERT (Bidirectional Representations from Transformers) based on transformer – an attention +mechanism that learns contextual connections in text between words. A simple transformation +has an encoder to read the text input and a decoder to generate the task prediction. Since the +goal of BERT is to create a model of the language representation, it only requires part of the +coder. The input to BERT encoder is a set of tokens that are first transformed into vectors and +then processed in a neural network. However, BERT requires the input to be decorated with +some additional metadata before processing can begin. Transformer essentially composes a +layer that maps onto sequences of sequences, so the output is also a vector sequence. +6.1 Generating BERT Embedding +Generating a large sentence embedding is created using BERT. BERT is a pre-trained model. +Let the sentence Si be tokenized into words W = {w1, w2, w3,..,wn}. Each w∈ W is fed to +Dataset +Semantic +Vector +Generation +using BERT +Training +Classifier over +word embeddings +User +Reviews +Saving best +performing +Model +Evaluation of +Results +Real +Fake + +BERT to obtain the word embeddings for ws ∈W. Let the loaded embedding for w1 be E1. +Similarly, we obtain embeddings for all wn words of Si, and so we have Vn embeddings for n +words of sentence Si. Then all embeddings of the Si theorem are combined to form the large +embeddings (BV) of the Si theorem. So BE = {E1 ∪ E1 ∪ E1 ∪ ... ∪ En }. Inserts are generated +for the hotel dataset. Once we have all large embeddings of all reviews and their corresponding +labels, we load these embeddings into the classification models for training and testing, in ratio +of 80:20 training/testing data. +7. Experimental evaluation: +In this section, we present the results of six experiments and their evaluation using six different +machine learning classifiers, namely: SVM, Random Forest, Bagging classifier, AdaBoost +classifier, Naïve Bayes and K-NN classifier. We experimented with our classification model +on the Hotel, Restaurant and Doctor datasets. However, due to space constraints, we only show +the experimental results performed on the hotel dataset. The confusion matrices for all +implemented classifiers are given below: + + + + + +Figure 2: Confusion matrix for SVM + + Figure 3: Confusion matrix for Random Forest + + + + + +Figure 4: Confusion matrix for Bagging classifier + Figure 5: Confusion matrix for Adaptive Boost + +Accuracy =87.812500% +precision +recall +fl-score +roddns +deceptive +0.87 +0.90 +0.89 +170 +truthful +0.88 +0.85 +0.87 +150 +accuracy +0.88 +320 +macro avg +0.88 +0.88 +0.88 +320 +weighted avg +0.88 +0.88 +0.88 +320 +Confusion Matrix +140 +deceptive +153 +17 +120 +100 +an +80 +60 +truthful +22 +128 +40 + 20 +deceptive +truthful +Predicted labelAccuracy =83.437500% +precision +recall +fl-score +support +deceptive +0.85 +0.84 +0.84 +170 +truthful +0.82 +0.83 +0.83 +150 +accuracy +0.83 +320 +macro avg +0.83 +0.83 +0.83 +320 +weighted avg +0.83 +0.83 +0.83 +320 +Confusion Matrix +-140 +120 +deceptive +142 +28 +100 +True label +80 +60 +truthful +125 +40 +deceptive +truthful +Predicted labelAccuracy =79.062500% +precision +recall +fl-gcore +support +deceptive +0.79 +0.82 +0.81 +170 +truthful +0.79 +0.75 +0.77 +150 +accuracy +0.79 +320 +macro avg +0.79 +0.79 +0.79 +320 +weighted avg +0.79 +0.79 +0.79 +320 +Confusion Matrix +140 +120 +deceptive +100 +True I +80 +truthful +87 +1 + 60 +40 +deceptive +truthfulAccuracy =78.437500% +precision +recall +fl-score +support +deceptive +0.80 +0.79 +0.80 +170 +truthful +0.76 +0.78 +0.77 +150 +accuracy +0.78 +320 +macro avg +0.78 +0.78 +0.78 +320 +weighted avg +0.78 +0.78 +0.78 +320 +Confusion Matrix +120 +deceptive +134 +100 + 80 +truthful +33 +60 +40 +deceptive +truthful + + +Figure 6: Confusion matrix for Naïve Bayes +Figure 7: Confusion matrix for K-NN classifier + + + + Table 2: Comparative performance analysis of proposed model + + +7. Conclusion and future work +In this article, we've shown the importance of reviews and how they affect almost everything +related to web data. It's clear that reviews play a vital role in people's decision-making. Thus, +detecting fake reviews is a lively and ongoing area of research. A machine learning approach +to detect fake reviews is presented, and review properties are considered in the proposed +approach. The Hotels dataset is used to present an experimental evaluation of the proposed +approach. Different classifiers were used on this dataset. The results reveal that the SVM +classifier outperforms other classifiers (with 87.81% accuracy) in the process of detecting fake +reviews and therefore can be used to effectively classify reviews as real or fake by considering +only the text content of the reviews and not necessarily the sentiment traits. However, future +work may consider including behavioural features of reviewers, such as features that depend +on the number of times reviewers perform reviews, the time it takes reviewers to complete + + Classifier + Accuracy (%) + F-Score +Previous +work[18] +This Study +Previous +work[18] +This Study +SVM +80.75 +87.81 +0.80 +0.88 +Random Forest +79.31 +83.43 +0.79 +0.83 +Bagging +78.19 +79.06 +0.78 +0.79 +K-NN +71.38 +77.18 +0.68 +0.78 +AdaBoost +77.06 +78.43 +0.77 +0.78 +Gaussian Naïve Bayes + 81.25 + 78.43 +0.81 +0.78 + +Accuracy =78.437500% +precision +recall +fl-score +support +deceptive +0.81 +0.77 +0.79 +170 +truthful +0.75 +0.80 +0.78 +150 +accuracy +0.78 +320 +macro avg +0.78 +0.79 +0.78 +320 +weighted avg +0.79 +0.78 +0.78 +320 +Confusion Matrix +120 +deceptive +131 +6E +100 +80 +120 + 60 +truthful +30 +40 +deceptive +truthful +Predicted labelAccuracy =77.187500% +precision +recall +fl-score +support +deceptive +0.75 +0.86 +0.80 +170 +truthful +0.81 +0.67 +0.73 +150 +accuracy +0.77 +320 +macro avg +0.78 +0.77 +0.77 +320 +weighted avg +0.78 +0.77 +0.77 +320 +Confusion Matrix +140 +146 +120 +deceptive +24 +100 +an +80 +60 +truthful +49 +40 +deceptive +truthful +Predicted labelreviews, and how often they submit positive or negative reviews. Considering behavioural +features is highly expected to enhance the performance of the presented fake review detection +approach. Also, using neural network models to perform this task would be equally beneficial +for detecting fake reviews from large datasets. + + References +1. TripAdvisor Homepage. http://ir.tripadvisor.com/. Accessed 21 January 2019. +2. Harris, C.: Detecting deceptive opinion spam using human computation. In: Workshops +at AAAI on Artificial Intelligence, pp. 87–93. AAAI (2012) +3. Heydari, A., ali Tavakoli, M., Salim, N., Heydari, Z.: Detection of review spam: a +survey. Expert Syst. Appl. 42(7), 3634–3642 (2015) +4. N. Jindal and B. Liu, “Review spam detection,” in Proceedings of the16th International +Conference on World Wide Web, ser. WWW ’07,2007. +5. Hajek, P., Barushka, A. and Munk, M., 2020. Fake consumer review detection using +deep neural networks integrating word embeddings and emotion mining. 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Ott M, Cardie C, Hancock JT (2013) Negative deceptive opinion spam. In: 2013 +conference of the North American chapter of the association for computational +linguistics: human language tech- nologies, ACL, pp 497–501 +31. Li J, Ott M, Cardie C, Hovy E (2014) Towards a general rule for identifying deceptive +opinion spam. In: Proceedings of the 52nd annual meeting of the association for +computational linguistics, ACL, vol 1, pp 1566–1576. +32. Garcia L (2018) Deception on Amazon—an NLP exploration. +https://medium.com/@lievgarcia/deception-on-amazon- c1e30d977cfd. Accessed 01 +Sept 2019 + + + + diff --git a/jdE1T4oBgHgl3EQfgASs/content/tmp_files/load_file.txt b/jdE1T4oBgHgl3EQfgASs/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..88fca972b38c8ea300cf16e72974f5cf2854724d --- /dev/null +++ b/jdE1T4oBgHgl3EQfgASs/content/tmp_files/load_file.txt @@ -0,0 +1,448 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf,len=447 +page_content='ONLINE FAKE REVIEW DETECTION USING SUPERVISED MACHINE LEARNING AND BERT MODEL 1Abrar Qadir Mir, 2Furqan Yaqub Khan, Mohammad Ahsan Chishti 1,3Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' of IT, CU Kashmir, J&K, India, 191131 2Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' of CSE, IIT Patna, Bihar, India 801106 Corresponding Author: furkaan309@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='com Abstract Online shopping stores have grown steadily over the past few years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' Due to the massive growth of these businesses, the detection of fake reviews has attracted attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' Fake reviews are seriously trying to mislead customers and thereby undermine the honesty and authenticity of online shopping environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' So far, various of fake review classifiers have been proposed that take into account the actual content of the review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' To improve the accuracies of existing fake review classification or detection approaches, we propose to use BERT (Bidirectional Encoder Representation from Transformers) model to extract word embeddings from texts (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' reviews).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' Word embeddings are obtained in various basic methods such as SVM (Support vector machine), Random Forests, Naive Bayes and others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' The confusion matrix method was also taken into account to evaluate and graphically represent the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' The results indicate that the SVM classifiers outperforms the others in terms of accuracy and f1-score with an accuracy of 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='81%, which is 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='6% higher than the classifier used in the previous study [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' Keywords: BERT Model ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' Machine Learning;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' Fake reviews;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' inserting words;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' Control detection 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' Introduction Online reviewing play vital role in various online shopping environments such as online hotel reservations, restaurant reservations and more importantly in the case of e-commerce sites as they make it possible for customers to judge the quality/price of different products in order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=" evaluate the seller's services and also make purchasing decisions." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' The users using online shopping platforms has been increasing significantly for quite some time now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' Let’s take an example of TripAdvisor, globally the most popular travel website, has approximately 455.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='5 million users and more than 600 million user reviews covering more than 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='5 million hotel, restaurant, and airline reservations [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' However, some unscrupulous businesses, motivated by self-interest, mislead some less cautions customers by writing fake reviews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' In order to over promote products of their choice while disparaging their alternate choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' Fake reviewing drastically affect and threaten the authenticity of online shopping environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' Therefore, detecting and removing such fake reviews has become vital and unconditionally important area of research for the present and if not dealt properly possibly for future as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' The Fake review detection can be manual or automatic but manual execution of the process is considered expensive, time-consuming and incorrect compared to automated detection methods [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' Over the past decade, many significant advances have been made in the automatic identification of fake reviews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' Many ML based tools and techniques have been effectively detecting fake reviews, some of the tools and techniques are SVM and Naïve Bayes[3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' This study mainly focuses on review content to detect fake reviews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' NLP is used to create some review features which are not directly related to data or text provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' We used BERT (Bidirectional Encoder Representation from Transformers) to perform NLP tasks and other text processing such as feature selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=" Related work The online shopping environment has started to gain massive interest in recent past and plays a important role in people's lives, saving time or saving money." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' People seem to rely quite heavily on such online shopping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' From this point of view, finding out online reviews is equally important and is one of the hot research topics today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' The problem of detecting fake reviewing had been addressed since 2007 [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' The basic categories for fake review detection research are categorized as;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' textual and behavioural features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' Text characters means characterization of text of reviews or to analyse and characterise the content of review text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' Behavioural traits characterise the non-verbal characteristics of review data corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' It mainly depends on the behaviour of reviewers, like style of writing, expressions which express emotions, and frequency of reviewing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' Liu [6] proposed the first review detection model using review similarity and products features as criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' The quality of fake reviews, also called "spammers", to post fake reviews of their products has been exploited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' Ott [7] proposed a classification model based on text polarity of review data corpus;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' created a dataset that contained four- hundred reviews divided into 2 groups real and fake or artificial and calculated the effectiveness of his model using different classifiers and on different system configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' Savage [8] improved the efficiency of Ott’s model by updating it with some new syntactic features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' Farris [9] proposed another model for classification of fake reviews using Genetic Algorithm with Random Weight Network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' This model specifically addressed spam detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' This model generated excellent results and it was concluded with this model that spam reviews can be classified with good precision, recall and accuracy using automated classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' Fake reviewer identification: [10] Although the above methods worked well in detecting fake comments and reviews, it was recognized that fake reviewer detection is an additional part and needs to be invented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' Huang [11] came up with another theory that inauthentic reviewers usually write comments frequently in particular time periods, so he defined time frame for comment upload time and estimated upload frequencies of fake comments which where unusually different from real reviews, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' If, uploads > threshold;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' The reviewer is said to be fake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' Lim [12] examined a large number of reviewers and similar data and pointed out two features of fake reviewers: 1) fake reviewers target review a “particular” product or a specific online purchase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' 2) The pattern and style of writing and the frequency of posting reviews is generally different for a real and a fake reviewer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=" Based on these two complimentary traits of reviewing, it modelled the reviewer's behaviour and calculated the reviewer's score using another algorithm." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' In [13], Supervised ML were used to classify reveiws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' The supervised classification techniques used are Naïve-Bayes, SVM, K-Star, KNN, and Decision trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' The experimentation was performed on three different movie review labelled datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' [14] having 1400, 2000, 10662 reviews per dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' In [15], the Naïve-Bayes, SVM, Decision Tree techniques were used along with classifiers like Random Forest and Maximum Entropy Classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' The dataset forming of Samsung products and services reviews with 10000 fake reviews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' In [16], Chicago hotel review dataset of 1600 reviews was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' In [17], The RNN, Average GRNN, GRNN, CNN, and Bi-directional Average GRNN’s were used to classify data corpuse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' The dataset used was from [18] having both fake and real reviews based on hotels, restaurants and doctors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' Only textual features were taken into consideration discarding behavioural features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' Background To parse the high volume data with velocity which can’t be dealt through routine algorithms for time critical tasks with accuracy machine learning is tool of choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' Machine Learning has an advantage of learning and developing algorithms on its own based on data patterns and improvise them while in use with further data[19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' The classification of machine learning algorithms is based on labels and action-reward theory[20];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' Based on Labels the classes are supervised, semi-supervised and unsupervised ML techniques while based on action-reward theory reinforced machine learning techniques are there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' Labels and data are both needed in supervised machine learning techniques[21] while in unsupervised ML only data is provided and relation is found between different data points based on desired attributes or functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' The semi-supervised techniques are hybrid of supervised and unsupervised ML techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' Finally, Reinforced learning deals with reacting to particular scenario or action if the reaction is in the right direction the priority is added to step taken that is the step is rewarded if not the priority is decreased for that particular step that means the step is punished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' Mainly Supervised ML classifiers are used in this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' Few prominent supervised classifiers used are SVM which separates the two needed classes via a hyperplane[22] Another one is naïve bayes classifier which uses bayes theorm to get the probability of particular review to be of one of the given classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' The equation for bayes theorm is P(A—B) = P(B—A)*P(A) P(B) [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' The K-Nearest Neighbour’s (KNN) algorithm [24] mostly used in statistical estimations and as well in pattern recognitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' The distance function is used to classify the least distant attributes into one cluster thus trying to minimize the intra-cluster distance and maximise the inter-cluster distances in order to classify more accurately[25] Decision Tree [26] classifies data based values of gini-ratio, entropy, information gain etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' Thus each decision point is represented as an internal node of tree while each class is represented by leaf node of tree giving the algorithm’s representation a tree like shape hence called decision tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' Random Forest [27] is an ensemble of more than one decision trees usually all unique thus eliminating the problem of overfitting which usually occurs in decision tree algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' Logistic regression [28] also develops a hyperplane between two different datatypes based on logistic function or log function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' Data files Primarily four datasets viz hotel, doctor, restaurant and amazon datasets are used in this article for experimentation(Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' Cornell Universities Positive review[29] and negative review[30] datasets are merged to form hotel dataset as described in[29-30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='These datasets are considered the gold standard mock review data corpuses[30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' Turks or assigned Fake reviewers were the ones responsible for creating fake reviews, one review per Turk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' The reviews were filtered as short and plagiarized review were removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' Turkers were instructed to write a review which seems realistic in nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' Correspondingly geniune reviews were taken from various online reviewing sites like Expedia, TripAdvisor and Hotels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' The dataset contains reviews of 20 hotels with 80 reviews of each one divided equally into 2 classes thus forming 1600 reviews half of class true reviews and half of class false reviews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' Corresponding actual true and false labels were also included for each review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' Thus each review in the dataset has a true/false label, hotel information, travel agency name, polarity (positive/negative), and review content with average word count of 152 per review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' Similarly two more datasets were created [31] namely restaurant review dataset and doctor review dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' Twenty fake reviews were created for 10 most famous restaurants in Chicago and 356 positive fake reviews for doctor review dataset were created by well rated American Turkers and true reviews of same number were taken from customer reviews of those hotels and doctors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=" Amazon's dataset consists of 21,000 reviews, of which 10,500 were identified as fake by Amazon." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' Some additional information was also provided about the reviews like class label, ratings, verified purchase (yes or no), product category, and product ID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' With average review rating as 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='13/5 as well with 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='7% of the data from verified purchases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' The reviews were form 30 product categories 700 reviews per category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=" These categories are identified as non-compliant with Amazon's policies." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' Table 1 Fake review datasets used in this study 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' The proposed system The proposed approach shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' The approach is divided into 3 stages and provides us with the best model for fake review classification given as follows: Dataset Fake/truthful reviews Polarity Aver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' review length (words) Hotel [29, 30] 400/400 Positive and negative 151.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='9 Restaurant [31] 200/200 Positive 137.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='1 Doctor[32] 356/200 Positive 102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='4 Fig 1: Working of the proposed model 1) The dataset is loaded into a BERT (Bidirectional Encoder Representation from Transformers) to generate word embeddings, which are large vector representations of the text words in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' 2) The input is then loaded into the classification models for their training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' Training and testing data are divided in the ratio: 80:20, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' : 80% for training and the rest for testing the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' 3) Results are evaluated using a confusion matrix representation for Precision, Recall, F1score and Accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' 4) The best performing classifier is then saved and later used to detect user reviews as fake or real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' Operation of the BERT model BERT (Bidirectional Representations from Transformers) based on transformer – an attention mechanism that learns contextual connections in text between words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' A simple transformation has an encoder to read the text input and a decoder to generate the task prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' Since the goal of BERT is to create a model of the language representation, it only requires part of the coder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' The input to BERT encoder is a set of tokens that are first transformed into vectors and then processed in a neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' However, BERT requires the input to be decorated with some additional metadata before processing can begin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' Transformer essentially composes a layer that maps onto sequences of sequences, so the output is also a vector sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='1 Generating BERT Embedding Generating a large sentence embedding is created using BERT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' BERT is a pre-trained model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' Let the sentence Si be tokenized into words W = {w1, w2, w3,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='.,wn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' Each w∈ W is fed to Dataset Semantic Vector Generation using BERT Training Classifier over word embeddings User Reviews Saving best performing Model Evaluation of Results Real Fake BERT to obtain the word embeddings for ws ∈W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' Let the loaded embedding for w1 be E1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' Similarly, we obtain embeddings for all wn words of Si, and so we have Vn embeddings for n words of sentence Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' Then all embeddings of the Si theorem are combined to form the large embeddings (BV) of the Si theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' So BE = {E1 ∪ E1 ∪ E1 ∪ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' ∪ En }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' Inserts are generated for the hotel dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' Once we have all large embeddings of all reviews and their corresponding labels, we load these embeddings into the classification models for training and testing, in ratio of 80:20 training/testing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' Experimental evaluation: In this section, we present the results of six experiments and their evaluation using six different machine learning classifiers, namely: SVM, Random Forest, Bagging classifier, AdaBoost classifier, Naïve Bayes and K-NN classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' We experimented with our classification model on the Hotel, Restaurant and Doctor datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' However, due to space constraints, we only show the experimental results performed on the hotel dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' The confusion matrices for all implemented classifiers are given below: Figure 2: Confusion matrix for SVM Figure 3: Confusion matrix for Random Forest Figure 4: Confusion matrix for Bagging classifier Figure 5: Confusion matrix for Adaptive Boost Accuracy =87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='812500% precision recall fl-score roddns deceptive 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='87 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='89 170 truthful 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='87 150 accuracy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='88 320 macro avg 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='88 320 weighted avg 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='88 320 Confusion Matrix 140 deceptive 153 17 120 100 an 80 60 truthful 22 128 40 20 deceptive truthful Predicted labelAccuracy =83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='437500% precision recall fl-score support deceptive 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='84 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='84 170 truthful 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='83 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='83 150 accuracy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='83 320 macro avg 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='83 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='83 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='83 320 weighted avg 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='83 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='83 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='83 320 Confusion Matrix 140 120 deceptive 142 28 100 True label 80 60 truthful 125 40 deceptive truthful Predicted labelAccuracy =79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='062500% precision recall fl-gcore support deceptive 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='79 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='81 170 truthful 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='79 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='77 150 accuracy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='79 320 macro avg 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='79 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='79 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='79 320 weighted avg 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='79 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='79 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='79 320 Confusion Matrix 140 120 deceptive 100 True I 80 truthful 87 1 60 40 deceptive truthfulAccuracy =78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='437500% precision recall fl-score support deceptive 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='79 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='80 170 truthful 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='77 150 accuracy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='78 320 macro avg 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='78 320 weighted avg 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='78 320 Confusion Matrix 120 deceptive 134 100 80 truthful 33 60 40 deceptive truthful Figure 6: Confusion matrix for Naïve Bayes Figure 7: Confusion matrix for K-NN classifier Table 2: Comparative performance analysis of proposed model 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=" Conclusion and future work In this article, we've shown the importance of reviews and how they affect almost everything related to web data." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=" It's clear that reviews play a vital role in people's decision-making." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' Thus, detecting fake reviews is a lively and ongoing area of research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' A machine learning approach to detect fake reviews is presented, and review properties are considered in the proposed approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' The Hotels dataset is used to present an experimental evaluation of the proposed approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' Different classifiers were used on this dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' The results reveal that the SVM classifier outperforms other classifiers (with 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='81% accuracy) in the process of detecting fake reviews and therefore can be used to effectively classify reviews as real or fake by considering only the text content of the reviews and not necessarily the sentiment traits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content=' However, future work may consider including behavioural features of reviewers, such as features that depend on the number of times reviewers perform reviews, the time it takes reviewers to complete Classifier Accuracy (%) F-Score Previous work[18] This Study Previous work[18] This Study SVM 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='75 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='81 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='88 Random Forest 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='31 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='43 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='79 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='83 Bagging 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='19 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='79 K-NN 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='38 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='78 AdaBoost 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='06 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='43 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='77 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='78 Gaussian Naïve Bayes 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='25 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='43 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='81 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='78 Accuracy =78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='437500% precision recall fl-score support deceptive 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='81 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='77 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='79 170 truthful 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='78 150 accuracy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='78 320 macro avg 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='79 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='78 320 weighted avg 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='79 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='78 320 Confusion Matrix 120 deceptive 131 6E 100 80 120 60 truthful 30 40 deceptive truthful Predicted labelAccuracy =77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='187500% precision recall fl-score support deceptive 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='86 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='80 170 truthful 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='81 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='67 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='73 150 accuracy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='77 320 macro avg 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='77 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='77 320 weighted avg 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='77 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE1T4oBgHgl3EQfgASs/content/2301.03225v1.pdf'} +page_content='77 320 Confusion Matrix 140 146 120 deceptive 24 100 an 80 60 truthful 49 40 deceptive truthful Predicted labelreviews, 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N.; +Papavassiliou, J. Gauge sector +dynamics in QCD. Particles 2023, 1, +1–57. https://doi.org/ +Received: January 9, 2023 +Accepted: +Published: +Publisher’s Note: MDPI stays neutral +with regard to jurisdictional claims in +published maps and institutional affil- +iations. +Copyright: +© 2023 by the authors. +Licensee MDPI, Basel, Switzerland. +This article is an open access article +distributed +under +the +terms +and +conditions of the Creative Commons +Attribution (CC BY) license (https:// +creativecommons.org/licenses/by/ +4.0/). +Review +Gauge Sector Dynamics in QCD +Mauricio Narciso Ferreira1,† +and Joannis Papavassiliou1,† +1 +Department of Theoretical Physics and IFIC, University of Valencia and CSIC, E-46100, Valencia, Spain. +* +Correspondence: ansonar@uv.es (M. N. Ferreira); Joannis.Papavassiliou@uv.es (J. Papavassiliou) +† +These authors contributed equally to this work. +Abstract: The dynamics of the gauge sector of QCD give rise to nonperturbative phenomena that are +crucial for the internal consistency of the theory; most notably, they account for the generation of a gluon +mass through the action of the Schwinger mechanism, the taming of the Landau pole and the ensuing +stabilization of the gauge coupling, and the infrared suppression of the three-gluon vertex. In the present +work, we review some key advances in the ongoing investigation of this sector within the framework of +the continuum Schwinger function methods, supplemented by results obtained from lattice simulations. +Keywords: continuum Schwinger function methods; emergence of hadron mass; gluon mass genera- +tion; lattice QCD; nonperturbative quantum field theory; quantum chromodynamics; Schwinger-Dyson +equations; Schwinger mechanism +Contents +1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +2 +2. Basic concepts and general theoretical framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +4 +3. Schwinger mechanism in Yang-Mills theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +10 +4. Dynamical formation of massless poles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +13 +5. Generation of the gluon mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +15 +5.1. Gluon mass from the qµqν component . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +17 +5.2. Gluon mass from the gµν component: seagull identity and Ward identity displacement +. . . . . . . . . +18 +6. Renormalization group invariant interaction strength +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +20 +7. Three-gluon vertex and its planar degeneracy +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +22 +8. Ghost dynamics from Schwinger-Dyson equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +25 +9. Divergent ghost loops and their impact on the QCD Green’s functions . . . . . . . . . . . . . . . . . . . . . +31 +10. Ward identity displacement of the three-gluon vertex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +34 +11. The ghost-gluon kernel contribution to the Ward identity +. . . . . . . . . . . . . . . . . . . . . . . . . . . . +36 +12. Displacement function from lattice inputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +40 +13. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +41 +A. Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +42 +References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +43 +Particles 2023, 1, 1–57. https://doi.org/10.3390/particles1010000 +https://www.mdpi.com/journal/particles +arXiv:2301.02314v1 [hep-ph] 5 Jan 2023 + +BYparticlesParticles 2023, 1 +2 +1. Introduction +The systematic exploration of the Green’s functions (n-point correlation functions) of +Quantum Chromodynamics (QCD) [1] by means of continuous Schwinger function meth- +ods [2–9], such as Schwinger-Dyson equations (SDEs) [10–21] and functional renormalization +group [22–31], together with a plethora of gauge-fixed lattice simulations [32–89], has afforded +ample access to the dynamical mechanisms responsible for the nonperturbative properties +of this remarkable theory. Particularly prominent in this quest is the notion of the emergent +hadron mass (EHM) [3,8,9,90–94], together with its three supporting pillars: first, the genera- +tion of a gluon mass [18,32,94–127] through the action of the Schwinger mechanism [128,129]; +second, the construction of the process-independent effective charge [3,16,20,80,97,130–132], +which arises as the QCD analogue of the Gell-Mann–Low charge known from Quantum Elec- +trodynamics (QED) [133,134], and has associated to it a renormalization-group invariant (RGI) +scale of about half of the proton mass [20,80]; and third, the dynamical breaking of chiral +symmetry and the generation of constituent quark masses [10,17,135–159]. +The dynamics of the gauge sector of QCD, which encompasses both gluonic and ghost +interactions, is instrumental for the physical picture of the EHM outlined above. In fact, the +basic concepts and pivotal mechanisms sustaining the first two pillars of the EHM have their +original inception and most genuine realization in the realm of pure Yang-Mills theories [18, +94,95,97,110,113,118,160–162]. Therefore, in the present review, we focus precisely on the rich +dynamical content of the gauge sector, especially in relation with the generation of a gluon +mass scale out of the intricate gluon self-interactions. +The formulation of the nonperturbative QCD physics in terms of the Green’s functions +of the fundamental degrees of freedom, such as gluon and ghost propagators and vertices, +provides an intuitive framework for unraveling a wide array of subtle mechanisms; in fact, +certain distinctive features of these functions have been inextricably connected with key +phenomena such as gluon mass generation, violation of reflection positivity, and confinement, +to name a few. Thus, the saturation of the gluon propagator in the deep infrared [37,45– +49,52,55–57,59,61,64,66–68,78,82] has been interpreted as the unequivocal signal of a gluon +mass [32,97–101,104,106,108–110,113,161–167]; and the existence of an inflection point in the +same function has been argued to lead to a non-positive gluon spectral density [8], and the +ensuing loss of reflection positivity [8,11,13,16,168–172] for the dressed gluons. Similarly, the +masslessness of the ghost induces [173] a maximum in the gluon propagator, and a zero crossing +in the form factors of the three-gluon vertex [28,50,69,70,72,73,82,85,173–181], followed by an +infrared divergence for vanishing momenta. The dynamical origin of these special traits will +be the focal point of the analysis presented in the main body of this article. +The integral equations that govern the full momentum evolution of the Green’s functions, +known as SDEs, constitute the indispensable formal and practical instrument for unraveling +the special characteristics mentioned above. In their primordial form, the SDEs are rigorously +derived from the generating functional of the theory [134,182], and encode all dynamical +information on the correlation functions, within the entire range of physical momenta. In +practice, due to the enormous complexity of these equations, approximations and truncations +need to be implemented; but, unlike perturbation theory, no expansion parameter is available +in the strongly coupled regime of the theory for carrying out such a task. Despite this intrinsic +shortcoming, in recent years the SDE predictions have become particularly robust, in part due +to various theoretical advances, and in part thanks to the intense synergy with gauge-fixed +lattice simulations, as will be evidenced in the subsequent sections. +Typically, the Green’s function of QCD are defined within the quantization scheme ob- +tained by implementing the linear covariant (Rξ) gauges [183]. The corresponding SDEs are +derived and solved within this same quantization scheme, and in particular in the Landau +gauge (ξ = 0), where lattice simulations are almost exclusively performed; for studies away + +Particles 2023, 1 +3 +from the Landau gauge, see e.g., [55,61,67,75,76,111,115,121,184–192]. A great deal may be +learned, however, by considering the Green’s functions and corresponding SDEs formulated +within the “PT-BFM”scheme [110,193], namely the framework that arises from the fusion of the +pinch technique (PT) [14,97,101,194–196] with the background field method (BFM) [197–207]. +The main advantage of the PT-BFM originates from the fact that certain appropriately chosen +Green’s functions satisfy Abelian Slavnov-Taylor identities (STIs), whose tree-level form does +not get modified by quantum corrections. This situation is to be contrasted to the standard +STIs [208,209] obtained in the conventional framework of the linear covariant gauges, which +are deformed by non-trivial contributions stemming from the gauge sector of the theory. In +the present work, we will carry out computations and develop arguments within both frame- +works (Rξ and PT-BFM), and will elaborate on their connection by means of the so-called +Background-Quantum identities (BQIs) [14,210–212]. +The article is organized as follows: +• In Section 2 we introduce some basic notation and review certain prominent features of +the Green’s functions within both the linear gauges and the PT-BFM formalism [110,193]. We +stress, in particular, the properties of the auxiliary function G(q) [16,132,213,214], which relates +the gluon propagators with quantum and background gluons, and is intimately connected with +the definition of the process-independent and RGI interaction strength [16], to be discussed in +detail in Section 6. In addition, we elucidate with a concrete example the important property of +“block-wise” transversality, displayed by the background gluon self-energy [18,110,113]. +• In Section 3 we review the general principles associated with the Schwinger mech- +anism [128,129] that endows gauge bosons with an effective mass, focusing on the details +associated with its realization in the context of Yang-Mills theories. We place particular empha- +sis on the pivotal requirement that must be satisfied by the fundamental vertices of the theory, +namely the appearance of massless poles in their form factors [18,94,110,112–114,118,160,215]. +• In Section 4 we examine the dynamical formation of colored composite excitations (bound +states) of vanishing mass, which provide the required structures in the vertices, in order for +the Schwinger mechanism to be activated [18,118,160,215]. The formation of these states out +of a pair of gluons or a ghost–antighost pair is controlled by a set of coupled Bethe-Salpeter +equations (BSEs) [18,118,125,215,216], which are found to have nontrivial solutions for the +corresponding Bethe-Salpeter (BS) amplitudes, to be denoted by C(r) and C(r), respectively. +• In Section 5 we explain in detail how the presence of the massless poles in the dressed +vertices that enter in the SDE of the gluon propagator gives rise to a gluon mass. The demon- +stration is carried out separately for the gµν and qµqν/q2 components of the gluon self-energy. +The former case requires the evasion of the so-called “seagull identity” [114,167]; this becomes +possible by virtue of the crucial Ward identity (WI) displacement, to be further considered in +Section 10. +• In Section 6 we go over the basic notions underpinning the PT [14,97,101,194,195], +and show how their application leads naturally to the definition of a dimensionful process- +independent RGI interaction strength [3,16,20,80,97,130–132], denoted by �d(q). The genuine +process-independence of this quantity is concretely exemplified by demonstrating its appear- +ance in two processes involving fundamentally different external fields. Next, �d(q) is computed +by combining lattice data for the gluon propagator and SDE results for the function G(q). +Finally, the dimensionless quantity is derived that constitutes the physical definition of the one- +gluon exchange interaction appearing in standard bound-state computations [15–17,217–223]. +• In Section 7 we focus on the structure of the “transversely projected” three-gluon +vertex [127,175,176,224], and discuss briefly the property of planar degeneracy [87], satisfied, +at a high level of accuracy [87–89,175,176,224], by the vertex form factors. This special property +induces a striking simplification to the structure of this vertex, captured by a particularly +compact expression [87], which will be extensively used in some of the following sections. + +Particles 2023, 1 +4 +• In Section 8 we take a close look at the ghost sector of the theory, and solve the coupled +system of SDEs governing the ghost propagator and ghost-gluon vertex [86,225–229]; as is +well-known, the ghost remains massless, but its dressing function saturates at the origin [21, +42,47,49,51,56,63,64,74,80,86,113,179,226,228–234], because the infrared-finite gluon propagator +used in the ghost SDE provides an effective infrared cutoff. In the SDE of the ghost-gluon +vertex, we employ as central input the compact expression for the three-gluon vertex presented +in the previous section. The results are in excellent agreement with the available lattice data for +the ghost dressing function [74,86] and the form factor of the ghost-gluon vertex evaluated in +the soft-gluon limit [42,43]. +• In Section 9 we discuss two important consequences of the masslessness of the ghost +propagator, which manifest themselves at the level of both the gluon propagator and the +three-gluon vertex. Specifically, the diagrams comprised by a ghost loop induce “unprotected” +logarithms, i.e., of the type ln q2; instead, gluonic loops give rise to “protected” logarithms, of +the type ln(q2 + m2), where m is the effective gluon mass [173,235]. As q2 → 0, the unprotected +contributions diverge, driving the appearance of a maximum in the gluon propagator and a +divergence in its first derivative, as well as a zero-crossing and a corresponding divergence +in the form factors of the three-gluon vertex. As we comment in this section, of particular +phenomenological importance [235–241] is the relative suppression that the above features +induce to the dominant vertex form factors in the intermediate range of momenta. +• In Section 10 we discuss an outstanding feature of the WI satisfied by the pole-free part +of the three-gluon vertex, namely the displacement induced by the presence of the aforemen- +tioned massless poles [94,125]. In this context, we introduce the key quantity denominated +“displacement function”, whose appearance serves as a smoking gun signal of the action of the +Schwinger mechanism in QCD; quite interestingly, it coincides [94,125] with the BS amplitude +C(r) for the formation of a massless scalar out of a pair of gluons, introduced in Section 4. In +addition, we derive a crucial relation, which ultimately permits the indirect determination +of C(r) from lattice QCD [94,125,127]; an important ingredient in this relation is a partial +derivative [125,242], denoted by W(r), of the ghost-gluon kernel [229], to be determined in the +next section. +• In Section 11 we set up and solve the SDE that governs the evolution of W(r) [125,127, +242,243]; the main component of this SDE is a special projection of the three-gluon vertex, which +is computed by appealing to formulas established in Section 7, and allows for the accurate +determination of W(r) in the entire range of relevant momenta [127]. +• In Section 12 we substitute into the central relation derived in Section 10 the solution +for W(r) found in the previous section, together with the lattice data [85,86] for the gluon +propagator, the ghost dressing function, and the form factor of the three-gluon vertex associated +with the soft-gluon limit, in order to obtain the form of the displacement function C(r) [125,127]. +As we discuss, the results exclude with nearly absolute certainty the null hypothesis (absence +of Schwinger mechanism, C(r) = 0), and corroborate the action of the Schwinger mechanism +in QCD [127]. In addition, we show that the form of C(r) found is statistically completely +compatible with that obtained from the BSE-based analysis presented in Section 4. +• In Section 13 we present our conclusions. +• Finally, in Appendix A we derive the BQIs relating the displacement functions of the +conventional and background vertices. +2. Basic concepts and general theoretical framework +We start by considering the Lagrangian density of an SU(N) Yang-Mills theory, comprised +of the classical part, Lcl, the contribution from the ghosts, Lgh, and the covariant gauge-fixing +term, Lgf, namely +LYM = Lcl + Lgh + Lgf , +(1) + +Particles 2023, 1 +5 +where +Lcl = −1 +4 Fa +µνFaµν , +Lgh = −ca∂µDab +µ cb , +Lgf = 1 +2ξ (∂µAa +µ)2 . +(2) +In the above formula, Aa +µ(x) denotes the gauge field, while ca(x) and ca(x) represent the ghost +and antighost fields, respectively, with a = 1, . . . , N2 − 1. +In addition, +Fa +µν = ∂µAa +ν − ∂νAa +µ + g f abcAb +µAc +ν , +(3) +is the antisymmetric field tensor, where f abc stands for the totally antisymmetric structure +constants of the SU(N) gauge group, and g is the gauge coupling, while +Dab +µ = ∂µδac + g f ambAm +µ , +(4) +denotes the covariant derivative in the adjoint representation. Finally, ξ represents the gauge- +fixing parameter; the choice ξ = 0 corresponds to the Landau gauge, while ξ = 1 specifies the +Feynman -´t Hooft gauge. +The transition from the pure Yang-Mills theory of Eq. (1) to QCD is implemented by sup- +plementing the corresponding kinetic and interaction terms for the quark fields. However, since +throughout this work we do not consider effects due to dynamical quarks, the aforementioned +terms will be omitted entirely. +The most fundamental correlation function is the gluon propagator, whose nonperturba- +tive features are inextricably connected with key dynamical properties of the theory. In the +Landau gauge that we will employ throughout, the gluon propagator, ∆ab +µν(q) = −iδab∆µν(q), is +completely transverse, i.e., +∆µν(q) = ∆(q)Pµν(q) , +Pµν(q) := gµν − qµqν/q2 . +(5) +In the continuum, the dynamical properties of the gluon propagator are encoded in the +corresponding SDE, given by +∆−1(q)Pµν(q) = q2Pµν(q) + iΠµν(q) , +(6) +where Πµν(q) is the gluon self-energy, shown diagrammatically in the first row of Fig. 1. The +fully-dressed vertices entering the diagrams are determined by their own SDEs, obtaining +finally a tower of coupled integral equations, which, for practical purposes, must be truncated +or treated approximately. +Given that, by virtue of the fundamental STI satisfied by the two-point function, the +self-energy Πµν(q) is transverse, +qµΠµν(q) = 0 , +(7) +we have that +Πµν(q) = Π(q)Pµν(q) , +(8) +and from Eq. (6) follows that +∆−1(q) = q2 + iΠ(q) . +(9) +Of particular importance is the exact way that Eq. (7) is enforced at the level of the SDE given +in Fig. 6, which governs the gluon evolution. In particular, note that, if we were to contract the +corresponding diagrams by qµ, the entire set of diagrams must be considered in order for Eq. (7) +to emerge from the SDE. This pattern manifests itself already at the one-loop level, where it +is known that the ghost-loop must be included in order to guarantee the transversality of the +self-energy. The main practical drawback stemming from this observation is that truncations, + +Particles 2023, 1 +6 ++ ++ +Πµν(q) = +(d1) +(d2) +(d3) ++ +(d4) ++ +(d5) +ν +q +µ +q +ν +q +ν +q +ν +q +ν +q +µ +q +µ +q +µ +q +µ +q ++ +�Πµν(q) = +(a1) +ν +q +µ +q ++ +(a3) +ν +q +µ +q ++ +(a2) +ν +q +µ +q ++ +(a4) +µ +q +ν +q +(a5) ++ +(a6) +ν +q +ν +q +µ +q +µ +q +�Π(1) +µν (q) +�Π(2) +µν (q) +�Π(3) +µν (q) +Figure 1. Upper panel: the diagrammatic representation of the conventional gluon self-energy, Πµν(q). +Bottom panel: the diagrammatic representation of the Qaµ(q)Bbν(−q), self-energy δab �Πµν(q); the grey +circles at the end of the gluon lines indicate a background gluon. The corresponding Feynman rules are +given in Appendix B of [14]. +in the form of omission of certain subsets of graphs, are likely to distort this fundamental +property. +Quite interestingly, within the PT-BFM framework the transversality property of Eq. (7) +is enforced in a very special way, which permits physically meaningful truncations. In what +follows we will employ predominantly the language of the BFM; for the basic principles of the +PT and its connection with the BFM, the reader is referred to the extended literature on the +subject [14,97,101,194,195,212,244], as well as to Section 6 of the present work. +The BFM is a powerful quantization procedure, where the gauge-fixing is implemented +without compromising explicit gauge invariance. Within this framework the gauge field +A appearing in the classical action is decomposed as A = B + Q, where B and Q are the +background and quantum (fluctuating) fields, respectively. Note that the variable of integration +in the generating functional Z(J) is the quantum field, which couples to the external sources, +as J · Q. The background field does not appear in loops. Instead, it couples externally to +the Feynman diagrams, connecting them with the asymptotic states to form elements of the +S-matrix. Then, if the gauge-fixing term +�Lgf = +1 +2ξQ +( �Dab +µ Qbµ)2 , +�Dab +µ = ∂µδab + g f ambBm +µ , +(10) +is used, the resulting gauge-fixed action retains its invariance under gauge transformations of +the background field. As a result of this invariance, when the Green’s functions are contracted +by the momentum carried by a background gluon, they satisfy Abelian (ghost-free) STIs, akin +to the Takahashi identities known from QED. In particular, the STIs of the BFM retain their +tree-level form to all orders, in contradistinction to the STIs of the Rξ gauges, whose form is +modified by contributions stemming from the ghost sector. +Within the BFM, one may consider three kinds of propagators, by choosing the type of +incoming and outgoing gluons [245]. In particular, we have: +(i) The propagator ⟨0| T[Qa +µ(q)Qb +ν(−q)]|0⟩ that connects two quantum gluons. Notice that +this propagator coincides with the conventional gluon propagator of the covariant gauges, +defined in Eq. (5), under the assumption that the corresponding gauge-fixing parameters, ξ +and ξQ, are identified, i.e., ξ = ξQ. +(ii) The propagator ⟨0| T[Qa +µ(q)Bb +ν(−q)]|0⟩ that connects a Qa +µ(q) with a Bb +ν(−q), to be +denoted by �∆ab +µν(q) = −iδab�∆µν(q). +(iii) The propagator ⟨0| T[Ba +µ(q)Bb +ν(−q)]|0⟩ that connects a Ba +µ(q) with a Bb +ν(−q), to be +denoted by �∆ab +µν(q) = −iδab�∆µν(q). Note that its full definition requires an additional gauge- +fixing term, with the associated “classical” gauge-fixing parameter, ξC [14,203,207]. + +Particles 2023, 1 +7 +Given that the relations captured by Eqs. (5) and (6) apply also in the cases of �∆µν(q) +and �∆µν(q), one may define the corresponding self-energies �Πµν(q) and �Πµν(q), as well as the +functions �∆(q) and �∆(q). +Table 1. The different types of gluon propagators of the background field method (BFM), together with +their diagrammatic representations, symbols, corresponding self-energies, and the background quantum +identities (BQIs) that relate them to the conventional propagator. +External legs +Diagrammatic +representation +Symbol +Self-energy +BQI +Qa +µ(q)Qb +ν(−q) +q +a +b +µ +ν +−iδab∆µν(q) +Πµν(q) +— +Qa +µ(q)Bb +ν(−q) +q +a +b +µ +ν +−iδab�∆µν(q) +�Πµν(q) +�∆(q) = +∆(q) +1 + G(q) +Ba +µ(q)Bb +ν(−q) +q +a +b +µ +ν +−iδab�∆µν(q) +�Πµν(q) +�∆(q) = +∆(q) +[1 + G(q)]2 +Quite interestingly, the three propagators defined in (i)-(iii) are related by a set of exact +identities, known as BQIs [14,210–212]. In particular, we have that (see also Table 1) +∆(q) = [1 + G(q)]�∆(q) = [1 + G(q)]2�∆(q) , +(11) +where the function G(q) is the gµν component of a particular two-point ghost function, Λµν(q), +given by [210,212,214,246] +Λµν(q) := ig2CA +� +k ∆ρ +µ(k)D(k + q)Hνρ(−q, k + q, −k) = gµνG(q) + qµqν +q2 L(q) , +(12) +where CA is the Casimir eigenvalue of the adjoint representation [N for SU(N)], Dab(q) = +iδabD(q) is the ghost propagator, and Hνµ(r, p, q) denotes the ghost-gluon kernel defined in +Fig. 2. +In the Landau gauge, a special identity relates the form factors of Λµν(q) to the ghost +dressing function, F(q), defined as F(q) = q2D(q), namely [16,132,214] +F−1(q) = 1 + G(q) + L(q) , +(13) +which is valid before renormalization. In fact, in this particular gauge, G(q) coincides with the +so-called Kugo-Ojima function [213,246–248]. + +Particles 2023, 1 +8 += −gf abcHνµ(r, p, q) +ν, b +k +p +µ, a +q +r +k + r +c +Figure 2. Diagrammatic definition of the ghost-gluon scattering kernel, Hνµ(r, p, q). At tree level, +H0νµ = gνµ. +To determine the renormalized form of Eq. (13), we introduce the renormalization con- +stants of the conventional Green’s functions +∆R(q) = Z−1 +A ∆(q) , +FR(q) = Z−1 +c +F(q) , +IΓR +µ(r, p, q) = Z1IΓµ(r, p, q) , +IΓR +αµν(q, r, p) = Z3IΓαµν(q, r, p) , +gR = Z−1 +g g , +� +gµν + ΛR +µν(q) +� += ZΛ +� +gµν + Λµν(q) +� +, +Z−1 +g += Z−1 +1 Z1/2 +A Zc = Z−1 +3 Z3/2 +A +, +(14) +where we denote by IΓabc +µ (r, p, q) = −g f abcIΓµ(r, p, q) and IΓabc +αµν(q, r, p) = g f abcIΓαµν(q, r, p) +the conventional ghost-gluon [Qa +µ(q)cc(p)¯cb(r)] and three-gluon [Qa +α(q)Qb +µ(r)Qc +ν(p)] vertices, +respectively. Note that, by virtue of Taylor’s theorem [208], Z1 is finite in the Landau gauge; its +precise value depends on the renormalization scheme adopted, see Sec. 8. Moreover, denoting +by �ZA the (wave-function) renormalization constant of �∆(q), the Abelian STIs of the BFM +impose the validity of the pivotal relation [14,203,207] +Zg = �Z−1/2 +A +, +(15) +which is the non-Abelian analogue of the textbook relation Ze = Z−1/2 +A +[134], relating the +renormalization constants of the electric charge and the photon propagator in QED. +Then, since the BQIs of Eq. (11) are direct consequences of the Becchi-Rouet-Stora-Tyutin +(BRST) symmetry [249–251] of the theory [210,212,214,246], their form is preserved by renor- +malization. Hence, combining Eqs. (11), (15) and (14) we obtain +ZΛ = Z−1 +1 Zc , +(16) +which yields1 +Z−1 +1 F−1(q) = 1 + G(q) + L(q) . +(17) +1 +In the original and widely used [3,8,16,20,80,132] version of Eq. (17) the renormalization is performed in the +so-called Taylor scheme, where Z1 = 1. + +Particles 2023, 1 +9 +As has been shown in [132], the dynamical equation governing L(q) yields L(0) = 0, +provided that the gluon propagator entering in it is finite at the origin. Thus, one obtains from +Eq. (17) the useful identity [213] +Z−1 +1 F−1(0) = 1 + G(0) . +(18) +According to numerous lattice simulations and studies in the continuum (see, e.g., [21,42, +47,49,51,56,63,64,74,80,86,113,179,226,228–234]), the ghost dressing function reaches a finite +(nonvanishing) value at the origin, which, due to Eq. (18), furnishes also the value of G(0). +The final upshot of the above considerations is that one may use the BQIs in Eq. (11) +to express the SDE given in Eq. (6) in terms of the �Πµν(q) or �Πµν(q), at the modest cost of +introducing in the dynamics the quantities 1 + G(q) or [1 + G(q)]2. Focusing on the former +possibility, Eq. (11) becomes +∆−1(q)Pµν(q) = q2Pµν(q) + i �Πµν(q) +1 + G(q) +, +(19) +where the diagrammatic representation of the self-energy �Πµν(q) is shown in the lower panel +of Fig. 1. +The principal advantage of this formulation is that the self-energy �Πµν(q) contains fully- +dressed vertices with a background gluon of momentum q exiting from them, which satisfy +Abelian STIs. In particular, denoting by �IΓµαβ(q, r, p), �IΓµ(r, p, q), and �IΓ +mnrs +µαβγ(q, r, p, t) the BQQ, +Bcc, and BQQQ vertices, respectively, we have that [14,101,110] +qµ �IΓµαβ(q, r, p) += +∆−1 +αβ (r) − ∆−1 +αβ (p) , +(20) +qµ �IΓµ(r, p, q) += +D−1(p) − D−1(r) , +(21) +qµ �IΓ +mnrs +µαβγ(q, r, p, t) += +f mse f ernIΓαβγ(r, p, q + t) + f mne f esrIΓβγα(p, t, q + r) ++ +f mre f ensIΓγαβ(t, r, q + p) . +(22) +In contrast, the conventional three-gluon and ghost-gluon vertices, IΓαµν(q, r, p) and +IΓα(r, p, q), respectively, satisfy the STIs [1,252–256] +qαIΓαµν(q, r, p) = F(q) +� +∆−1(p)Pσ +ν (p)Hσµ(p, q, r) − ∆−1(r)Pσ +µ (r)Hσν(r, q, p) +� +, +(23) +qµF−1(q)IΓµ(r, p, q) + pµF−1(p)IΓµ(r, q, p) = −r2F−1(r)U(r, q, p) , +(24) +where U(r, q, p) is an interaction kernel containing only ghost fields; its tree level value is +U0(r, q, p) = 1. The STI for the conventional four-gluon vertex is given in Eq. (C.24) of [14]. +The special STIs listed in Eqs. (20), (21) and (22) are responsible for the remarkable property +of “block-wise” transversality [110,193,245], displayed by �Πµν(q). To appreciate this point, +notice that the diagrams comprising �Πµν(q) in Fig. 1 have been separated into three different +subsets (blocks) comprised of: (i) one-loop dressed diagrams containing only gluons, (ii) one- +loop dressed diagrams containing a ghost loop, and (iii) two-loop dressed diagrams containing +only gluons. The corresponding contributions of each block to �Πµν(q) are denoted by �Π(i) +µν(q), +with i = 1, 2, 3. +The block-wise transversality is a stronger version of the standard transversality relation +qµ �Πµν(q) = 0; it states that each block of diagrams mentioned above is individually transverse, +namely +qµ �Π(i) +µν(q) = 0 , +i = 1, 2, 3. +(25) + +Particles 2023, 1 +10 +In order to appreciate in detail the reason why the STIs in Eqs. (20), (21) and (22) are +instrumental for the block-wise transversality, we will consider the case of �Π(2) +µν (q); the relevant +diagrams are enclosed by the blue box of Fig. 1. +The diagrams (a3) and (a4) are given by +(a3)µν(q) = g2CA +� +k(k + q)µD(k + q)D(k)�IΓν(−k, k + q, −q) , +(26) +(a4)µν(q) = g2CA gµν +� +k D(k) , +(27) +where a color factor δab has been suppressed in both expressions. In addition, for the formal +manipulations of integrals, we employ dimensional regularization [257]; to that end, we +introduce the short-hand notation +� +k := +µϵ +0 +(2π)d +� +∞ +−∞ ddk , +(28) +where d = 4 − ϵ is the dimension of the space-time, and µ0 denotes the ’t Hooft mass. +The contraction of graph (a3)µν(q) by qν triggers the STI satisfied by �Γν(−k, k + q, −q) +[given by Eq. (21)], and we obtain +qν(a3)µν(q) += +g2CA +� +k(k + q)µD(k + q)D(k) +� +D−1(k) − D−1(k + q) +� += +g2CA +� +k(k + q)µ[D(k + q) − D(k)] += +−g2CA qµ +� +k D(k) , +(29) +which is precisely the negative of the contraction qν(a4)µν(q). Hence, +qν�(a3)µν(q) + (a4)µν(q) +� = 0 . +(30) +3. Schwinger mechanism in Yang-Mills theories +The BRST symmetry of the Yang-Mills Lagrangian given in Eq. (1) prohibits the inclusion +of a mass term of the form m2A2 +µ. Moreover, a symmetry-preserving regularization scheme, +such as dimensional regularization, prevents the generation of a mass term at any finite order in +perturbation theory. Nonetheless, as affirmed four decades ago [95–100], the nonperturbative +Yang-Mills dynamics endow the gluons with an effective mass, which sets the scale for all +dimensionful quantities, and tames the instabilities originating from the infrared divergences +of the perturbative expansion (e.g., Landau pole). In addition, the presence of this mass +causes the effective decoupling (screening) of the gluonic modes beyond a “maximum gluon +wavelength” [258], and leads to the dynamical suppression of the Gribov copies, see, e.g., +[16,259,260] and references therein. +The generation of a gluon mass proceeds through the nonperturbative realization of +the Schwinger mechanism [128,129]. Even though the technical details associated with the +implementation of this mechanism in a four-dimensional non-Abelian setting are particularly +elaborate, the general underlying idea is relatively easy to convey. +To that end, consider the dimensionless vacuum polarization Π(q), defined through +Π(q) = q2Π(q), such that +∆−1(q) = q2[1 + iΠ(q)] . +(31) +The Schwinger mechanism is based on the fundamental observation that, if Π(q) develops +a pole at q2 = 0 (to be referred to as “massless pole”) then the vector meson (gluon) picks up + +Particles 2023, 1 +11 +a mass, regardless of any “prohibition” imposed by the gauge symmetry at the level of the +original Lagrangian. Thus, in Euclidean space, the above sequence of ideas leads to +lim +q→0 Π(q) = m2/q2 =⇒ lim +q→0 ∆−1(q) = lim +q→0 (q2 + m2) =⇒ ∆−1(0) = m2 , +(32) +and the gauge boson propagator saturates to a non-zero value at the origin. This effect of +infrared saturation of the propagator signifies the generation of a mass, which is identified +with the positive residue of the pole. +At this descriptive level, Schwinger’s argument is completely general, making no par- +ticular reference to the specific dynamics that would lead to the appearance of the required +massless pole inside Π(q). In fact, depending on the particular theory, the field-theoretic +circumstances that trigger the crucial sequence captured by Eq. (32) may be very distinct, see, +e.g., [261,262]. In the case of Yang-Mills theories, the origin of the massless poles is purely +nonperturbative [160]: the strong dynamics produce scalar composite excitations, which carry +color and have vanishing masses. These poles are carried by the fully-dressed vertices of the +theory; and since these vertices enter in the gluon SDE shown in Fig. 1 [upper (lower) panel +for the QQ (QB) propagator], the massless poles find their way into the gluon self-energy (or, +equivalently, the gluon vacuum polarization). The detailed implementation of this idea has +been presented in a series of works [18,94,97,113,117,118,118,119,160–162,167,190,263], and will +be summarized in the rest of this section. +Let us focus for now on the conventional three-gluon and ghost-gluon vertices, IΓαµν(q, r, p) +and IΓα(r, p, q), respectively, introduced below Eq. (14). When the formation of massless poles +is triggered, these vertices assume the general form (see Fig. 3) +IΓαµν(q, r, p) += +Γαµν(q, r, p) + Vαµν(q, r, p) , +IΓα(r, p, q) += +Γα(r, p, q) + Vα(r, p, q) , +(33) +where Γαµν(q, r, p) and Γα(r, p, q) are their pole-free components, while Vαµν(q, r, p) and Vα(q, r, p) +contain longitudinally coupled poles, whose special tensorial structure is given by +Vαµν(q, r, p) += +qα +q2 Cµν(q, r, p) + rµ +r2 Aαν(q, r, p) + pν +p2 Bαµ(q, r, p) , +Vα(r, p, q) += +qα +q2 C(r, p, q) , +(34) +such that +Pα +α′(q)Pµ +µ′(r)Pν +ν′(p)Vαµν(q, r, p) = 0 , +Pα +α′(q)Vα(r, p, q) = 0 . +(35) +We emphasize that the reason why Vαµν(q, r, p) and Vα(q, r, p) are longitudinally coupled +may be directly inferred from their special decomposition, shown in Fig. 3. In particular, let +us denote by Iα(q) the transition amplitude that connects a gluon with a massless composite +scalar, depicted as a gray circle in Fig. 3. Since Iα(q) depends solely on the momentum q, and +carries a single Lorentz index, α, its general form is given by Iα(q) = qαI(q), where I(q) is a +scalar form factor [118,215]. This observation accounts directly for the form of Vα(q, r, p) given +in Eq. (34); to deduce the form of Vαµν(q, r, p), one must, in addition, appeal to Bose symmetry, +which imposes the structures rµ/r2 and pν/p2 in the remaining two channels. +Returning to the SDE of Eq. (1), the component Vαµν(q, r, p) will enter in it through graphs +(d1) and (d4), while the component Vα(q, r, p) through graph (d3). Since Vαµν(q, r, p) has poles +for each one of its three momenta, let us point out that only the pole associated with the +q-channel, i.e., the channel that carries the momentum entering in the gluon propagator, is +relevant for the Schwinger mechanism that will generate mass for ∆(q). In fact, in the Landau + +Particles 2023, 1 +12 += ++ +q +q +a, α +q +a, α +a, α +i/q2 +Vαµν +IΓαµν +Γαµν +� �� � +Iα(q) +µ, b +ν, c +r +p +µ, b +ν, c +r +p +µ, b +ν, c +r +p += +q +a, α ++ +i/q2 +qq +q +a, α +a, α +Vα +IΓα +Γα +� �� � +Iα(q) +b +c +r +p +b +c +r +p +b +c +r +p +Figure 3. The diagrammatic representation of the three-gluon and ghost-gluon vertices introduced in +Eq. (33): IΓαµν(q, r, p) (first row) and IΓα(r, p, q) (second row). The first term on the r.h.s. indicates the +pole-free part, Γαµν(q, r, p) or Γα(r, p, q), while the second denotes the pole term Vαµν(q, r, p) or Vα(r, p, q). +gauge that we employ, the gluon propagators inside the diagrams (d1) and (d4) are transverse, +leading to a considerable reduction in the number of the form factors of Vαµν(q, r, p) that +participate actively, since +Pµ +µ′(r)Pν +ν′(p)Vαµν(q, r, p) = qα +q2 Pµ +µ′(r)Pν +ν′(p)Cµν(q, r, p) . +(36) +Consequently, for the ensuing analysis, one requires only the tensorial decomposition of the +component Cµν(q, r, p) in Eq. (34), which is given by +Cµν(q, r, p) = C1 gµν + C2 rµrν + C3 pµpν + C4 rµpν + C5 pµrν , +(37) +where Cj := Cj(q, r, p). Then, the substitution of Eq. (37) into Eq. (36), and use of the relation +q + p + r = 0, reveals that only two form factors survive inside (d1) and (d4), namely +Pµ +µ′(r)Pν +ν′(p)Vαµν(q, r, p) = qα +q2 Pµ +µ′(r)Pν +ν′(p) +� +C1 gµν + C5qµqν +� +. +(38) +Since the main function of the Schwinger mechanism is to make the gluon propagator +saturate at the origin, it is important to explore the properties of the structures appearing in +Eq. (38) near q = 0. To that end, we expand the r.h.s. of Eq. (38), keeping terms at most linear in +q. After noticing that the term proportional to C5 in Eq. (38) is of order O(q2), we end up with a +single relevant form factor associated with Vαµν(q, r, p), namely C1(q, r, p), which survives the +q → 0 limit of graphs (d1) and (d4). As for Vα(r, p, q), its unique component, C(q, r, p), enters +directly in (d3). +The continuation of this analysis entails the Taylor expansion of C1(q, r, p) and C(r, p, q) +around q = 0. In carrying out this expansion, one employs the following two key relations, +C1(0, r, −r) = 0 , +C(r, −r, 0) = 0 . +(39) +The first one follows directly from the Bose symmetry of the three-gluon vertex, which implies +that C1(q, r, p) = −C1(q, p, r); as we will see in Sec. 10, it may also be derived in a completely +independent way from the fundamental STIs satisfied by the three-gluon vertex. The justi- +fication of the second relation in Eq. (39) is less straightforward; its derivation, presented in +Appendix A, relies on the BQI [14,212] linking the conventional ghost-gluon vertex, IΓα(r, p, q), +with its background counterpart, �IΓα(r, p, q). + +Particles 2023, 1 +13 += ++ +ν, n +r +p +µ, m +q +α, a +ν, n +r +p +µ, m +q +α, a +α, a ++ +r +µ, m +p +ν, n +r +µ, m +p +ν, n ++ · · · +k +k + q +k +k + q +K11 +K12 +q +α, a +q +(a) +(b) += ++ +n +r +p +m +q +α, a +n +r +p +m +q +α, a +α, a ++ +r +m +p +n +r +m +p +n ++ · · · +k +k + q +k +k + q +q +α, a +q +(c) +(d) +K21 +K22 +Figure 4. The coupled system of Schwinger-Dyson equations (SDEs) for the three-gluon and ghost-gluon +vertices, IΓαµν(q, r, p) and IΓα(r, p, q), respectively. The orange ellipses represent four-point scattering +kernels, denoted by Kij. We omit diagrams containing five-point scattering kernels. +Thus, after taking Eq. (39) into account, the Taylor expansion of C1(q, r, p) and C(r, p, q) +around q = 0 yields +lim +q→0 C1(q, r, p) = 2(q · r)C(r) + · · · , +lim +q→0 C(r, p, q) = 2(q · r)C(r) + · · · , +(40) +with +C(r) := +�∂C1(q, r, p) +∂p2 +� +q=0 +, +C(r) := +�∂C(r, p, q) +∂p2 +� +q=0 +. +(41) +The functions C(r) and C(r) are of central importance for the rest of this review. In particular, +there are three key points related to them that will be elucidated in detail in what follows: +1. +C(r) and C(r) are the BS amplitudes describing the formation of gluon-gluon and ghost- +antighost colored composite bound states, respectively, see Sec. 4. +2. +The gluon mass is determined by certain integrals that involve C(r) and C(r), given +explicitly in Sec. 5. +3. +C(r) and C(r) lead to smoking-gun displacements of the WIs. In fact, the displacement +induced by C(r), has been confirmed by lattice QCD, by combining judiciously the results +of several lattice simulations, see subsection 5.2. +We end this section by emphasizing that the BFM vertices develop poles in exactly the +same way as their conventional counterparts. In particular, the main relations Eqs. (33), (34), (39) +and (41) remain valid, with the only modification that all quantities carry hats or tildes; these +BFM vertices will be used extensively in Sec. 5. Note that the conventional and background +vertices, including their pole content, are related through appropriate BQIs, see e.g., Eqs. (A3) +and (A6). +4. Dynamical formation of massless poles +One crucial aspect of the implementation of the Schwinger mechanism in a Yang-Mills +context is that the poles that comprise the components Vαµν(q, r, p) and Vα(q, r, p) in Eq. (34) +are not introduced by hand; rather, they are generated dynamically, as massless composite +excitations that carry color. In fact, this subtle process is controlled by a system of coupled linear +BSEs for the functions C(r) and C(r), which play the role of the BS amplitudes for generating +composite massless scalars out of two gluons and a ghost-antighost pair, respectively. + +Particles 2023, 1 +14 +The starting point for the derivations of the aforementioned BSEs are the SDEs for +IΓαµν(q, r, p) and IΓα(r, p, q), shown diagrammatically in Fig. 4, and given by [125] +IΓαµν += +Γαµν +0 +− λ +� +k IΓαβγ∆βρ∆γσKµνσρ +11 ++ 2λ +� +k IΓαDDKµν +12 , +IΓα += +Γα +0 − λ +� +k IΓαβγ∆βρ∆γσKσρ +21 − λ +� +k IΓαDDK22 , +(42) +where +λ := ig2CA/2 , +(43) +and the tree-level expressions for the vertices IΓαµν and IΓα are given by +Γαµν +0 +(q, r, p) = (q − r)νgαµ + (r − p)αgµν + (p − q)µgνα , +Γα +0 (r, p, q) = rα . +(44) +Note that, for compactness, all momentum arguments have been suppressed; they may be +easily restored by appealing to Fig. 4. +The following steps are subsequently implemented: +1. Substitute into both sides of Eq. (42) the expressions for the fully-dressed vertices given +in Eq. (33). +2. In order to exploit Eq. (38), multiply the first equation by the factor Pµ′µ(r)Pµ′ +ν (p). +3. Take the limit of the system as q → 0: this activates Eq. (40) and introduces the functions +C(r) and C(r). +4. Isolate the tensorial structures proportional to qα, and match the terms on both sides. +5. Employ the “one-particle exchange” approximation for the kernels Kij, to be denoted +by K0 +ij, shown in Fig. 5. +Thus, we arrive at a system of homogeneous equations involving C(r) and C(r), +C(r) += +−λ +3 +� +k C(k)∆2(k)Pρσ(k)Pµν(r) �Kµνσρ +11 ++ 2λ +3 +� +k C(k)D2(k)Pµν(r) �Kµν +12 , +C(r) += +−λ +� +k C(k)∆2(k)Pσρ(k) �Kσρ +21 − λ +� +k C(k)D2(k) �K22 , +(45) +where �Kij := (r · k/r2) K0 +ij(r, −r, k, −k); the system is diagrammatically depicted in Fig. 6. +Before turning to the numerical analysis, the BSE system must be passed to the Euclidean +space, following standard conversion rules. In doing so we note that the integral measure is +modified according to d4k → id4kE; this extra factor of i combines with the λ defined in Eq. (43) +to give real expressions. +As announced, the system of coupled equations given in Eq. (45) represents the BSEs that +govern the formation of massless colored bound states out of two gluons and a ghost-antighost +pair. The functions C(r) and C(r) are the corresponding BS amplitudes; finding nontrivial +solutions for them, i.e., something other than C(r) = C(r) = 0 identically, is crucial for the +implementation of the Schwinger mechanism. +The equations in Eq. (45) are linear and homogeneous in the unknown functions. There are +two main consequences arising from this fact. First, the numerical solution of the system will be +reduced to an eigenvalue problem. Second, the overall scale of the solutions is undetermined, +since the multiplication of a given solution by an arbitrary real constant produces another +solution 2. +2 +The ambiguity originates from considering only leading terms in the expansion around q = 0, and may be resolved +if further orders in q are kept, see, e.g., [220,264,265]. + +Particles 2023, 1 +15 += +r +r +k +k +K0 +11 +r +k +k +r +r − k += +r +r +k +k +K0 +12 +k +r +k +r +r − k += +k +r +r +k +K0 +21 +k +r +k +r +r − k += +k +r +k +r +r − k +k +r +r +k +K0 +22 +Figure 5. The one-particle exchange approximations, K0 +ij, of the kernels Kij appearing in Fig. 4. +It turns out that the condition for obtaining nontrivial solutions, when expressed in +terms of the strong coupling, αs := g2/4π, states that they exist for αs = 0.63, when the +renormalization point µ = 4.3 GeV. The solutions obtained when αs acquires this special +value are shown in Fig. 7; they have undergone scale fixing3, and are denoted by C⋆(r) and +C⋆(r). Observe that C⋆(r) is significantly larger in magnitude than C⋆(r), implying that the +three-gluon vertex accounts for the bulk of the gluon mass, as originally claimed in [216]. +It is important to compare the value of αs = 0.63, imposed by the BSE eigenvalue, with +the expected value for αs for the renormalization scheme employed: within the asymmetric +momentum subtraction (MOM) scheme (see Sec. 8), we have that αs = 0.27 [72]. This numerical +discrepancy in the values of αs is clearly an artifact of the truncation employed, and concretely +of the approximation of the kernels Kij by their one-particle exchange diagrams, K0 +ij. A +preliminary analysis reveals that mild modifications of the kernels Kij lead to considerable +variations in the value of αs, but leave the form of the solutions for C⋆(r) and C⋆(r) practically +unaltered. This observation suggests that, while a more complete knowledge of the BSE kernels +is required in order to bring αs closer to its MOM value, the solutions obtained with the present +approximations should be considered as particularly stable. +5. Generation of the gluon mass +We next demonstrate in detail how the presence of the massless poles in the vertices that +enter in the SDE of the gluon propagator generate a gluon mass. +We start by pointing out that, since the fundamental STIs of the theory remain intact +under the action of the Schwinger mechanism, Eqs. (7) and (8) remain valid, and the mass +term m2 = ∆−1(0) will appear in the transverse combination ∆−1(0)Pµν(q). However, the +determination of the mass proportional to gµν exposes an entirely different array of principles +compared to the corresponding computation for the qµqν/q2 component. +The calculation with respect to the qµqν/q2 component is rather direct; since the massless +poles in the vertices are themselves longitudinally coupled, their contribution to the qµqν/q2 +component of Πµν(q) is easily worked out, as will be illustrated in Subsec. 5.1. In contrast, +the emergence of a mass proportional to gµν is intimately connected with a powerful relation, +3 +The scale was fixed by requiring the best possible matching with the result obtained for C(r) from the WI +displacement, see Sec. 12. + +Particles 2023, 1 +16 +k +r +− 2 +q = 0 +α, a +µ, m +ν, n +k +r +r − k +r +r − k +q = 0 +α, a +r +µ, m +ν, n +k +k +C +ν, n += +r +r +µ, m +q = 0 +α, a +C +C +q = 0 +n +m +α, a +C +r +r +k +r ++ +q = 0 +α, a +m +n +k +r +r − k += +r − k +m +n +k +r +r +k +α, a +q = 0 +C +C +Figure 6. The diagrammatic representation of the coupled system of Bethe-Salpeter equations (BSEs) that +governs the evolution of the functions C(r2) and C(r2). +0.0 +1.0 +2.0 +3.0 +4.0 +5.0 +-0.4 +-0.3 +-0.2 +-0.1 +0.0 +0.1 +Displacement functions +C⋆(r) +C⋆(r) +Figure 7. The solutions for C⋆(r) (purple dot-dashed) and C⋆(r) (red dashed) obtained from the coupled +BSE system of Eq. (45). +known as seagull identity [114,167], which in the absence of the Schwinger mechanism would +enforce the masslessness of the propagator, as will be discussed in Subsec. 5.2. In fact, one +main conceptual difference between the two approaches is that in the gµν case, the use of the +PT-BFM-based version of the SDE given in Eq. (19) is crucial for the emergence of the correct +result. +In order to simplify the technical aspects of the calculation without compromising its +conceptual content, we will determine the contribution to the gluon mass due the pole in the +ghost-gluon vertex, namely Vα(r, p, q) in the case of IΓα(r, p, q), and �Vα(r, p, q) in the case of +�IΓα(r, p, q). To that end, we will focus on the subset of self-energy graphs containing only ghost +loops, i.e., graph (d3) in the case of Πµν(q), and graphs (a3) and (a4) in the case of �Πµν(q), +shown in the upper and lower row of Fig. 1, respectively. + +Particles 2023, 1 +17 +5.1. Gluon mass from the qµqν component +Let us calculate the contribution to the gluon mass stemming from the ghost loop, i.e., the +diagram (d3) of Fig. 1, which, for general values of q, reads +(d3)µν(q) = g2CA +� +k(k + q)µD(k + q)D(k)IΓν(−k, k + q, −q) . +(46) +To isolate the qµqν/q2 component of Eq. (46) at the origin, we first decompose the full +vertex IΓν(−k, k + q, −q) as in Eqs. (33) and (34), and drop directly the pole-free part, since it +does not contribute at q = 0. Then, denoting by (dV +3 )µν(q) the contribution of Vν(−k, k + q, −q) +to (d3)µν(q), we obtain +(dV +3 )µν(q) = −g2CA +qν +q2 +� +k(k + q)µD(k + q)D(k)C(−k, k + q, −q) . +(47) +Next, a Taylor expansion around q = 0, using Eqs. (39) and (40), yields +(dV +3 )µν(q) = −2g2CA +qνqρ +q2 +� +k kµkρD2(k)C(k) . +(48) +Evidently, the integral above can only be proportional to gµρ, such that +(dV +3 )µν(q) = −2g2CA +d +�qµqν +q2 +� � +k k2D2(k)C(k) , +(49) +where the tensor structure qµqν/q2 is already isolated. +Then, let us denote by ∆−1 +gh (0) the contribution to the mass originating in the qµqν/q2 +of the ghost loop. Noting that the contribution of (dV +3 )µν(q) to the propagator is i times the +negative of its qµqν/q2 form factor, we obtain that +∆−1 +gh (0) = 4λ +d +� +k k2D2(k)C(k) . +(50) +At this point, we set d = 4 and renormalize Eq. (50). This leads to the appearance of the finite +renormalization constant of the ghost-gluon vertex, Z1. +Next, we express the result in terms of the ghost dressing function F, pass to Euclidean +space, and employ hyperspherical coordinates, to obtain the final expression +∆−1 +gh (0) = ˆλ Z1 +� ∞ +0 dy F2(y) C(y) , +(51) +where ˆλ := CAαs/8π. +The derivation of the contributions from the diagrams (d1) and (d4) proceeds in a com- +pletely analogous way, but is algebraically more involved, see [167] for details. +It is instructive to consider how the result of Eq. (51) emerges in the context of Eq. (19). To +this end, we consider the ghost block �Π(2) +µν (q) of Fig. 1, whose diagrams have the expressions +given in Eq. (27); clearly, only diagram (a3)µν(q) can contribute to the qµqν component of +�Π(2) +µν (q). +Then, we decompose �IΓα(r, p, q) in complete analogy with Eqs. (33) and (34), i.e., +�IΓα(r, p, q) = �Γα(r, p, q) + qα +q2 �C(r, p, q) , +(52) + +Particles 2023, 1 +18 +and expand the (a3)µν(q) of Eq. (27) around q = 0, isolating its qµqν/q2 component. These +steps eventually lead to +�∆−1 +gh (0) = 4λ +d +� +k k2D2(k) �C(k) , +(53) +where �C(q) is defined in the exact same way as C(q), namely through Eq. (41) but with tildes +over all relevant quantities. It is now easy to establish that Eq. (53) is completely equivalent to +Eq. (50), simply by multiplying both of its sides by Z1F(0), and then using Eq. (A4) on the r.h.s. +and Eqs. (19) and (18) on the l.h.s. +Hence, when the mass is computed through the qµqν/q2 component of the self-energy, the +contributions originating from the ghost diagrams of either the BQ or the QQ propagator furnish +the same result. The same is not true for the calculation through the gµν component, since +the ghost diagram (d3)µν of the QQ propagator is not by itself transverse, and a meaningful +analysis is preferably carried out within the BFM. +5.2. Gluon mass from the gµν component: seagull identity and Ward identity displacement +The fact that the activation of the Schwinger mechanism is crucial for the self-consistent +generation of a gluon mass may be best appreciated in conjunction with the so-called seagull +identity [114,167]. The content of this identity is that +� +k k2 ∂ f (k) +∂k2 ++ d +2 +� +k f (k) = 0 , +(54) +for functions f (k) that satisfy Wilson’s criterion [266]; the cases of physical interest are +f (k) = ∆(k), D(k). The general demonstration of the validity of Eq. (54) has been given in [167]; +for a detailed discussion of how Eq. (54) prevents the photon from acquiring a mass in scalar +electrodynamics, see [18]. +What is so special about Eq. (54) is that, within the PT-BFM formalism, the l.h.s. of Eq. (54) +coincides with the contributions of loop diagrams to the gµν component of the gluon mass. +Therefore, Eq. (54) enforces the nonperturbative masslessness of the gluon in the absence of +the Schwinger mechanism: even if a massive gluon propagator (made “massive” through +a procedure other than the Schwinger mechanism) were to be substituted inside Eq. (54), +one would obtain zero as contribution to the gluon mass! For example, the simple choice +f = (k2 − m2)−1, reduces the l.h.s of Eq. (54) to (dimensionally regularized) text-book integrals, +which add up to give precisely zero [18]. +In order to appreciate in some detail how the seagull identity prevents the gµν component +of the propagator from acquiring a mass in the absence of the Schwinger mechanism, let us +consider once again the ghost block �Π(2) +µν (q) of Fig. 1; now both graphs, (a3) and (a4), contribute +to the gµν component. +Let us assume that the Schwinger mechanism is turned off; at the level of the Bcc vertex +this means that �Vα(r, p, q) vanishes identically, and �IΓα(r, p, q) = �Γα(r, p, q). Consequently, +�Γα(r, p, q) saturates the STI of Eq. (21), +qα�Γα(r, p, q) = D−1(p) − D−1(r) . +(55) +Since the form-factors of the vertex �Γα(r, p, q) do not contain any poles, the derivation from +Eq. (55) of the corresponding WI proceeds in the standard text-book way: both sides of Eq. (55) +undergo a Taylor expansion around q = 0, and terms at most linear in q are retained. Thus, one +arrives at the simple QED-like WI +�Γα(r, −r, 0) = ∂D−1(r) +∂rα +=⇒ +D2(r)�Γα(r, −r, 0) = −2rα +∂D(r) +∂r2 +. +(56) + +Particles 2023, 1 +19 +We now compute the gµν component of �Π(2) +µν (q) at q = 0, or, equivalently, �∆−1 +gh (0). From +Eq. (27), we see that (a4)µν is proportional to gµν in its entirety. On the other hand, (a3)µν(q) +contains both gµν and qµqν components; however, the latter vanishes in the limit q → 0 if the +vertex is pole-free. Then, it is straightforward to show that, as q → 0, +�∆−1 +gh (0) = 2λ +d +�� +k kµD2(k)�Γµ(−k, k, 0) + d +� +k D(k) +� +. +(57) +At this point, employing the WI of Eq. (56) (with r → −k), we get +�∆−1 +gh (0) = 4λ +d +�� +k k2 ∂D−1(k) +∂k2 ++ d +2 +� +k D(k) +� +� +�� +� +seagull identity += 0 . +(58) +Hence, the WI satisfied by the vertex in the absence of the Schwinger mechanism triggers the +seagull identity, which, in turn, enforces the masslessness of the propagator. +When the Schwinger mechanism gets activated, the STIs satisfied by the vertices of the +theory retain their original form, but are resolved through the nontrivial participation of the +terms containing the massless poles [97,113,160–162,167,263,267]. In particular, the full vertex +�IΓα(r, p, q) satisfies precisely Eq. (21), namely +qα �IΓα(r, p, q) += +qα�Γα(r, p, q) + �C(r, p, q) += +D−1(p) − D−1(r) . +(59) +Notice in particular that the contraction of �IΓα(r, p, q) by qα cancels the massless pole in q2, +leading to a completely pole-free result. Therefore, the WI obeyed by �Γα(r, p, q) may be derived +as before, through a standard Taylor expansion, leading to +qα�Γα(r, −r, 0) = − �C(r, −r, 0) + qα +� +� +� +∂D−1(r) +∂rα +− +� +∂ �C(r, p, q) +∂qα +� +q=0 +� +� +� . +(60) +Evidently, the unique zeroth-order contribution appearing in Eq. (60), namely �C(r, −r, 0), must +vanish, +�C(r, −r, 0) = 0 . +(61) +Note that this particular property may be independently derived from the antisymmetry of +�C(r, p, q) under r ↔ p, �C(r, p, q) = − �C(p, r, q), which is a consequence imposed by the ghost- +antighost symmetry of the B(q)¯c(r)c(p) vertex. The above result, together with Eq. (A3), is +used to prove Eq. (39) in App. A. +Thus, Eq. (60) becomes +qα�Γα(r, −r, 0) = qα +�∂D−1(r) +∂rα +− 2rα �C(r) +� +, +�C(r) := +� +∂ �C(r, p, q) +∂p2 +� +q=0 +, +(62) +and the matching of the terms linear in q yields the WI +�Γα(r, −r, 0) = ∂D−1(r) +∂rα +− +2rα �C(r) +� �� � +WI displacement +. +(63) + +Particles 2023, 1 +20 +Comparing Eqs. (56) and (63), it becomes clear that the Schwinger mechanism induces a char- +acteristic displacement to the WIs that are satisfied by the pole-free parts of the vertices [167]. +Returning to Eq. (57), but now substituting in it the displaced version of Eq. (56), namely +D2(k)�Γµ(−k, k, 0) = 2kµ +�∂D(k) +∂k2 ++ D2(k) �C(k) +� +. +(64) +When Eq. (64) is substituted into Eq. (57), the first term of its r.h.s. triggers the seagull identity +and vanishes, exactly as before; however, the second term survives, furnishing precisely the +result given in Eq. (53). +Completely analogous procedures may be applied to the remaining two blocks, �Π(1) +µν (q) +and �Π(3) +µν (q), by exploiting the Abelian STIs of Eqs. (20) and (22), respectively [162]. +6. Renormalization group invariant interaction strength +The PT-BFM formalism provides the natural framework for the construction of the RGI +version of the naive one-gluon exchange interaction. +To fix the ideas, recall that in QED, the one-photon exchange interaction, defined as α∆A(q), +where α := e2/4π is the hyper-fine structure constant and ∆A(q) the photon propagator, is +an RGI combination, by virtue of the relation Ze = Z−1/2 +A +; see comments following Eq. (15). +Moreover, this particular combination is universal (process-independent) because it may be +identified within any two-to-two scattering process, regardless of the nature of the initial and +final states (electrons, muons, taus, etc). Instead, in QCD, the corresponding combination αs∆(q) +is (trivially) universal but not RGI. When the vertices that connect the gluon to the external +particles are “dressed” (Γ0 → Γ), the combination Γ αs∆ Γ becomes RGI; however, it is no +longer process-independent, because the vertices Γ contain information on the characteristics +of the external particles, e.g., the Γ is not the same if the external particles are quarks or +gluons. This apparent conundrum may be resolved by resorting to the PT, which reconciles +harmoniously the notions of RGI and process-independence. +Within the PT framework, the starting point of the construction are “on-shell” pro- +cesses [14,97,101,194,195], such as those depicted in Fig. 8. The fundamental observation +is that the dressed vertices appearing there contain propagator-like contributions, which may +be unambiguously identified by means of a well-defined diagrammatic procedure. After dis- +carding terms that vanish on shell, the contributions extracted from a vertex have a two-fold +effect: (i) the genuine vertex contributions left behind form a new vertex, �Γ, which satisfies +Abelian STIs, and (ii) when the propagator-like pieces from both vertices are allotted to the +conventional propagator, ∆µν(q), the resulting effective propagator, �∆µν(q), captures all RG +logarithms associated with the running of the coupling; for example, at one loop and for large +q2, one has +�∆−1(q) ≈ q2� +1 + bg2 ln(q2/µ2) +� +, +(65) +where b = 11CA/48π2 is the first coefficient of the Yang-Mills β function. We emphasize that the +PT construction goes through to all orders in perturbation theory, as well as nonperturbatively, +and all key properties of the PT Green’s function persist unaltered [195,196]. +The correspondence between the PT and the BFM may be summarized by stating that the +PT rearrangement outlined above amounts effectively to replacing the Q-type gluon that is +being exchanged (carrying momentum q) by a B-type gluon [194,268–270]; external (on-shell) +fields are always of the Q-type. Thus, the notation used above for the PT effective Green’s +functions (“tildes” and “hats”) corresponds precisely to the BFM notation introduced in Sec- +tion 2. Note that the formal expression of all PT rearrangements implemented diagrammatically +are the BQIs that relate conventional Green’s functions to their BFM counterparts [14]. For + +Particles 2023, 1 +21 +example, in the case of the quark-gluon vertex, we have that the vertices Γµ(q, k1, −k2) [with +external fields Qa +µ(q)qb(k1) ¯qc(−k2)] and �Γµ(q, k1, −k2) [Ba +µ(q)qb(k1) ¯qc(−k2)] are related by the +BQI [271] +�Γµ(q, k1, −k2) = [1 + G(q)]Γµ(q, k1, −k2) + · · · , +(66) +where the ellipsis denotes terms that vanish on shell. Similarly, the BQI of Eq. (A5), when +evaluated on-shell, yields a completely analogous result, to wit, +�IΓµαρ(q, k1, −k2) = [1 + G(q)]IΓµαρ(q, k1, −k2) + · · · . +(67) +It is now clear how the PT gives rise to a process-independent propagator-like component: +regardless of the process (i.e., the type of vertex connecting the internal gluon to the external +states), each vertex contributes to the conventional ∆(q) a factor of [1 + G(q)]−1, finally leading +to the BQI of Eq. (11) [16]. +The culmination of the above sequence of ideas is reached by noting that, by virtue of +Eq. (15), the combination +�d(q) := αs�∆(q) = +αs∆(q) +[1 + G(q)]2 , +(68) +is RGI: it retains exactly the same form before and after renormalization, and, consequently, +does not depend on the renormalization point µ [97]. The quantity �d(q) has mass dimension of +−2, and is known in the literature as the “RGI running interaction strength” [16]. +PT +==⇒ +∆ +g2 �∆ +gΓν +�Γν +�Γµ +gΓµ +µ +ν +µ +ν +q +k1 +k2 +k3 +k4 +q +k1 +k2 +k3 +k4 +PT +==⇒ +∆ +g2 �∆ +�Γβνσ +�Γαµρ +gΓβνσ +gΓαµρ +µ +ν +q +k1 +k2 +k3 +k4 +q +k1 +k2 +k3 +k4 +β +σ +ρ +α +β +σ +ρ +α +µ +ν +Figure 8. Diagrammatic representation of the basic PT rearrangement in the case of quark-antiquark +scattering, corresponding to the S-matrix element Tq ¯q→q ¯q of Eq. (69) (left), and gluon-gluon scattering, +corresponding to Tgg→gg of Eq. (70) (right). +The steps leading to the natural appearance of �d(q) within any given process may be +summarized in the case of quark-antiquark, or gluon-gluon scattering. +Consider the S-matrix elements Tq ¯q→q ¯q, for the scattering of a quark and an antiquark, and +Tgg→gg, for the scattering of two gluons. The quark-antiquark scattering is depicted in the left +panel of Fig. 8. Using the BQI of Eq. (11) we obtain +Tq ¯q→q ¯q = +� +gΓµ(q, k1, −k2) +� +∆(q)Pµν(q)[gΓν(−q, k3, −k4)] +PT += +� +g[1 + G(q)]−1�Γµ(q, k1, −k2) +� +∆(q)Pµν(q) +� +g[1 + G(q)]−1�Γν(−q, k3, −k4) +� +PT += �Γµ(q, k1, −k2) +� +g2[1 + G(q)]−2∆(q) +� +Pµν(q)�Γν(−q, k3, −k4) +PT += �Γµ(q, k1, −k2) +� +g2�∆(q) +� +� +�� +� +4π �d(q) +Pµν(q)�Γν(−q, k3, −k4) , +(69) + +Particles 2023, 1 +22 +where we omit color structures. +Similarly, the scattering of two gluons, depicted in the right panel of Fig. 8, yields +Tgg→gg = +� +gΓαµρ(k1, q, −k2) +� +∆(q)Pµν(q) +� +gΓβνσ(k3, −q, −k4) +� +PT += +� +g[1 + G(q)]−1�Γαµρ(k1, q, −k2) +� +∆(q)Pµν(q) +� +g[1 + G(q)]−1�Γβνσ(k3, −q, −k4) +� +PT += �Γαµρ(k1, q, −k2) +� +g2[1 + G(q)]−2∆(q) +� +Pµν(q)�Γβνσ(k3, −q, −k4) +PT += �Γαµρ(k1, q, −k2) +� +g2�∆(q) +� +� +�� +� +4π �d(k) +Pµν(q)�Γβνσ(k3, −q, −k4) . +(70) +Evidently, the same �d(q), defined in Eq. (68), appears naturally in both Eqs. (69) and (70): it +is, in that sense, a process-independent RGI interaction capturing faithfully the one-gluon +exchange dynamics [3,16,20,80,97,130–132]. +The actual determination of �d(q) proceeds by means of the second equality in Eq. (68), i.e., +by combining the standard gluon propagator, ∆(q), together with the function 1 + G(q). In the +top left panel of Fig. 9 we show lattice data for the conventional gluon propagator from [86] +(points) and a physically motivated fit (blue continuous), given by Eq. (C11) of [125]. In the +top right panel of the same figure we show the 1 + G(q) auxiliary function, which can be +computed by contracting Eq. (12) with Pµν(q)/3 (see, e.g., [132]), using the results of [229] for +the ghost-gluon kernel, Hνµ(r, p, q). Then, in the bottom left panel of Fig. 9 we show the �d(q) +that results from combining the fit for ∆(q) and the 1 + G(q) shown in the top panels of the +same figure and using αs = 0.27 [72] and Z1 = 0.9333 [see Sec. 8]. +From the �d(q) of Eq. (68) one may define the dimensionless RGI interaction [16], I(q), +I(q) := q2 �d(q) . +(71) +As explained in [16], this quantity provides the strength required in order to describe ground- +state hadron observables using SDEs in the matter sector of the theory. In that sense, I(q) +bridges a longstanding gap that has existed between nonperturbative continuum QCD and ab +initio predictions of basic hadron properties. +7. Three-gluon vertex and its planar degeneracy +The three-gluon vertex, IΓαµν(q, r, p), plays a pivotal role in the dynamics of QCD [235], +manifesting its non-Abelian nature through the gluon self-interaction. In fact, the most cel- +ebrated perturbative feature of QCD, namely asymptotic freedom, hinges on the properties +of this particular interaction vertex. Its importance in the nonperturbative domain has led +to an intense effort for unveiling its elaborate features [21,28,33–36,41,50,69,70,72,72,79,82,87, +88,123,173–180,180,181,272]. Indeed, as we have seen in Secs. 3 and 4, the pole structure +of the three-gluon vertex is crucial for the onset of the Schwinger mechanism and the dy- +namical generation of a gluon mass. Moreover, its pole-free part provides highly nontrivial +contributions to the SDEs of several Green’s functions, most notably the gluon propagator +(cf. Fig. 1), as well as in the Bethe-Salpeter and Faddeev equations that determine the properties +of glueballs [236,237,239–241] and hybrid mesons [238], respectively. +For general momenta, IΓαµν(q, r, p) is a particularly complicated function, comprised by +14 tensor structures and their associated form factors [252]. Fortunately, in the Landau gauge, +considerable simplifications take place, making the treatment of the three-gluon vertex less + +Particles 2023, 1 +23 +0 +1.0 +2.0 +3.0 +4.0 +5.0 +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +0 +1.0 +2.0 +3.0 +4.0 +5.0 +0.25 +0.50 +0.75 +1.00 +0 +1.0 +2.0 +3.0 +4.0 +5.0 +0 +3 +6 +9 +12 +15 +18 +0 +1.0 +2.0 +3.0 +4.0 +5.0 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +Figure 9. Top left: Gluon propagator, ∆(q), from lattice simulations of Ref. [86] (points) and a fit given by +Eq. (C11) of [125] (blue continuous). Top right: The auxiliary function 1 + G(q), defined in Eq. (12). Bottom +left: The renormalization group invariant (RGI) running interaction strength �d(q) defined in Eq. (68), +computed using the ∆(q) and 1 + G(q) shown in the top panels, with αs = 0.27 [72] and Z1 = 0.9333 [see +Sec. 8]. Bottom right: The corresponding dimensionless RGI interaction I(q), defined in Eq. (71). +cumbersome. Indeed, in the latter gauge, quantities of interest require only the knowledge of +the transversely projected three-gluon vertex [127,175,176,224], Γαµν(q, r, p), defined as +Γαµν(q, r, p) = IΓα′µ′ν′(q, r, p)Pα′α(q)Pµ′µ(r)Pν′ν(p) += Γα′µ′ν′(q, r, p)Pα′α(q)Pµ′µ(r)Pν′ν(p) . +(72) +Note that Γαµν(q, r, p) does not contain massless poles, by virtue of Eq. (35). Furthermore, +Γαµν(q, r, p) can be parametrized in terms of only 4 independent tensor structures, i.e., +Γαµν(q, r, p) = +4 +∑ +i=1 +�Γi(q2, r2, p2) �λαµν +i +(q, r, p) . +(73) +Due to the Bose symmetry of Γαµν(q, r, p), the �λαµν +i +(q, r, p) can be chosen to be individually +Bose symmetric, such that its form factors �Γi(q2, r2, p2) are symmetric under the exchange of +any two arguments [87]. In fact, they can only depend on three totally symmetric combinations +of momenta. +Quite remarkably, lattice [87–89] and continuum [175,176,224] studies alike, have demon- +strated that, to a very good level of accuracy, the �Γi depend exclusively on a single judiciously + +Particles 2023, 1 +24 +chosen variable. Specifically, the �Γi computed on the lattice in [87–89] can be parametrized in +terms of the special Bose symmetric combination +s2 = 1 +2 +� +q2 + r2 + p2� +. +(74) +Thus, the �Γi are the same for any combination of q2, r2, and p2 that fulfils Eq. (74) for a given +value of s2. This property has been denominated planar degeneracy, because Eq. (74) with fixed +s defines a plane, normal to the vector (1, 1, 1), in the first octant of the coordinate system +(q2, r2, p2). +In particular, the form factor �Γ1(q2, r2, p2) of the classical tensor structure is rather accu- +rately approximated by +�Γ1(q2, r2, p2) ≈ �Γ1(s2, s2, 0) ≈ Lsg(s) . +(75) +In the above equation, Lsg is the single transverse form factor of the three-gluon vertex in the +soft gluon limit [125], and is obtained in lattice simulations as the q = 0 limit of the following +totally transverse projection [85] +Lsg(r) = +Γαµν +0 +(q, r, p)Pαα′(q)Pµµ′(r)Pνν′(p)IΓα′µ′ν′(q, r, p) +Γαµν +0 +(q, r, p)Pαα′(q)Pµµ′(r)Pνν′(p)Γα′µ′ν′ +0 +(q, r, p) +������ +q→0 +. +(76) +A particular realization of the planar degeneracy property is shown in Fig. 10, where +we show the classical form factor �Γ1(q2, r2, p2), obtained from the lattice simulation of [87]; +we consider three different kinematic configurations, characterized by a single momentum. +Specifically, the orange stars correspond to the soft-gluon limit, q = 0, which implies p2 = r2; +the green diamonds denote the symmetric limit, where all of the momenta have the same +magnitude, q2 = p2 = r2; and the purple circles represent points with p2 = r2 and q2 = 2r2. +When plotted against the momentum r, the three configurations of �Γ1(q2, r2, p2) produce three +clearly distinct curves; however, when plotted in terms of the Bose symmetric variable s of +Eq. (74), they become statistically indistinguishable, manifesting the validity of Eq. (75). +0 +1 +2 +3 +4 +5 +6 +0 +0.5 +1.0 +1.5 +0 +1 +2 +3 +4 +5 +6 +0 +0.5 +1.0 +1.5 +Figure 10. Lattice data from Ref. [87] for the classical form factor, �Γ1(q2, r2, p2), of the transversely +projected three-gluon vertex in three different kinematic configurations: the soft-gluon (q = 0, p2 = r2, +orange stars), the symmetric limit (q2 = p2 = r2, green diamonds), and the case p2 = r2 with q2 = 2r2 +(purple circles). In the left panel �Γ1(q2, r2, p2) is plotted as a function of r, while in the right it is plotted as +a function of the Bose symmetric variable s defined in Eq. (74). + +Particles 2023, 1 +25 +In addition to the planar degeneracy property, lattice [85,87–89] and continuum [175,176, +180,224] results show a clear dominance of the classical form factor �Γ1 over the remaining ones. +Based on these considerations, the special approximation +Γαµν(q, r, p) ≈ Lsg(s)Γαµν +0 +(q, r, p) , +(77) +has been put forth, where Γαµν +0 +(q, r, p) is the tree-level value of Γαµν(q, r, p), i.e., Eq. (72) with +Γα′µ′ν′(q, r, p) → Γα′µ′ν′ +0 +(q, r, p). Eq. (77) provides an accurate and exceptionally compact +approximation for Γαµν(q, r, p) in general kinematics. +We emphasize that the shape of Lsg(r) has been very precisely determined through +dedicated lattice studies with large-volume simulations [69,72,85,86]. The outcome of this +exploration is shown in Fig. 11, where we plot the lattice data of [85] for Lsg(r), together with a +physically motivated fit given by Eq. (C12) of [125] (blue continuous curve). +The approximation given by Eq. (77), with the fit for Lsg shown in Fig. 11, will be used +explicitly in Secs. 8 and 11, where the Γαµν(q, r, p) in general kinematics will be needed as input +for the determination of other physically important quantities. +0.0 +1.0 +2.0 +3.0 +4.0 +5.0 +-0.4 +0.0 +0.4 +0.8 +1.2 +Figure 11. Lattice data from Ref. [85] for Lsg(q), compared to the fit for it given by Eq. (C12) of [125] (blue +continuous curve). +8. Ghost dynamics from Schwinger-Dyson equations +We next turn our attention to the ghost sector of the theory, whose scrutiny is important for +several reasons. First, it has been connected to particular scenarios of color confinement [273, +274]. Second, the Green’s functions associated with the ghost sector appear as ingredients in +the SDEs of several key functions, such as the gluon propagator and the three-gluon vertex [41, +50,69,70,72,82,123,173–180,275], affecting their nonperturbative behavior in nontrivial ways, +as will be discussed in Sec. 9. Third, the SDEs governing the ghost sector are simpler than +their gluonic counterparts, because they are comprised by fewer diagrams; in fact, the SDE of +the ghost propagator contains a single diagram, see Fig. 12. Fourth, in the Landau gauge, the +validity of Taylor’s theorem [208] facilitates considerably the task of renormalization. +Consequently, the SDEs of the ghost sector are an excellent testing ground for (a) probing +the impact of the gluonic Green’s functions that contribute to them [86]; (b) assessing the +reliability of truncation schemes [276,277]; and (c) testing the agreement between lattice and +continuum approaches. +One of the central results of numerous studies in the continuum [21,63,86,113,179,226,228– +234] as well as a variety of lattice simulations [42,47,49,51,56,64,74,80] may be summarized by +stating that the ghost propagator, D(q), remains massless, while the corresponding dressing +function, F(q), saturates at the origin. As we will discuss in Sec. 9, the nonperturbative + +Particles 2023, 1 +26 +masslessness of the ghost has important implications for the infrared behavior of the gluon +propagator and the three-gluon vertex. +In what follows we provide a concrete example of the state-of-the-art SDE analysis of +the ghost sector, by solving the coupled system of equations that governs the ghost-dressing +function and the ghost-gluon vertex. In order to obtain a closed system of equations, we use +lattice results for the gluon propagator, the three-gluon vertex, and the value of the coupling +constant in the particular renormalization scheme employed. +The main points of this analysis may be summarized as follows. +(i) We begin by considering the coupled system of SDEs given by Fig. 12, which determines +the ghost propagator and ghost-gluon vertex. The treatment will be simplified by neglecting +diagram (dν +3) of Fig. 12, thus eliminating the dependence on the ghost-ghost-gluon-gluon +vertex, Γµσ. This is a particularly robust truncation, because the impact of the neglected +diagram on the ghost-gluon vertex has been shown to be less than 2% [276]. +(ii) Note that due to the fully transverse nature of the gluon propagators in the Landau +gauge, in conjunction with the fact that various projections need to be implemented during the +treatment of this system, the pole parts V of all fully dressed vertices appearing in Fig. 12 will +be annihilated; thus, we will have throughout the replacement IΓ → Γ. +(iii) We proceed by decomposing the pole-free part, Γν(r, q, p), of the ghost-gluon vertex +into its most general Lorentz structure, namely +Γν(r, q, p) = rνB1(r, q, p) + pνB2(r, q, p) , +(78) +whose scalar form factors reduce to B0 +1 = 1 and B0 +2 = 0 at tree level. Evidently, due to the +transversality of the gluon propagator, only the classical tensor rν, accompanied by the form +factor B1, will survive in all SDE diagrams of Fig. 12. ++ +k +q +q +k + q += +( +−1 +q +) +) +q +−1 +( +Γµσ += +rν ++ ++ ++ +k − p +k +k + r +(g ν +1 ) +k + r +k − p +p +k +ν, a +q +(g ν +2 ) +c +q +k +ν, a +p +(g ν +3 ) +k + r +c +b +r +b +r +p +ν, a +q +c +b +r +b +q +c +r +p +ν, a +Figure 12. Top: SDE governing the momentum evolution of the ghost propagator. Bottom: SDE for the +ghost-gluon vertex, IΓν(r, q, p). +(iv) The SDE of Fig. 12 is given by +F−1(r) = 1 + 2λ +� +k f (k, r)B1(−r, k + r, −k)∆(k)D(k + r) , +(79) + +Particles 2023, 1 +27 +where λ is given by Eq. (43), and we define +f (k, r) := 1 − (r · k)2 +r2k2 +. +(80) +(v) Next, we note that the form factor B1(r, q, p) can be extracted from Γν(r, q, p) through +the projection +B1(r, q, p) = ενΓν(r, q, p) , +εν := p2rν − (r · p)pν +r2p2 − (r · p)2 . +(81) +Hence, acting with εν on the diagrams in the second line of Fig. 12, we obtain +B1(r, q, p) = 1 − λ[a(r, q, p) − b(r, q, p)] , +(82) +where +a(r, q, p) = qαrµεν +� +k D(k)D(k − p)∆(k + r)B1(p − k, q, k + r)B1(−k, k − p, p)Pαµ(k + r)kν , +b(r, q, p) = qαrµεν +� +k ∆(k)∆(k − p)D(k + r)B1(k + r, q, p − k)Γνµα(p, −k, k − p) . +(83) +(vi) At this point, we invoke the property of the planar degeneracy of Γαµν(q, r, p), dis- +cussed in Sec. 7. Employing Eq. (77) into the SDE for B1, the term b(r, q, p) of Eq. (83) becomes +b(r, q, p) = qαrµεν +� +k ∆(k)∆(k − p)D(k + r)B1(k + r, q, p − k)Γ0 +νµα(p, −k, k − p)Lsg(¯s) , +(84) +with ¯s2 = p2 + k2 − 2(k · p). +We emphasize that although Eq. (77) constitutes in general an approximation, there is one +particular kinematic limit in which the expression for b(r, q, p) given in Eq. (84) becomes exact. +Specifically, in the soft gluon limit (p = 0), it can be shown exactly that [86] +Pµ′ +µ (k)Pν′ +ν (k)Γαµ′ν′(0, k, −k) = 2Lsg(k)kαPµν(k) . +(85) +Then, starting from either the general expression for b(r, q, p) of Eq. (83) and using Eq. (85), or +from the approximate version given by Eq. (84), it can easily be shown that the p = 0 limit is +the same. As such, the use of Eq. (77) yields not only an excellent approximation in general +kinematics, but also the exact soft gluon limit for the contribution of the three-gluon vertex to +the form factor B1. +(vii) Now we consider the renormalization of the coupled system of equations. Since +the ghost-gluon vertex is finite in the Landau gauge [208], most SDE treatments [86,225–229] +of the ghost sector employ the so-called Taylor renormalization scheme, defined in such a +way that the finite renormalization constant of the ghost-gluon vertex has the exact value +Z1 = 1 [54,60,81,86,208]. +However, in order to employ Eq. (77) most expeditiously, it is more convenient to renor- +malize in the so-called asymmetric MOM scheme, because this is precisely the scheme employed +in the lattice calculations of Lsg [69,72,85,86]. Specifically, this scheme is defined by imposing +the normalization conditions [85,86] +∆−1 +R (µ) = µ2 , +FR(µ) = 1 , +LR +sg(µ) = 1 . +(86) +Past this point, we denote by �Z1 the finite value of the ghost-gluon renormalization constant in +the asymmetric MOM scheme. Evidently, Eqs. (14) and (78) imply that BR +1 = �Z1B1. + +Particles 2023, 1 +28 +The renormalization of Eqs. (79) and (82) proceeds by substitution of the unrenormalized +quantities by their renormalized counterparts, following Eq. (14), and imposing Eq. (86) for +F(µ2). +Note that, in principle, �Z1, may be determined from the relation �Z1 = Z3ZcZ−1 +A , imposed +by the corresponding STI [278]; however, these renormalization constants are not available to +us, given that the associated Green’s functions have been obtained from the lattice. Therefore, +�Z1 is treated as an adjustable parameter, whose value is determined by requiring that the +solution of the SDE for F(q) reproduces the corresponding lattice data of [74,86] as well as +possible. +(viii) Finally, we transform Eqs. (79) and (82) from Minkowski to Euclidean space, using +standard conversion rules. Note that, once in Euclidean space, we will express the functional +dependence of B1(r, q, p) in terms of the squared momenta of the antighost and gluon legs, r2 +and p2, and the angle, θ, between them, i.e., B1(r, q, p) ≡ B1(r2, p2, θ). +The result of these manipulations is that Eqs. (79) and (82) become +F−1(r) = 1 − αsCA �Z1 +2π2 +� ∞ +0 +dk2k2∆(k) +� π +0 +dφ s4 +φ +× +� +B1(r2, k2, φ) F(√z) +z +− B1(µ2, k2, φ) F(√u) +u +� +, +(87) +and +B1(r2, p2, θ) = �Z1 − αsCA �Z1 +8π2 +� +a + 2b +� +, +(88) +respectively, with +a = 1 +sθ +� ∞ +0 +dk2k2F(k) +� π +0 +dφs3 +φ +∆(√z) +z +� π +0 +dωsω +F(√v) +v +B1(k2, p2, α)B1(v, z, β)Ka , +(89) +b = 1 +sθ +� ∞ +0 +dk2k2∆(k) +� π +0 +dφs3 +φ +F(√z) +z +� π +0 +dωsω +∆(√v) +v +B1(z, v, β)Lsg(s)Kb . +In the above equations we employ the notation cx := cos x and sx := sin x, and define the +following variables +r · k := rkcφ , +p · k := pk(cθcφ + sθsφcω) , +z := r2 + k2 + 2rkcφ , +u := µ2 + k2 + 2µkcφ , +s2 := (p2 + k2 + v)/2 , +v := p2 + k2 − 2pk(cθcφ + sθsφcω) , +α := π − cos−1� +cθcφ + sθsφcω +� +, +β := cos−1 +� +k(pcθcφ + psθsφcω − rcφ) + prcθ − k2 +√vz +� +. + +Particles 2023, 1 +29 +Finally, the kernels Ka and Kb are given by +Ka =(cθcωsφ − cφsθ) +� +ksφ(pcθ + r) − pcθcω(kcφ + r) +� +, +Kb =cω +� +k2pcφ +� +cθp +� +s2 +θ(s2 +φs2 +ω − 4s2 +φ + 1) + s2 +φ +� ++ r +� +s2 +φ − s2 +θ(2s2 +φ + 1) +�� +− k3� +s2 +φ +� +rcθ − 2ps2 +θ + p +� ++ ps2 +θ +� ++ kp2� +s2 +φ +� +2s2 +θ(p − rcθ) − rcθ − p +� ++ s2 +θ(rcθ − p) +� +−cφp3rs2 +θ +� ++ sθsφ +� +cθp +� +r +� +p2 − k2(s2 +ω + s2 +φs2 +ω − 2s2 +φ) +� +− cφk(s2 +ω − 2)(k2 + p2) +� ++ k +� +cφk2r − cφp2r +� +s2 +θ(s2 +ω − 2) + s2 +ω +� ++ kp2� +3s2 +θs2 +φs2 +ω − 2s2 +θs2 +ω − 4s2 +θs2 +φ + 3s2 +θ ++(3 − 2s2 +ω)s2 +φ − 2 +��� +. +We are now in position to solve Eqs. (87) and (88) numerically. We choose the renormal- +ization point at µ = 4.3 GeV, and employ for ∆(q) and Lsg(q) the fits to lattice data shown in +Figs. 9 and 11, respectively. Note that for large momenta these fits recover the behavior dictated +by the corresponding anomalous dimensions [125]. For the strong coupling, we use the value +αs(4.3 GeV) = 0.27, determined from the lattice simulations of [72]. +Below we discuss the main results of this analysis: +The value of �Z1 was obtained by solving the SDE system for various values of this constant +until the χ2 of the comparison between the solution for F(q) and the lattice data of [74,86] was +minimized. This procedure yields �Z1 = 0.9333 ± 0.0075. +In the left panel of Fig. 13 we show as a blue continuous line the SDE result for F(q), with +the above value of �Z1. The result is compared to the lattice data of [74,86], which have been +cured from discretization artifacts. As it turns out, the SDE and lattice results for F agree within +1%. +We next consider the form factor B1. In the right panel of Fig. 13 we show B1(r2, p2, θ) as +a surface, for arbitrary values of the magnitudes of the momenta r and p, and for the angle θ +formed between them at θ = 2π/3. In the same panel, we highlight as a red dot-dashed curve +the soft gluon limit4 B1(r2, 0, 2π/3) of the general kinematics B1(r2, p2, 2π/3). +The only available SU(3) lattice data for B1 have been obtained in the soft gluon limit [42, +43], and have sizable error bars. Furthermore, they have been computed within the Taylor +scheme, while in the present work we used the asymmetric MOM scheme. Nevertheless, we +can meaningfully compare our SDE results with those of the lattice, and perform a statistical +analysis to assess their agreement. +Specifically, denoting by BT +1 the Taylor scheme value of the form factor B1, it can easily be +shown that +B1(r2, p2, θ) = �Z1BT +1(r2, p2, θ) , +(90) +which allows us to carry out the desired comparison. +Then, we use Eq. (90) to compute BT +1(r2, 0, θ) from the B1(r2, 0, 2π/3) slice (red dot-dashed +curve) in the right panel of Fig. 13, and compare the result to the lattice data of [42,43] (points) +in Fig. 14. Evidently, the SDE determination agrees with the lattice results. +In order to quantify this agreement, we next conduct a χ2 analysis. To this end, we +consider only the 22 lattice points ri in the interval ri ∈ [0.3, 2.5] GeV, where the signal is most +pronounced. Then, we compute the χ2 of the data through +χ2 +j = ∑ +i +[Blat +1 (r2 +i , 0, θ) − gj(ri)]2 +ϵB1(r2 +i , 0, θ) +, +(91) +4 +The soft gluon limit is approached by taking p → 0 in B1(r2, p2, θ); in the nonperturbative case, this limit is +independent of the value of θ. + +Particles 2023, 1 +30 +0 +0.5 +1.0 +1.5 +2.0 +2.5 +1.0 +1.5 +2.0 +2.5 +3.0 +Figure 13. Left: ghost dressing function F(q) obtained from the coupled system of SDEs of Eqs. (79) +and (82) (blue continuous line) compared to the lattice data of Ref. [74,86]. Right: The corresponding +result for B1(r2, p2, θ) for arbitrary magnitudes of the antighost and gluon momenta, r and p, respectively, +and a representative value of θ = 2π/3 for the angle between them. The red dot-dashed curve highlights +the soft gluon limit (p = 0). +0 +1 +2 +3 +4 +5 +1.0 +1.1 +1.2 +1.3 +Figure 14. Soft gluon limit, BT +1(r2, 0, θ), of the classical form factor of the ghost-gluon vertex in Taylor +scheme. The points correspond to the lattice data of Ref. [42,43]. The red dot-dashed line shows the SDE +solution with the three-gluon vertex dressed according to Eq. (77), while the green dashed represents the +SDE solution with tree-level three-gluon vertex. +where Blat +1 (r2 +i , 0, θ) are the lattice points shown in Fig. 14, ϵB1(r2 +i , 0, θ) their respective errors, +and gj(ri) are three hypotheses which we will compare to the lattice data. Specifically, for the +gj we consider the three cases +gj(ri) = +� +� +� +� +� +1 +if j = 1 , +SDE with Γαµν = Γαµν +0 +Lsg(s) +if j = 2 , +SDE with Γαµν = Γαµν +0 +if j = 3 , +(92) +i.e., g1 is the tree-level value of B1, g2 the solution of the SDE using Eq. (77) for dressing +the three-gluon vertex, corresponding to the red dot-dashed curve of Fig. 14, and g3 is the +solution of the SDE obtained setting the three-gluon vertex to tree-level, which amounts to the +substitution Lsg → 1 in Eq. (88), and is represented by a green dashed curve in Fig. 14. + +[Vo] +b [GeA] +0 +T +士 +8 +3 +5 +于 +1 +G +Q.0 +B(3) +T +I'S +1'3Particles 2023, 1 +31 +Then, for each χ2 +j we compute the probability Pj that normally distributed errors would +yield a χ2 at least as large as χ2 +j , through +Pj = +� ∞ +χ2 +j +χ2 +PDF(22, x)dx = Γ(nr/2, χ2/2) +Γ(nr/2) +���� +χ2=χ2 +j +nr=22 +. +(93) +In the above equation, χ2 +PDF(n, x) = xn/2−1e−x/2/[2n/2Γ(n/2)] denotes the χ2 probability +distribution function with n degrees of freedom, while Γ(z, x) is the incomplete Γ function. +The results of the above analyses are collected in Table 2. We note that the case g1, i.e., the +tree-level value of B1, is discarded at 5.1σ confidence level. As for case g3, it is discarded at the +3.4σ level. On the other hand, the SDE result with dressed three-gluon vertex, g2, is statistically +indistinguishable from the lattice data. +Table 2. Statistical results of the χ2 analysis for the three hypotheses given in Eq. (92) for the form factor +B1. For each case (first column), we give the corresponding χ2 +j computed from Eq. (91) (second column), +probability Pj computed from Eq. (93) (third row), and the same Pj expressed in terms of confidence levels +σ (fourth row). +Case (j) +χ2 +j +Pj +Confidence +level in σ +1 +71.37 +4.0 × 10−7 +5.1 +2 +3.399 +1 - 1.8 × 10−6 +2.2 × 10−6 +3 +50.03 +5.8 × 10−4 +3.4 +Lastly, we point out that for both F and B1 we find a good qualitative agreement with +various related studies [21,29,179,180,225,227–229,279,280], including kinematics other than +the soft gluon limit considered in Fig. 14. +9. Divergent ghost loops and their impact on the QCD Green’s functions +The masslessness of the ghost propagator, discussed in Sec. 8, has important implications +for the infrared behavior of other Green’s functions. Specifically, while the saturation of the +gluon propagator renders gluon loops infrared finite, ghost loops furnish infrared divergent +contributions [173], akin to those encountered in perturbation theory. In this section, we +highlight with two characteristic examples how the effects of ghost loops manifest themselves +at the level of the two- and three-point functions. Specifically, the ghost loops induce the +appearance of a moderate maximum in the gluon propagator and are responsible for the +zero-crossing and the logarithmic divergence at the origin displayed by the dominant form +factors of the three-gluon vertex. +The basic observation at the level of the gluon SDE shown in Fig. 1 is that, the ghost loop +of (d3), due to the masslessness of its ingredients, furnishes “unprotected” logarithms, i.e., +terms of the type ln q2, which diverge as q2 → 0. Instead, gluonic loops contain infrared finite +gluon propagators, and, therefore, give rise to contributions that remain finite as q2 → 0, i.e., +they may be described in terms of “protected” logarithms of the type ln(q2 + m2). +The circumstances described above may be modeled by +∆−1(q) = q2 + m2 + c1q2 ln +�q2 + ρm2 +Λ2 +� +� +�� +� +f (q) ++c2q2 ln +� q2 +Λ2 +� +, +(94) + +Particles 2023, 1 +32 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +6.0 +6.5 +7.0 +7.5 +8.0 +8.5 +9.0 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +6.0 +6.5 +7.0 +7.5 +8.0 +8.5 +9.0 +Figure 15. Lattice data for the gluon propagator in the deep infrared. The data displayed correspond to +the two lattice setups with the largest volumes of [49], namely, V = 724 (left) and V = 804 (right). The red +dashed lines are smooth fits from which the position of the maximum can be estimated. +where m is the gluon mass, Λ the mass scale of QCD, and c1, c2 and ρ are constants; note that +∆−1(0) = f (0) = m2 +Differentiating Eq. (94) with respect to q2 we obtain +d∆−1(q) +dq2 += d f (q) +dq2 ++ c2 +� +1 + ln +� q2 +Λ2 +�� +. +(95) +The second term on the r.h.s. of Eq. (95) is infrared divergent, and necessarily dominates +the behavior of the derivative of the propagator for sufficiently small q. Moreover, the value +of the coefficient c2 can be computed explicitly by expanding the ghost block �Π(2) +µν (q) of Fig. 1 +around q = 0 and using Eq. (19), which yields +c2 = αsCA �Z2 +1F2(0) +48π +. +(96) +Therefore, d∆−1(q)/dq2 has the asymptotic behavior +lim +q→0 +d∆−1(q) +dq2 += +� +αsCA �Z2 +1F2(0) +48π +� +ln +� q2 +Λ2 +� +, +(97) +which diverges to −∞ as q → 0. Now, since the gluon propagator is a decreasing function in +the ultraviolet, we have that d∆−1(q)/dq2 is positive for large momenta. Therefore, there must +exist a special momentum, denoted by q⋆, such that [d∆(q)/dq2]q=q⋆ = 0, which corresponds +to a maximum5 of ∆(q). +The maximum of ∆(q), predicted by means of the simple arguments presented above, is +observed in lattice simulations of the gluon propagator [49,56,86]. In particular, it is clearly +visible in Fig. 15, where the data from the two largest volume lattice setups of [49] are shown. +The red dashed lines represent smooth functions, fitted to each of the data sets, in the window +q ∈ [0, 0.5] GeV. For each of the volumes considered, V = 724 (left panel) and V = 804 (right +panel), the estimate obtained for q⋆ is q⋆ = 140 MeV. +5 +Note that d∆−1(q)/dq2 is an increasing function, since it is negative in the infrared and positive in the ultraviolet, i.e., +d2∆−1(q)/d(q2)2 > 0. Therefore, assuming that d∆−1(q)/dq2 only crosses zero once, q = q⋆ must be a maximum +of ∆(q). + +Particles 2023, 1 +33 +It is interesting to observe in passing that the existence of a maximum of ∆(q) has an +interesting implication on the form of the spectral function of the gluon propagator [281–286]. +In particular, the standard Källén-Lehmann representation [287,288] states that +∆(q) = +� ∞ +0 dλ2 ρ(λ2) +q2 + λ2 , +(98) +where ρ(λ2) is the gluon spectral function (with a factor 1/π absorbed in it). Thus, the +differentiation of both sides of Eq. (98) with respect to q2 yields +d∆(q) +dq2 += − +� ∞ +0 dλ2 +ρ(λ2) +(q2 + λ2)2 . +(99) +Then, from Eq. (99) follows that the existence of a maximum for ∆(q) at q = q⋆ leads necessarily +to the violation of reflection positivity [11,168,169,172], because the condition +� ∞ +0 dλ2 +ρ(λ2) +(q2⋆ + λ2)2 = 0 , +(100) +may be fulfilled only if ρ(λ2) reverses its sign. Note that an analogous argument based on the +existence of an inflection point has been presented recently in [8]. +Turning to the three-gluon vertex, it is well known that the corresponding ghost loops +induce characteristic features to the form factors associated with its classical (tree-level) tensors. +There are two complementary continuum descriptions of the dynamics that determine the +behavior of these form factors: (i) the SDE of the three-gluon vertex [175–177,227], depicted +diagrammatically in Fig. 16, and (ii) the STI of Eq. (23) [173], which, in the limit of vanishing +gluon momentum, and when the displacement function and the ghost sector are neglected, +yields the approximate WI +IΓαµν(0, r, −r) ≈ +∂∆−1 +µν (r) +∂rα +, +(101) +which transmits the properties of the propagator derivative to the vertex form factors, as shown +schematically in Fig. 18. +In the simplified kinematic circumstances where only a single representative momentum +is considered, to be denoted by r, the conclusions drawn by either method may be qualitatively +described in terms of a simple model, namely +L(r) = b0 + bgl ln +�r2 + m2 +Λ2 +� ++ bgh ln +� r2 +Λ2 +� +, +(102) +where L(r) denotes the particular combination of form factors, such that, at tree level, L0(r) = 1, +and b0, bgl, and bgh are positive constants. The model in Eq. (102) encompasses two important +cases studied on the lattice [69,70,72,82], namely (i) the soft gluon limit, L(r) → Lsg(r), corre- +sponding to the kinematic choice q → 0 , +p = −r , +θ := �pr = π, defined in Eq. (76), and (ii) +totally symmetric limit, L(r) → Lsym(r), corresponding to q2 = p2 = r2 , +θ := �qr = � +qp = �rp = +2π/3. +Upon inspection of Eq. (102) we note that, as r → 0, the term with the unprotected +logarithm will eventually dominate, forcing L(r) to reverse its sign (zero crossing), and finally +display a logarithmic divergence, L(0) → −∞. Given that, in practice, bgl is considerably larger +than bgh, the unprotected logarithm overtakes the protected one rather deep in the infrared: the +location of the zero-crossing is at about 160 MeV [72]. Consequently, in the intermediate region +of momenta, which is considered relevant for the onset of nonperturbative dynamics, we have + +Particles 2023, 1 +34 +L(r) < 1; this effect is known in the literature as the infrared suppression of the three-gluon +vertex. ++ ++ · · · +(e1) +(e2) +µ, b +α, a +p += ++ +α, a +ν, c +q +p +r +ν, c +µ, b +q +r +Figure 16. The SDE of the three-gluon vertex at the one-loop dressed level. The diagrams (e1) and (e2) +are the gluon and the ghost triangle contributions entering in the skeleton expansion of the three-gluon +vertex. +Most importantly, the special features of infrared suppression, zero-crossing, and log- +arithmic divergence at the origin have been corroborated through a variety of lattice re- +sults [50,69,70,72,73,82,85], as shown, e.g., in Fig. 11. The central curve of this figure is presented +as the blue line in Fig. 17, where the aforementioned characteristics have been explicitly marked +for the benefit of the reader. Note the close proximity of the blue curve to the d∆−1(r)/dr2 (red +dashed line), especially below 1 GeV. +0 +1.0 +2.0 +3.0 +4.0 +5.0 +-0.4 +0.0 +0.4 +0.8 +1.2 +Figure 17. Comparison of Lsg(r) (blue continuous) from Fig. 11 and d∆−1(r)/dr2 (red dashed) resulting +from the fit for ∆(r) of Fig. 9. Note that both display the characteristic features of infrared suppression +with respect to their tree-level values (which is 1 for both quantities), zero-crossing, and logarithmic +divergence at the origin. +We end this section by pointing out that, in the case of Yang-Mills in d = 3 [28,173,224,289– +303], the situation is qualitatively similar to the one described above, but the divergences +induced due to the masslessness of the ghost are stronger. Specifically, as may be already +established at the level of a simple one-loop calculation [303], the first derivative of the gluon +propagator diverges at the origin as 1/q rather than a ln q2. Consequently, the corresponding +effects are significantly enhanced; in particular, the maximum of the gluon propagator is +considerably more pronounced, becoming plainly visible on the lattice [53]. Similarly, an +abrupt negative divergence is observed in the corresponding vertex form factors [41,83]. +10. Ward identity displacement of the three-gluon vertex +In complete analogy to the case of the ghost-gluon vertex discussed in subsection 5.2, the +WI satisfied by the pole-free part of the three-gluon vertex is also displaced in the presence of + +Particles 2023, 1 +35 +⊃ +⊃ +STI +Figure 18. The ghost triangle present in the three-gluon vertex SDE (top) and the ghost loop contributing +to the gluon propagator in the corresponding equation (middle). The infrared divergences arising from +these diagrams are connected through the Slavnov-Taylor identity (STI) of Eq. (23), as shown schematically +in the bottom panel. +longitudinally coupled massless poles. Quite importantly, the associated displacement function, +C(r), coincides with the BS amplitude that controls the formation of a (colored) scalar bound +state with vanishing mass out of a gluon pair. The displacement of the WI circumvents the +seagull cancellation involving the gluon propagator [i.e., f = ∆ in Eq. (54)], furnishing to the +gµν component the mass originating from graphs (d1) and (d4) in Fig. 1. In addition, it permits +the indirect determination of the displacement function C(r) from the lattice; this is particularly +important, given that, by virtue of Eq. (35), the lattice “observables” do not perceive directly +the presence of the massless poles. +The starting point of the analysis is the STI satisfied by the three-gluon vertex, IΓαµν(q, r, p), +given by Eq. (23). In order to eliminate the poles in r and p, thus isolating the displacement of +the WI originating from the pole in the channel q, we contract that equation with Pµ +µ′(r)Pν +ν′(p). +Note that this procedure also eliminates any longitudinal pole terms in the Hσµ(p, q, r) and +Hσν(r, q, p). +Then, we decompose IΓαµν(q, r, p) into pole-free and longitudinally coupled massless pole +parts, as in Eq. (33), and use Eq. (38), to obtain +Pµ +µ′(r)Pν +ν′(p) +� +qαΓαµν(q, r, p) + gµνC1(q, r, p) + qµqνC5(q, r, p) +� = Pµ +µ′(r)Pν +ν′(p)Rνµ(p, q, r) , +(103) +where +Rνµ(p, q, r) := F(q) +� +∆−1(p)Pσ +ν (p)Hσµ(p, q, r) − ∆−1(r)Pσ +µ (r)Hσν(r, q, p) +� +. +(104) +At this point, we expand Eq. (103) around q = 0 and match coefficients of equal orders. At +zeroth order in this expansion we obtain immediately that +C1(0, r, −r) = 0 , +(105) +in exact analogy to Eq. (61). Note that we have arrived once again at the result of Eq. (39), +but through an entirely different path: while Eq. (39) is enforced by the Bose symmetry of the +three-gluon vertex, Eq. (105) is a direct consequence of the STI that this vertex satisfies. + +Particles 2023, 1 +36 +We next gather the terms in the expansion of Eq. (103) that are of first order in q. Evidently, +the term C5 does not contribute to this order. Then, the expansion leads to the appearance of +derivatives of the gluon propagator, in analogy to Eq. (63), but also of the ghost-gluon kernel. +Specifically, we obtain for the WI of the three-gluon vertex and its displacement the expression +Lsg(r) = F(0) +� +�Z1 +d∆−1(r) +dr2 ++ W(r) +r2 +∆−1(r) +� +− C(r) . +(106) +In the above equation, Lsg(r) is the form factor of the three-gluon vertex defined in Eq. (76) and +with lattice results shown in Fig. 11, while W(r) is a particular derivative of the ghost-gluon +kernel, namely [125,242] +W(r) = − 1 +3r2 Pµν(r) +�∂Hνµ(p, q, r) +∂qα +� +q=0 +. +(107) +For the detailed derivation of Eq. (106), we refer to [94,125]. +In the following section, we will use Eq. (106) to determine the displacement amplitude +C(r) from lattice inputs. To this end, we must first pass to Euclidean space, where we note that +CE(r2 +E) = −C(r)|r2=−r2 +E , +(108) +with the extra sign originating from the fact that C is a derivative [see Eq. (41)]. Then, suppress- +ing the indices “E” and solving for C(r2), we obtain the central relation +C(r) = Lsg(r) − F(0) +�W(r) +r2 +∆−1(r) + �Z1 +d∆−1(r) +dr2 +� +. +(109) +For the determination of C(r), we use lattice inputs for all the quantities that appear on +the r.h.s. of Eq. (109), with the exception of the function W(r), which will be computed from +the SDE satisfied by the ghost-gluon kernel, derived and analyzed in the next section. +11. The ghost-gluon kernel contribution to the Ward identity +In this section, we derive the SDE that determines the key function W(r); the resulting +SDE will be solved using lattice inputs for the various quantities entering in it. In addition, the +infrared behavior of W(r) will be analyzed in detail, following an analytic procedure. +Our discussion starts with the SDE of the ghost-gluon kernel, Hµν(r, q, p), shown diagram- +matically in Fig. 19, from which W(r) can be obtained using Eq. (107). +Note that the similarity between the diagrams shown in Fig. 19 and those in the bottom +panel of Fig. 12, depicting the SDE of the ghost-gluon vertex, is a simple reflection of the +fundamental STI relating the ghost-gluon kernel with the ghost-gluon vertex, +Γν(r, q, p) = rµHµν(r, q, p) . +(110) +Specifically, Eq. (110) is preserved by the SDEs of Γν(r, q, p) and Hµν(r, q, p); indeed, contraction +of each diagram (hµν +i ) of Fig. 19 by rµ yields the corresponding diagram (gν +i ) of Fig. 12 (up to a +shift of k → −k − r for i = 1, introduced to simplify certain expressions). Note that, in Fig. 19, +the diagram corresponding to the (g3) of Fig. 12 has been omitted, for the reason explained in +the item (i) of Section 8. +It is well known that, in the Landau gauge, the momentum q of the ghost field in +Hµν(r, q, p) factors out of its quantum corrections [1], allowing us to write [125,229,242] +Hµν(r, q, p) = gµν + qαKµνα(r, q, p) , +(111) + +Particles 2023, 1 +37 += +gµν ++ ++ +k − q +q +p +k + r +r +k +ν, a +µ, b +(h µν +1 ) +k + r +k − p +p +k +r +ν, a +q +µ, b +(h µν +2 ) +c +c +µ, b +k +q +ν, a +p +r +k + r +c +Figure 19. SDE for the ghost-gluon scattering kernel, Hµν(r, q, p). We omit a diagram containing a 1PI +four-point function. +where no particular assumptions are made about the structure of the function Kµνα(r, q, p). +Following Eq. (107), we differentiate the r.h.s. of Eq. (111) with respect to q and subsequently +set q = 0, to obtain +W(r) = −1 +3rαPµν(r)Kµνα(r, 0, −r) . +(112) +Lastly, the finite renormalization of W proceeds through the use of Eqs. (14) and (86), +which leads to the appearance of an overall factor of �Z1 in the equations. +The outcome of the above steps is that W(r) can be written as +W(r) = W1(r) + W2(r) , +(113) +where the Wi(r) are the contributions originating from the diagrams (hµν +i ) of Fig. 19, respec- +tively, and read +W1(r) += +λ�Z1 +3 +� +k ∆(k)D(k)D(k + r)(r · k) f (k, r)B1(k + r, −k, −r)B1(k, 0, −k) , +W2(r) += +λ�Z1 +3 +� +k ∆(k)∆(k + r)D(k + r)B1(k + r, 0, −k − r)IW(−r, −k, k + r) , +(114) +where f (k, r) is given by Eq. (80), and we define the specific contribution of the three-gluon +vertex to the kernel of W(r2) as +IW(q, r, p) := 1 +2(q − r)νΓα +αν(q, r, p) . +(115) +Note that, from Eq. (115) and the Bose symmetry of the Γαµν(q, r, p) under the exchange +{q, α} ↔ {r, µ}, it follows that +IW(q, r, p) = IW(r, q, p) . +(116) +At this point, by capitalizing on the planar degeneracy of Γαµν(q, r, p) discussed in Sec. 7, +we obtain a compact, and yet accurate, approximation for IW. Specifically, using Eq. (77), we +find +IW(q, r, p) ≈ I0 +W(q, r, p)Lsg(s) , +(117) + +Particles 2023, 1 +38 +where I0 +W(q, r, p) is the tree-level value of IW, given by +I0 +W(q, r, p) := 2f (q, r) +p2 +� +2q2r2 − (q2 + r2)(q · r) − (q · r)2� +. +(118) +We remark that the approximation given by Eq. (117) becomes exact in the limit p = 0. +Using the above approximation for IW, the contribution W2(r) reads +W2(r) =2λ�Z1 +3 +� +k ∆(k)∆(k + r)D(k + r) +(k + r)2 +B1(k + r, 0, −k − r) f (k, r) +× +� +2r2k2 − (r2 + k2)(r · k) − (r · k)2� +Lsg(ˆs) , +(119) +where we now have ˆs2 = r2 + k2 + (r · k). +Lastly, we transform W1 of Eq. (114) and W2 of Eq. (119) to Euclidean space to obtain the +final expression to be used for the numerical determination of W, +W1(r) = −rαsCA �Z1 +12π2 +� ∞ +0 dk2k∆(k)F(k)B1(k2, k2, π) +� π +0 dφs4 +φcφ +F(√z) +z +B1(z, r2, χ) , +W2(r) = −rαsCA �Z1 +6π2 +� ∞ +0 dk2 k3∆(k) +� π +0 dφ s4 +φ∆(√ +z)B1(z, z, π) F(√z) +z2 +� +kr(2 + c2 +φ) − zcφ +� +× Lsg +� +r2 + k2 + rkcφ +� +, +(120) +where z has been defined below Eq. (89) and +χ := cos−1 +� +−(r + kcφ) +√z +� +. +(121) +We emphasize that we have used into the SDEs of both B1 and W, given by Eqs. (88) +and (120), respectively, the same approximation for the three-gluon vertex, namely Eq. (77). +Therefore, our analyses of B1 and W are self-consistent, in the sense that the STI in Eq. (110) is +strictly preserved. +Before embarking on the numerical determination of W(r) for the entire range of Euclidean +momenta, we discuss the infrared behavior of this function, and demonstrate an important +self-consistency proof involving C(r). +Specifically, as discussed in Sec. 9, the Lsg(r) and d∆−1(r)/dr2 that appear in Eq. (109) are +infrared divergent, due to massless ghost loops present in their SDEs. Nevertheless, the BSE +solutions for the amplitude C(r) are all found to be finite at r = 0, (cf. Fig. 7) [118,122,125,216]. +Therefore, in order for the WI displacement of Eq. (109) to be consistent with the finite C(0) +obtained from BSE solutions, the infrared divergences of the ingredients appearing in Eq. (109) +must cancel against each other. +Indeed, a careful analysis of diagram (e2) of Fig. 16 yields +lim +r→0 Lsg(r) = +� +αsCA �Z3 +1F3(0) +96π +� +ln +� r2 +µ2 +� +, +(122) +up to infrared finite terms6. Then, assuming that only Lsg(r) and d∆−1/dr2 diverge and using +the asymptotic form of d∆(r)/dr2 given in Eq. (97) into Eq. (109), we find that the divergences +6 +We note that results identical to Eqs. (97) and (122) for the infrared divergences of d∆−1(r)/dr2 and Lsg(r), +respectively, have been previously derived within the Curci-Ferrari model [181]. + +Particles 2023, 1 +39 +do not fully cancel. Therefore, the finiteness of C(0) demands that the term W(r)/r2 appearing +in the WI must be infrared divergent. +Now, it is evident from Eq. (120) that W(r) vanishes as r → 0. Nevertheless, the ratio +W(r)/r2 is found to diverge at the origin. Specifically, expanding Eq. (120) around r = 0, it can +be shown that W(r)/r2 has the asymptotic behavior +lim +r→0 +W(r) +r2 += − +� +αsCA �Z3 +1∆(0)F2(0) +96π +� +ln +� r2 +µ2 +� +. +(123) +Then, combining Eqs. (123), (122) and (123) we find that the infrared divergences in Eq. (109) +cancel out exactly, leaving a finite C(0), in full agreement with the BSE results. +r +r +0 +µ +ν +ρ +− F(0) �Z1 d +dr2 +r +r +0 +µ +ν +ρ +− F(0) +∆(0) +lim +r2→0 += IR finite +r +µ +ν +� +�� +� +Kµνρ(r, 0, −r) +Figure 20. Diagrammatic representation of the cancellation of the infrared divergences originating from +massless ghost loops in Eq. (109) to yield a finite C(0). The red cross indicates that the overall ghost +momentum is factored out before being set to zero. +We finish the discussion of the infrared finiteness of C(0) with a remark. In the absence +of the Schwinger mechanism, i.e., for an identically zero C(r), the infared divergences of +Lsg(r), W(r)/r2 and d∆−1(r)/dr2 must also cancel in Eq. (109). For instance, this cancellation +can be explicitly verified at the one loop level7, where, evidently, C(r) = 0. In that case, +however, the gluon propagator is also massless, causing the gluonic loops contributing to the +functions entering Eq. (109) to also diverge, such that the cancellation occurs among all radiative +diagrams. In contrast, in the presence of a gluon mass, the cancellation of the remaining infrared +divergences takes place at the level of the ghost loops only, as illustrated diagrammatically in +Fig. 20. +Figure 21. W(r) obtained using the approximation Eq. (117) based on the observed planar degeneracy of +the three-gluon vertex in its kernel (blue solid curve) together with uncertainty estimate (blue band). +7 +In the perturbative realization of Eq. (109) F(0) also diverges, participating in the overall cancellation of infrared +divergences. + +0.0 +-0.2-0.4 +-0.6 +-0.8 +0.0 +1.0 +2.0 +3 +r[GeV].0 +4.0 +5.0Particles 2023, 1 +40 +We now return to the numerical determination of W(r) from Eq. (120). To this end, we +employ the fits to lattice data of [85] for ∆(q) and Lsg(q) shown in Figs. 9 and 11, respectively, +and the SDE solution for F(q) shown in the left panel of Fig. 13. All of the fits employed +are constructed so as to reproduce the correct ultraviolet behavior of the respective Green’s +functions. For the value of the coupling in the asymmetric MOM scheme we employ g2 = 4παs, +with αs(4.3 GeV) = 0.27, as determined in the lattice study of [72]. Lastly, for B1 we use the +SDE result of Sec. 8, shown in the right panel of Fig. 13, which reproduces accurately the +available lattice data for the ghost-gluon vertex. +Using the above ingredients into Eq. (120) we obtain the W(r) shown as the blue solid +curve in Fig. 21. The blue band in Fig. 21 represents the error estimate on our results; the +procedures followed to obtain it are described in detail in [127]. +12. Displacement function from lattice inputs +In this section we determine C(r) from the main relation given in Eq. (109). +For W(r) we use the result shown in Fig. 21, together with the curves for Lsg(r) from +Fig. 11, ∆(r) and d∆−1(r)/dr2 from Figs. 9 and 17, respectively, and the F(r) of Fig. 13. The +C(r) obtained is shown as a black solid curve in the left panel of Fig. 22. In the same panel, we +show as points the estimates of C(r) obtained by using into Eq. (109) the lattice data points of +Ref. [85] directly, rather than a fit. To estimate the uncertainty in the resulting C(r), we combine +the error estimate of W(r), represented by the blue band in Fig. 21, with the statistical error of +the lattice data points for Lsg(r) of [85], and neglect the error in the gluon propagator, which is +much smaller than the errors in Lsg and W. Then, a conservative error propagation analysis +produces the error bars shown in Fig. 22. +0.3 +1.0 +2.0 +3.0 +4.3 +-0.6 +-0.4 +-0.2 +0.0 +0.2 +C(r) +C(r) +0.3 +1.0 +2.0 +3.0 +4.3 +-0.6 +-0.4 +-0.2 +0.0 +0.2 +C(r) +C(r) +C⋆(r) +Figure 22. Left: Result for C(r) (black continuous line) obtained from Eq. (109) using the W(r) shown +in Fig. 21, the fits to lattice data for ∆(r) and Lsg(r) shown in Figs. 9 and 17, respectively, and the SDE +solution for F(r) shown in Fig. 13. The points are obtained using for Lsg(r) the data in Ref. [85], with error +bars denoting the error propagated from Lsg and W. The green band is obtained by fitting the upper and +lower bounds of the points and guide the eye to the typical error associated with C(r). Right: The C(r) of +the left panel is compared to the BSE prediction C⋆(r) (purple dot-dashed and error band) of Fig. 7. +At this point, we quantify the significance of the C(r) obtained above, in comparison to +the null hypothesis result; evidently, in the absence of the Schwinger mechanism, this latter +quantity, to be denoted by C0 in what follows, vanishes identically, namely C0 = 0. To this end, +we first compute the χ2 of our points as +χ2 = ∑ +i +[C(ri) − C0(ri)]2 +ϵ2 +C(ri) +, +(124) + +Particles 2023, 1 +41 +i.e., the null hypothesis is taken as the estimator for our data. The sum runs over the nr = 515 +indices i such that ri ∈ [0.3, 4.3] GeV, the interval of momenta for which the systematic error in +our calculation of W(r) is best known, and ϵC(ri) denotes the error estimate of C(ri). Then we +obtain χ2 = 2 630, corresponding to χ2 +d.o.f. = 5.11. The probability PC0 that our result for C is +consistent with the null hypothesis is vanishingly small, given by the formula +PC0 = +� ∞ +χ2=2 630 χ2 +PDF(515, x)dx = Γ(nr/2, χ2/2) +Γ(nr/2) +���� +χ2=2 630 +nr=515 += 7.3 × 10−280 . +(125) +In fact, even if the errors were 0.95% larger, i.e., nearly doubled, we could still discard C0 at the +5σ confidence level. +In the right panel of Fig. 22 we compare C(r) to the BSE prediction, C⋆(r), of Fig. 7, shown +as a purple dot-dashed curve and corresponding error band. In that panel, we observe an +excellent qualitative agreement between the two results. The most noticeable quantitative +difference is in the position of the minimum. Specifically, C reaches the minimum value of +−0.36 ± 0.11 at r = 1.93+0.09 +−0.06 GeV, while the minimum of C⋆ is −0.341 ± 0.003 at r = 1.5 ± 0.1. +Nevertheless, it is clear in the right panel of Fig. 22 that the BSE prediction lies within the +error estimate of the lattice-derived C(r). In fact, defining a χ2 measure for the discrepancy +between C and C⋆ as +χ2 +⋆ = ∑ +i +[C(ri) − C⋆(ri)]2 +ϵ2 +C(ri) +, +(126) +we obtain χ2⋆ = 258.5, which is smaller than the number of degrees of freedom. Then, this +value of χ2⋆ amounts to a probability of +PC⋆ = Γ(nr/2, χ2⋆/2) +Γ(nr/2) +���� +χ2⋆=258.5 +nr=515 += 1 − 2.0 × 10−23 , +(127) +showing that C⋆ is statistically compatible with the lattice derived C, with probability extremely +near unit. +13. Conclusions +The gauge sector of QCD is host to a wide array of subtle mechanisms that are of vital +importance for the self-consistency and infrared stability of the theory. In the present work, +we have offered a comprehensive review of the intricate dynamics that account for some of +the most prominent infrared phenomena, such as the generation of a gluon mass through +the action of the Schwinger mechanism, the nonperturbative masslessness of the ghost, and +the characteristic features induced by this particular mass pattern to the form factors of the +three-gluon vertex. +The SDEs, supplemented by the judicious use of certain key results from lattice QCD, +provide a robust continuum framework for carrying out such a demanding investigation. In +fact, the results obtained from the SDEs are increasingly reliable, passing successfully all sorts +of tests imposed on them. A particularly impressive, and certainly not isolated, case of such a +success has been outlined in detail in Section 6. +Symmetry and dynamics are tightly interwoven; therefore, the information encoded in +the STIs and WIs of the theory is particularly decisive for unraveling basic dynamical patterns. +A striking manifestation of the profound connection between symmetry and dynamics is +provided by the dual role played by the function C(r): it is both the BS amplitude of the +massless states composed by a pair of gluons, and the quantity that embodies the displacement +induced to the WIs by the presence of these states. + +Particles 2023, 1 +42 +In our opinion, the determination of C(r) described in Section 12 represents a major +success of the entire set of concepts and techniques surrounding the generation of a gluon mass +through the action of the Schwinger mechanism. Thus, fifty years after the genesis of QCD, we +seem to be closing in on the mechanism that the theory uses for curing the infrared instabilities +endemic to perturbation theory. We hope to be able to report further progress in this direction +in the near future. +Funding: The authors are supported by the Spanish MICINN grant PID2020-113334GB-I00. M. N. F. +acknowledges financial support from Generalitat Valenciana through contract CIAPOS/2021/74. J. P. also +acknowledges funding from the regional Prometeo/2019/087 from the Generalitat Valenciana. +Data Availability Statement: Not applicable. +Acknowledgments: The authors thank A.C. Aguilar, D. Binosi, D. Ibáñez, J. Pawlowski, C.D. Roberts, +and J. Rodríguez-Quintero for several collaborations. +Conflicts of Interest: The authors declare no conflict of interest. +Abbreviations +The following abbreviations are used in this work: +BFM +background field method +BQI +background-quantum identity +BRST +Becchi-Rouet-Stora-Tyutin +BS +Bethe-Salpeter +BSE +Bethe-Salpeter equation +EHM +emergent hadron mass +MOM +momentum subtraction (renormalization schemes) +PT +pinch technique +QCD +Quantum Chromodynamics +QED +Quantum Electrodynamics +RGI +renormalization group invariant +SDE +Schwinger-Dyson equation +STI +Slavnov-Taylor identity +WI +Ward identity +Appendix A. BQIs for the BSE amplitudes +In this Appendix, we use two particular BQIs in order to relate the displacement functions +C and C with their BFM counterparts �C and �C, respectively. +The ghost-gluon vertices IΓµ(r, p, q) and �IΓµ(r, p, q) are related by a BQI [14], which reads +�IΓµ(r, p, q) += +� +[1 + G(q)]gν +µ + L(q)qµqν +q2 +� +IΓν(r, p, q) ++F−1(p)pνKµν(r, q, p) + r2F−1(r)Kµ(r, q, p) , +(A1) +where Kµ and Kµν are two auxiliary functions, shown diagrammatically in Fig. A1, while G(q) +and L(q) are the form factors of Λµν(q), defined in Eq. (12). +Next, we decompose the �IΓµ(r, p, q) and IΓµ(r, p, q) in Eq. (A1) into their regular and pole +parts, using Eqs. (33) and (52), respectively. Note that the second and third terms in Eq. (A1) do +not contain poles in q2; this is so because Kµν(r, q, p) can contain (longitudinally coupled) poles +only in the pν channel, whereas Kµ(r, q, p) has no external gluon legs, and hence no poles. + +Particles 2023, 1 +43 +−gf amnKµ(r, q, p) = +n +µ, a +p +m +q +r +−gf anmKµν(r, q, p) = gf amngµν + +p +ν, m +n +µ, a +q +r +Figure A1. The auxiliary functions Kµ(q, r, p) and Kµν(q, r, p), appearing in the BQI of Eq. (A1). +Then, multiplying Eq. (A1) by q2 we obtain +qµ �C(r, p, q) = qµ[1 + G(q) + L(q)]C(r, p, q) + O(q2) . +(A2) +Setting q = 0 in Eq. (A2) and using Eq. (18), we find +C(r, −r, 0) = Z1F(0) �C(r, −r, 0) . +(A3) +Hence, using Eq. (61), we obtain the result in Eq. (39). +Then, expanding Eq. (A2) to first order in q, using Eq. (41) for C(r, p, q) and Eq. (62) for +�C(r, p, q), entails +C(r) = Z1F(0) �C(r) , +(A4) +which is one of the main results of this Appendix. +A relation identical to Eq. (A4) can be obtained for C(r) and its BFM analog, �C(r). The +starting point of the derivation is the BQI [14] +�IΓαµν(q, r, p) += +� +[1 + G(q)]gρ +α + L(q)qαqρ +q2 +� +IΓρµν(q, r, p) +(A5) ++Kρνα(r, q, p)Pρ +µ(r)∆−1(r) − Kρµα(p, q, r)Pρ +ν (p)∆−1(p) , +where Kµνα(r, q, p) is the function defined in Eq. (111). +Then, we note that the only longitudinal poles at q = 0 present in Eq. (A5) are those +contained in the IΓαµν(q, r, p) and �IΓαµν(q, r, p) vertices, with the auxiliary functions Kανρ(q, p, r) +having poles only in the rµ and pν channels. As such, isolating the qαgµν/q2 pole and expanding +around q = 0, one eventually finds +�C1(0, r, −r) = Z−1 +1 F−1(0)C1(0, r, −r) = 0 , +(A6) +and +C(r) = Z1F(0)�C(r) , +(A7) +where �C1(q, r, p) and �C(r2) are defined in analogy to the Eqs. (37) and (41), and we used +Eq. (39). +References +1. +Marciano, W.J.; Pagels, H. Quantum Chromodynamics: A Review. +Phys. Rept. 1978, 36, 137. +https://doi.org/10.1016/0370-1573(78)90208-9. +2. +Qin, S.x.; Roberts, C.D. 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Rev. 2010, D81, 125025, [arXiv:hep-ph/1004.2011]. https://doi.org/10.1103/ +PhysRevD.81.125025. + diff --git a/ltE0T4oBgHgl3EQfZAA9/content/tmp_files/load_file.txt b/ltE0T4oBgHgl3EQfZAA9/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..bbffb2ba3c5b3c7ab8a495a087e587a6bba0e0f4 --- /dev/null +++ b/ltE0T4oBgHgl3EQfZAA9/content/tmp_files/load_file.txt @@ -0,0 +1,6876 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf,len=6875 +page_content='���������� ������� Citation: Ferreira, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Papavassiliou, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Gauge sector dynamics in QCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Particles 2023, 1, 1–57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='org/ Received: January 9, 2023 Accepted: Published: Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Copyright: © 2023 by the authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Licensee MDPI, Basel, Switzerland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='org/licenses/by/ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='0/).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Review Gauge Sector Dynamics in QCD Mauricio Narciso Ferreira1,† and Joannis Papavassiliou1,† 1 Department of Theoretical Physics and IFIC, University of Valencia and CSIC, E-46100, Valencia, Spain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Correspondence: ansonar@uv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='es (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Ferreira);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Joannis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='Papavassiliou@uv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='es (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Papavassiliou) † These authors contributed equally to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Abstract: The dynamics of the gauge sector of QCD give rise to nonperturbative phenomena that are crucial for the internal consistency of the theory;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' most notably, they account for the generation of a gluon mass through the action of the Schwinger mechanism, the taming of the Landau pole and the ensuing stabilization of the gauge coupling, and the infrared suppression of the three-gluon vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' In the present work, we review some key advances in the ongoing investigation of this sector within the framework of the continuum Schwinger function methods, supplemented by results obtained from lattice simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Keywords: continuum Schwinger function methods;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' emergence of hadron mass;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' gluon mass genera- tion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' lattice QCD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' nonperturbative quantum field theory;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' quantum chromodynamics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Schwinger-Dyson equations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Schwinger mechanism Contents 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Introduction .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 41 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Appendix A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 42 References .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 43 Particles 2023, 1, 1–57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='3390/particles1010000 https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='mdpi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='com/journal/particles arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='02314v1 [hep-ph] 5 Jan 2023 BYparticlesParticles 2023, 1 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Introduction The systematic exploration of the Green’s functions (n-point correlation functions) of Quantum Chromodynamics (QCD) [1] by means of continuous Schwinger function meth- ods [2–9], such as Schwinger-Dyson equations (SDEs) [10–21] and functional renormalization group [22–31], together with a plethora of gauge-fixed lattice simulations [32–89], has afforded ample access to the dynamical mechanisms responsible for the nonperturbative properties of this remarkable theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Particularly prominent in this quest is the notion of the emergent hadron mass (EHM) [3,8,9,90–94], together with its three supporting pillars: first, the genera- tion of a gluon mass [18,32,94–127] through the action of the Schwinger mechanism [128,129];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' second, the construction of the process-independent effective charge [3,16,20,80,97,130–132], which arises as the QCD analogue of the Gell-Mann–Low charge known from Quantum Elec- trodynamics (QED) [133,134], and has associated to it a renormalization-group invariant (RGI) scale of about half of the proton mass [20,80];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' and third, the dynamical breaking of chiral symmetry and the generation of constituent quark masses [10,17,135–159].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The dynamics of the gauge sector of QCD, which encompasses both gluonic and ghost interactions, is instrumental for the physical picture of the EHM outlined above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' In fact, the basic concepts and pivotal mechanisms sustaining the first two pillars of the EHM have their original inception and most genuine realization in the realm of pure Yang-Mills theories [18, 94,95,97,110,113,118,160–162].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Therefore, in the present review, we focus precisely on the rich dynamical content of the gauge sector, especially in relation with the generation of a gluon mass scale out of the intricate gluon self-interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The formulation of the nonperturbative QCD physics in terms of the Green’s functions of the fundamental degrees of freedom, such as gluon and ghost propagators and vertices, provides an intuitive framework for unraveling a wide array of subtle mechanisms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' in fact, certain distinctive features of these functions have been inextricably connected with key phenomena such as gluon mass generation, violation of reflection positivity, and confinement, to name a few.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Thus, the saturation of the gluon propagator in the deep infrared [37,45– 49,52,55–57,59,61,64,66–68,78,82] has been interpreted as the unequivocal signal of a gluon mass [32,97–101,104,106,108–110,113,161–167];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' and the existence of an inflection point in the same function has been argued to lead to a non-positive gluon spectral density [8], and the ensuing loss of reflection positivity [8,11,13,16,168–172] for the dressed gluons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Similarly, the masslessness of the ghost induces [173] a maximum in the gluon propagator, and a zero crossing in the form factors of the three-gluon vertex [28,50,69,70,72,73,82,85,173–181], followed by an infrared divergence for vanishing momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The dynamical origin of these special traits will be the focal point of the analysis presented in the main body of this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The integral equations that govern the full momentum evolution of the Green’s functions, known as SDEs, constitute the indispensable formal and practical instrument for unraveling the special characteristics mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' In their primordial form, the SDEs are rigorously derived from the generating functional of the theory [134,182], and encode all dynamical information on the correlation functions, within the entire range of physical momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' In practice, due to the enormous complexity of these equations, approximations and truncations need to be implemented;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' but, unlike perturbation theory, no expansion parameter is available in the strongly coupled regime of the theory for carrying out such a task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Despite this intrinsic shortcoming, in recent years the SDE predictions have become particularly robust, in part due to various theoretical advances, and in part thanks to the intense synergy with gauge-fixed lattice simulations, as will be evidenced in the subsequent sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Typically, the Green’s function of QCD are defined within the quantization scheme ob- tained by implementing the linear covariant (Rξ) gauges [183].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The corresponding SDEs are derived and solved within this same quantization scheme, and in particular in the Landau gauge (ξ = 0), where lattice simulations are almost exclusively performed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' for studies away Particles 2023, 1 3 from the Landau gauge, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=', [55,61,67,75,76,111,115,121,184–192].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' A great deal may be learned, however, by considering the Green’s functions and corresponding SDEs formulated within the “PT-BFM”scheme [110,193], namely the framework that arises from the fusion of the pinch technique (PT) [14,97,101,194–196] with the background field method (BFM) [197–207].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The main advantage of the PT-BFM originates from the fact that certain appropriately chosen Green’s functions satisfy Abelian Slavnov-Taylor identities (STIs), whose tree-level form does not get modified by quantum corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' This situation is to be contrasted to the standard STIs [208,209] obtained in the conventional framework of the linear covariant gauges, which are deformed by non-trivial contributions stemming from the gauge sector of the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' In the present work, we will carry out computations and develop arguments within both frame- works (Rξ and PT-BFM), and will elaborate on their connection by means of the so-called Background-Quantum identities (BQIs) [14,210–212].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The article is organized as follows: In Section 2 we introduce some basic notation and review certain prominent features of the Green’s functions within both the linear gauges and the PT-BFM formalism [110,193].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' We stress, in particular, the properties of the auxiliary function G(q) [16,132,213,214], which relates the gluon propagators with quantum and background gluons, and is intimately connected with the definition of the process-independent and RGI interaction strength [16], to be discussed in detail in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' In addition, we elucidate with a concrete example the important property of “block-wise” transversality, displayed by the background gluon self-energy [18,110,113].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' In Section 3 we review the general principles associated with the Schwinger mech- anism [128,129] that endows gauge bosons with an effective mass, focusing on the details associated with its realization in the context of Yang-Mills theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' We place particular empha- sis on the pivotal requirement that must be satisfied by the fundamental vertices of the theory, namely the appearance of massless poles in their form factors [18,94,110,112–114,118,160,215].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' In Section 4 we examine the dynamical formation of colored composite excitations (bound states) of vanishing mass, which provide the required structures in the vertices, in order for the Schwinger mechanism to be activated [18,118,160,215].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The formation of these states out of a pair of gluons or a ghost–antighost pair is controlled by a set of coupled Bethe-Salpeter equations (BSEs) [18,118,125,215,216], which are found to have nontrivial solutions for the corresponding Bethe-Salpeter (BS) amplitudes, to be denoted by C(r) and C(r), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' In Section 5 we explain in detail how the presence of the massless poles in the dressed vertices that enter in the SDE of the gluon propagator gives rise to a gluon mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The demon- stration is carried out separately for the gµν and qµqν/q2 components of the gluon self-energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The former case requires the evasion of the so-called “seagull identity” [114,167];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' this becomes possible by virtue of the crucial Ward identity (WI) displacement, to be further considered in Section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' In Section 6 we go over the basic notions underpinning the PT [14,97,101,194,195], and show how their application leads naturally to the definition of a dimensionful process- independent RGI interaction strength [3,16,20,80,97,130–132], denoted by �d(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The genuine process-independence of this quantity is concretely exemplified by demonstrating its appear- ance in two processes involving fundamentally different external fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Next, �d(q) is computed by combining lattice data for the gluon propagator and SDE results for the function G(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Finally, the dimensionless quantity is derived that constitutes the physical definition of the one- gluon exchange interaction appearing in standard bound-state computations [15–17,217–223].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' In Section 7 we focus on the structure of the “transversely projected” three-gluon vertex [127,175,176,224], and discuss briefly the property of planar degeneracy [87], satisfied, at a high level of accuracy [87–89,175,176,224], by the vertex form factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' This special property induces a striking simplification to the structure of this vertex, captured by a particularly compact expression [87], which will be extensively used in some of the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Particles 2023, 1 4 In Section 8 we take a close look at the ghost sector of the theory, and solve the coupled system of SDEs governing the ghost propagator and ghost-gluon vertex [86,225–229];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' as is well-known, the ghost remains massless, but its dressing function saturates at the origin [21, 42,47,49,51,56,63,64,74,80,86,113,179,226,228–234], because the infrared-finite gluon propagator used in the ghost SDE provides an effective infrared cutoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' In the SDE of the ghost-gluon vertex, we employ as central input the compact expression for the three-gluon vertex presented in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The results are in excellent agreement with the available lattice data for the ghost dressing function [74,86] and the form factor of the ghost-gluon vertex evaluated in the soft-gluon limit [42,43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' In Section 9 we discuss two important consequences of the masslessness of the ghost propagator, which manifest themselves at the level of both the gluon propagator and the three-gluon vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Specifically, the diagrams comprised by a ghost loop induce “unprotected” logarithms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=', of the type ln q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' instead, gluonic loops give rise to “protected” logarithms, of the type ln(q2 + m2), where m is the effective gluon mass [173,235].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' As q2 → 0, the unprotected contributions diverge, driving the appearance of a maximum in the gluon propagator and a divergence in its first derivative, as well as a zero-crossing and a corresponding divergence in the form factors of the three-gluon vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' As we comment in this section, of particular phenomenological importance [235–241] is the relative suppression that the above features induce to the dominant vertex form factors in the intermediate range of momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' In Section 10 we discuss an outstanding feature of the WI satisfied by the pole-free part of the three-gluon vertex, namely the displacement induced by the presence of the aforemen- tioned massless poles [94,125].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' In this context, we introduce the key quantity denominated “displacement function”, whose appearance serves as a smoking gun signal of the action of the Schwinger mechanism in QCD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' quite interestingly, it coincides [94,125] with the BS amplitude C(r) for the formation of a massless scalar out of a pair of gluons, introduced in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' In addition, we derive a crucial relation, which ultimately permits the indirect determination of C(r) from lattice QCD [94,125,127];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' an important ingredient in this relation is a partial derivative [125,242], denoted by W(r), of the ghost-gluon kernel [229], to be determined in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' In Section 11 we set up and solve the SDE that governs the evolution of W(r) [125,127, 242,243];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' the main component of this SDE is a special projection of the three-gluon vertex, which is computed by appealing to formulas established in Section 7, and allows for the accurate determination of W(r) in the entire range of relevant momenta [127].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' In Section 12 we substitute into the central relation derived in Section 10 the solution for W(r) found in the previous section, together with the lattice data [85,86] for the gluon propagator, the ghost dressing function, and the form factor of the three-gluon vertex associated with the soft-gluon limit, in order to obtain the form of the displacement function C(r) [125,127].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' As we discuss, the results exclude with nearly absolute certainty the null hypothesis (absence of Schwinger mechanism, C(r) = 0), and corroborate the action of the Schwinger mechanism in QCD [127].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' In addition, we show that the form of C(r) found is statistically completely compatible with that obtained from the BSE-based analysis presented in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' In Section 13 we present our conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Finally, in Appendix A we derive the BQIs relating the displacement functions of the conventional and background vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Basic concepts and general theoretical framework We start by considering the Lagrangian density of an SU(N) Yang-Mills theory, comprised of the classical part, Lcl, the contribution from the ghosts, Lgh, and the covariant gauge-fixing term, Lgf, namely LYM = Lcl + Lgh + Lgf , (1) Particles 2023, 1 5 where Lcl = −1 4 Fa µνFaµν , Lgh = −ca∂µDab µ cb , Lgf = 1 2ξ (∂µAa µ)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (2) In the above formula, Aa µ(x) denotes the gauge field, while ca(x) and ca(x) represent the ghost and antighost fields, respectively, with a = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' , N2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' In addition, Fa µν = ∂µAa ν − ∂νAa µ + g f abcAb µAc ν , (3) is the antisymmetric field tensor, where f abc stands for the totally antisymmetric structure constants of the SU(N) gauge group, and g is the gauge coupling, while Dab µ = ∂µδac + g f ambAm µ , (4) denotes the covariant derivative in the adjoint representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Finally, ξ represents the gauge- fixing parameter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' the choice ξ = 0 corresponds to the Landau gauge, while ξ = 1 specifies the Feynman -´t Hooft gauge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The transition from the pure Yang-Mills theory of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (1) to QCD is implemented by sup- plementing the corresponding kinetic and interaction terms for the quark fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' However, since throughout this work we do not consider effects due to dynamical quarks, the aforementioned terms will be omitted entirely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The most fundamental correlation function is the gluon propagator, whose nonperturba- tive features are inextricably connected with key dynamical properties of the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' In the Landau gauge that we will employ throughout, the gluon propagator, ∆ab µν(q) = −iδab∆µν(q), is completely transverse, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=', ∆µν(q) = ∆(q)Pµν(q) , Pµν(q) := gµν − qµqν/q2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (5) In the continuum, the dynamical properties of the gluon propagator are encoded in the corresponding SDE, given by ∆−1(q)Pµν(q) = q2Pµν(q) + iΠµν(q) , (6) where Πµν(q) is the gluon self-energy, shown diagrammatically in the first row of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The fully-dressed vertices entering the diagrams are determined by their own SDEs, obtaining finally a tower of coupled integral equations, which, for practical purposes, must be truncated or treated approximately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Given that, by virtue of the fundamental STI satisfied by the two-point function, the self-energy Πµν(q) is transverse, qµΠµν(q) = 0 , (7) we have that Πµν(q) = Π(q)Pµν(q) , (8) and from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (6) follows that ∆−1(q) = q2 + iΠ(q) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (9) Of particular importance is the exact way that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (7) is enforced at the level of the SDE given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 6, which governs the gluon evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' In particular, note that, if we were to contract the corresponding diagrams by qµ, the entire set of diagrams must be considered in order for Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (7) to emerge from the SDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' This pattern manifests itself already at the one-loop level, where it is known that the ghost-loop must be included in order to guarantee the transversality of the self-energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The main practical drawback stemming from this observation is that truncations, Particles 2023, 1 6 + + Πµν(q) = (d1) (d2) (d3) + (d4) + (d5) ν q µ q ν q ν q ν q ν q µ q µ q µ q µ q + �Πµν(q) = (a1) ν q µ q + (a3) ν q µ q + (a2) ν q µ q + (a4) µ q ν q (a5) + (a6) ν q ν q µ q µ q �Π(1) µν (q) �Π(2) µν (q) �Π(3) µν (q) Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Upper panel: the diagrammatic representation of the conventional gluon self-energy, Πµν(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Bottom panel: the diagrammatic representation of the Qaµ(q)Bbν(−q), self-energy δab �Πµν(q);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' the grey circles at the end of the gluon lines indicate a background gluon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The corresponding Feynman rules are given in Appendix B of [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' in the form of omission of certain subsets of graphs, are likely to distort this fundamental property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Quite interestingly, within the PT-BFM framework the transversality property of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (7) is enforced in a very special way, which permits physically meaningful truncations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' In what follows we will employ predominantly the language of the BFM;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' for the basic principles of the PT and its connection with the BFM, the reader is referred to the extended literature on the subject [14,97,101,194,195,212,244], as well as to Section 6 of the present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The BFM is a powerful quantization procedure, where the gauge-fixing is implemented without compromising explicit gauge invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Within this framework the gauge field A appearing in the classical action is decomposed as A = B + Q, where B and Q are the background and quantum (fluctuating) fields, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Note that the variable of integration in the generating functional Z(J) is the quantum field, which couples to the external sources, as J · Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The background field does not appear in loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Instead, it couples externally to the Feynman diagrams, connecting them with the asymptotic states to form elements of the S-matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Then, if the gauge-fixing term �Lgf = 1 2ξQ ( �Dab µ Qbµ)2 , �Dab µ = ∂µδab + g f ambBm µ , (10) is used, the resulting gauge-fixed action retains its invariance under gauge transformations of the background field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' As a result of this invariance, when the Green’s functions are contracted by the momentum carried by a background gluon, they satisfy Abelian (ghost-free) STIs, akin to the Takahashi identities known from QED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' In particular, the STIs of the BFM retain their tree-level form to all orders, in contradistinction to the STIs of the Rξ gauges, whose form is modified by contributions stemming from the ghost sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Within the BFM, one may consider three kinds of propagators, by choosing the type of incoming and outgoing gluons [245].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' In particular, we have: (i) The propagator ⟨0| T[Qa µ(q)Qb ν(−q)]|0⟩ that connects two quantum gluons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Notice that this propagator coincides with the conventional gluon propagator of the covariant gauges, defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (5), under the assumption that the corresponding gauge-fixing parameters, ξ and ξQ, are identified, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=', ξ = ξQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (ii) The propagator ⟨0| T[Qa µ(q)Bb ν(−q)]|0⟩ that connects a Qa µ(q) with a Bb ν(−q), to be denoted by �∆ab µν(q) = −iδab�∆µν(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (iii) The propagator ⟨0| T[Ba µ(q)Bb ν(−q)]|0⟩ that connects a Ba µ(q) with a Bb ν(−q), to be denoted by �∆ab µν(q) = −iδab�∆µν(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Note that its full definition requires an additional gauge- fixing term, with the associated “classical” gauge-fixing parameter, ξC [14,203,207].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Particles 2023, 1 7 Given that the relations captured by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (5) and (6) apply also in the cases of �∆µν(q) and �∆µν(q), one may define the corresponding self-energies �Πµν(q) and �Πµν(q), as well as the functions �∆(q) and �∆(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The different types of gluon propagators of the background field method (BFM), together with their diagrammatic representations, symbols, corresponding self-energies, and the background quantum identities (BQIs) that relate them to the conventional propagator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' External legs Diagrammatic representation Symbol Self-energy BQI Qa µ(q)Qb ν(−q) q a b µ ν −iδab∆µν(q) Πµν(q) — Qa µ(q)Bb ν(−q) q a b µ ν −iδab�∆µν(q) �Πµν(q) �∆(q) = ∆(q) 1 + G(q) Ba µ(q)Bb ν(−q) q a b µ ν −iδab�∆µν(q) �Πµν(q) �∆(q) = ∆(q) [1 + G(q)]2 Quite interestingly, the three propagators defined in (i)-(iii) are related by a set of exact identities, known as BQIs [14,210–212].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' In particular,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' we have that (see also Table 1) ∆(q) = [1 + G(q)]�∆(q) = [1 + G(q)]2�∆(q) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (11) where the function G(q) is the gµν component of a particular two-point ghost function,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Λµν(q),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' given by [210,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='212,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='214,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='246] Λµν(q) := ig2CA � k ∆ρ µ(k)D(k + q)Hνρ(−q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' k + q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' −k) = gµνG(q) + qµqν q2 L(q) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (12) where CA is the Casimir eigenvalue of the adjoint representation [N for SU(N)],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Dab(q) = iδabD(q) is the ghost propagator,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' and Hνµ(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' q) denotes the ghost-gluon kernel defined in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' In the Landau gauge, a special identity relates the form factors of Λµν(q) to the ghost dressing function, F(q), defined as F(q) = q2D(q), namely [16,132,214] F−1(q) = 1 + G(q) + L(q) , (13) which is valid before renormalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' In fact, in this particular gauge, G(q) coincides with the so-called Kugo-Ojima function [213,246–248].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Particles 2023, 1 8 = −gf abcHνµ(r, p, q) ν, b k p µ, a q r k + r c Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Diagrammatic definition of the ghost-gluon scattering kernel, Hνµ(r, p, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' At tree level, H0νµ = gνµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' To determine the renormalized form of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (13),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' we introduce the renormalization con- stants of the conventional Green’s functions ∆R(q) = Z−1 A ∆(q) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' FR(q) = Z−1 c F(q) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' IΓR µ(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' q) = Z1IΓµ(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' q) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' IΓR αµν(q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' p) = Z3IΓαµν(q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' p) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' gR = Z−1 g g ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' � gµν + ΛR µν(q) � = ZΛ � gµν + Λµν(q) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Z−1 g = Z−1 1 Z1/2 A Zc = Z−1 3 Z3/2 A ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (14) where we denote by IΓabc µ (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' q) = −g f abcIΓµ(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' q) and IΓabc αµν(q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' p) = g f abcIΓαµν(q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' p) the conventional ghost-gluon [Qa µ(q)cc(p)¯cb(r)] and three-gluon [Qa α(q)Qb µ(r)Qc ν(p)] vertices,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Note that, by virtue of Taylor’s theorem [208], Z1 is finite in the Landau gauge;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' its precise value depends on the renormalization scheme adopted, see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Moreover, denoting by �ZA the (wave-function) renormalization constant of �∆(q), the Abelian STIs of the BFM impose the validity of the pivotal relation [14,203,207] Zg = �Z−1/2 A , (15) which is the non-Abelian analogue of the textbook relation Ze = Z−1/2 A [134], relating the renormalization constants of the electric charge and the photon propagator in QED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Then, since the BQIs of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (11) are direct consequences of the Becchi-Rouet-Stora-Tyutin (BRST) symmetry [249–251] of the theory [210,212,214,246], their form is preserved by renor- malization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Hence, combining Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (11), (15) and (14) we obtain ZΛ = Z−1 1 Zc , (16) which yields1 Z−1 1 F−1(q) = 1 + G(q) + L(q) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (17) 1 In the original and widely used [3,8,16,20,80,132] version of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (17) the renormalization is performed in the so-called Taylor scheme, where Z1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Particles 2023, 1 9 As has been shown in [132], the dynamical equation governing L(q) yields L(0) = 0, provided that the gluon propagator entering in it is finite at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Thus, one obtains from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (17) the useful identity [213] Z−1 1 F−1(0) = 1 + G(0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (18) According to numerous lattice simulations and studies in the continuum (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=', [21,42, 47,49,51,56,63,64,74,80,86,113,179,226,228–234]), the ghost dressing function reaches a finite (nonvanishing) value at the origin, which, due to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (18), furnishes also the value of G(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The final upshot of the above considerations is that one may use the BQIs in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (11) to express the SDE given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (6) in terms of the �Πµν(q) or �Πµν(q), at the modest cost of introducing in the dynamics the quantities 1 + G(q) or [1 + G(q)]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Focusing on the former possibility, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (11) becomes ∆−1(q)Pµν(q) = q2Pµν(q) + i �Πµν(q) 1 + G(q) , (19) where the diagrammatic representation of the self-energy �Πµν(q) is shown in the lower panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The principal advantage of this formulation is that the self-energy �Πµν(q) contains fully- dressed vertices with a background gluon of momentum q exiting from them, which satisfy Abelian STIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' In particular, denoting by �IΓµαβ(q, r, p), �IΓµ(r, p, q), and �IΓ mnrs µαβγ(q, r, p, t) the BQQ, Bcc, and BQQQ vertices, respectively, we have that [14,101,110] qµ �IΓµαβ(q, r, p) = ∆−1 αβ (r) − ∆−1 αβ (p) , (20) qµ �IΓµ(r, p, q) = D−1(p) − D−1(r) , (21) qµ �IΓ mnrs µαβγ(q, r, p, t) = f mse f ernIΓαβγ(r, p, q + t) + f mne f esrIΓβγα(p, t, q + r) + f mre f ensIΓγαβ(t, r, q + p) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (22) In contrast, the conventional three-gluon and ghost-gluon vertices, IΓαµν(q, r, p) and IΓα(r, p, q), respectively, satisfy the STIs [1,252–256] qαIΓαµν(q, r, p) = F(q) � ∆−1(p)Pσ ν (p)Hσµ(p, q, r) − ∆−1(r)Pσ µ (r)Hσν(r, q, p) � , (23) qµF−1(q)IΓµ(r, p, q) + pµF−1(p)IΓµ(r, q, p) = −r2F−1(r)U(r, q, p) , (24) where U(r, q, p) is an interaction kernel containing only ghost fields;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' its tree level value is U0(r, q, p) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The STI for the conventional four-gluon vertex is given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='24) of [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The special STIs listed in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (20), (21) and (22) are responsible for the remarkable property of “block-wise” transversality [110,193,245], displayed by �Πµν(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' To appreciate this point, notice that the diagrams comprising �Πµν(q) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 1 have been separated into three different subsets (blocks) comprised of: (i) one-loop dressed diagrams containing only gluons, (ii) one- loop dressed diagrams containing a ghost loop, and (iii) two-loop dressed diagrams containing only gluons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The corresponding contributions of each block to �Πµν(q) are denoted by �Π(i) µν(q), with i = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The block-wise transversality is a stronger version of the standard transversality relation qµ �Πµν(q) = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' it states that each block of diagrams mentioned above is individually transverse, namely qµ �Π(i) µν(q) = 0 , i = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (25) Particles 2023, 1 10 In order to appreciate in detail the reason why the STIs in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (20), (21) and (22) are instrumental for the block-wise transversality, we will consider the case of �Π(2) µν (q);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' the relevant diagrams are enclosed by the blue box of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The diagrams (a3) and (a4) are given by (a3)µν(q) = g2CA � k(k + q)µD(k + q)D(k)�IΓν(−k, k + q, −q) , (26) (a4)µν(q) = g2CA gµν � k D(k) , (27) where a color factor δab has been suppressed in both expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' In addition, for the formal manipulations of integrals, we employ dimensional regularization [257];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' to that end, we introduce the short-hand notation � k := µϵ 0 (2π)d � +∞ −∞ ddk , (28) where d = 4 − ϵ is the dimension of the space-time, and µ0 denotes the ’t Hooft mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The contraction of graph (a3)µν(q) by qν triggers the STI satisfied by �Γν(−k, k + q, −q) [given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (21)], and we obtain qν(a3)µν(q) = g2CA � k(k + q)µD(k + q)D(k) � D−1(k) − D−1(k + q) � = g2CA � k(k + q)µ[D(k + q) − D(k)] = −g2CA qµ � k D(k) , (29) which is precisely the negative of the contraction qν(a4)µν(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Hence, qν�(a3)µν(q) + (a4)µν(q) � = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (30) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Schwinger mechanism in Yang-Mills theories The BRST symmetry of the Yang-Mills Lagrangian given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (1) prohibits the inclusion of a mass term of the form m2A2 µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Moreover, a symmetry-preserving regularization scheme, such as dimensional regularization, prevents the generation of a mass term at any finite order in perturbation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Nonetheless, as affirmed four decades ago [95–100], the nonperturbative Yang-Mills dynamics endow the gluons with an effective mass, which sets the scale for all dimensionful quantities, and tames the instabilities originating from the infrared divergences of the perturbative expansion (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=', Landau pole).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' In addition, the presence of this mass causes the effective decoupling (screening) of the gluonic modes beyond a “maximum gluon wavelength” [258], and leads to the dynamical suppression of the Gribov copies, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=', [16,259,260] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The generation of a gluon mass proceeds through the nonperturbative realization of the Schwinger mechanism [128,129].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Even though the technical details associated with the implementation of this mechanism in a four-dimensional non-Abelian setting are particularly elaborate, the general underlying idea is relatively easy to convey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' To that end, consider the dimensionless vacuum polarization Π(q), defined through Π(q) = q2Π(q), such that ∆−1(q) = q2[1 + iΠ(q)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (31) The Schwinger mechanism is based on the fundamental observation that, if Π(q) develops a pole at q2 = 0 (to be referred to as “massless pole”) then the vector meson (gluon) picks up Particles 2023, 1 11 a mass, regardless of any “prohibition” imposed by the gauge symmetry at the level of the original Lagrangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Thus, in Euclidean space, the above sequence of ideas leads to lim q→0 Π(q) = m2/q2 =⇒ lim q→0 ∆−1(q) = lim q→0 (q2 + m2) =⇒ ∆−1(0) = m2 , (32) and the gauge boson propagator saturates to a non-zero value at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' This effect of infrared saturation of the propagator signifies the generation of a mass, which is identified with the positive residue of the pole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' At this descriptive level, Schwinger’s argument is completely general, making no par- ticular reference to the specific dynamics that would lead to the appearance of the required massless pole inside Π(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' In fact, depending on the particular theory, the field-theoretic circumstances that trigger the crucial sequence captured by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (32) may be very distinct, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=', [261,262].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' In the case of Yang-Mills theories, the origin of the massless poles is purely nonperturbative [160]: the strong dynamics produce scalar composite excitations, which carry color and have vanishing masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' These poles are carried by the fully-dressed vertices of the theory;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' and since these vertices enter in the gluon SDE shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 1 [upper (lower) panel for the QQ (QB) propagator], the massless poles find their way into the gluon self-energy (or, equivalently, the gluon vacuum polarization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The detailed implementation of this idea has been presented in a series of works [18,94,97,113,117,118,118,119,160–162,167,190,263], and will be summarized in the rest of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Let us focus for now on the conventional three-gluon and ghost-gluon vertices, IΓαµν(q, r, p) and IΓα(r, p, q), respectively, introduced below Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' When the formation of massless poles is triggered, these vertices assume the general form (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 3) IΓαµν(q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' p) = Γαµν(q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' p) + Vαµν(q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' p) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' IΓα(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' q) = Γα(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' q) + Vα(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' q) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (33) where Γαµν(q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' p) and Γα(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' q) are their pole-free components,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' while Vαµν(q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' p) and Vα(q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' p) contain longitudinally coupled poles,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' whose special tensorial structure is given by Vαµν(q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' p) = qα q2 Cµν(q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' p) + rµ r2 Aαν(q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' p) + pν p2 Bαµ(q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' p) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Vα(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' q) = qα q2 C(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' q) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (34) such that Pα α′(q)Pµ µ′(r)Pν ν′(p)Vαµν(q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' p) = 0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Pα α′(q)Vα(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' q) = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (35) We emphasize that the reason why Vαµν(q, r, p) and Vα(q, r, p) are longitudinally coupled may be directly inferred from their special decomposition, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' In particular, let us denote by Iα(q) the transition amplitude that connects a gluon with a massless composite scalar, depicted as a gray circle in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Since Iα(q) depends solely on the momentum q, and carries a single Lorentz index, α, its general form is given by Iα(q) = qαI(q), where I(q) is a scalar form factor [118,215].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' This observation accounts directly for the form of Vα(q, r, p) given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (34);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' to deduce the form of Vαµν(q, r, p), one must, in addition, appeal to Bose symmetry, which imposes the structures rµ/r2 and pν/p2 in the remaining two channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Returning to the SDE of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (1), the component Vαµν(q, r, p) will enter in it through graphs (d1) and (d4), while the component Vα(q, r, p) through graph (d3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Since Vαµν(q, r, p) has poles for each one of its three momenta, let us point out that only the pole associated with the q-channel, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=', the channel that carries the momentum entering in the gluon propagator, is relevant for the Schwinger mechanism that will generate mass for ∆(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' In fact, in the Landau Particles 2023, 1 12 = + q q a, α q a, α a, α i/q2 Vαµν IΓαµν Γαµν � �� � Iα(q) µ, b ν, c r p µ, b ν, c r p µ, b ν, c r p = q a, α + i/q2 qq q a, α a, α Vα IΓα Γα � �� � Iα(q) b c r p b c r p b c r p Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The diagrammatic representation of the three-gluon and ghost-gluon vertices introduced in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (33): IΓαµν(q, r, p) (first row) and IΓα(r, p, q) (second row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The first term on the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' indicates the pole-free part, Γαµν(q, r, p) or Γα(r, p, q), while the second denotes the pole term Vαµν(q, r, p) or Vα(r, p, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' gauge that we employ, the gluon propagators inside the diagrams (d1) and (d4) are transverse, leading to a considerable reduction in the number of the form factors of Vαµν(q, r, p) that participate actively, since Pµ µ′(r)Pν ν′(p)Vαµν(q, r, p) = qα q2 Pµ µ′(r)Pν ν′(p)Cµν(q, r, p) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (36) Consequently, for the ensuing analysis, one requires only the tensorial decomposition of the component Cµν(q, r, p) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (34), which is given by Cµν(q, r, p) = C1 gµν + C2 rµrν + C3 pµpν + C4 rµpν + C5 pµrν , (37) where Cj := Cj(q, r, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Then, the substitution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (37) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (36), and use of the relation q + p + r = 0, reveals that only two form factors survive inside (d1) and (d4), namely Pµ µ′(r)Pν ν′(p)Vαµν(q, r, p) = qα q2 Pµ µ′(r)Pν ν′(p) � C1 gµν + C5qµqν � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (38) Since the main function of the Schwinger mechanism is to make the gluon propagator saturate at the origin, it is important to explore the properties of the structures appearing in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (38) near q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' To that end, we expand the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (38), keeping terms at most linear in q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' After noticing that the term proportional to C5 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (38) is of order O(q2), we end up with a single relevant form factor associated with Vαµν(q, r, p), namely C1(q, r, p), which survives the q → 0 limit of graphs (d1) and (d4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' As for Vα(r, p, q), its unique component, C(q, r, p), enters directly in (d3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The continuation of this analysis entails the Taylor expansion of C1(q, r, p) and C(r, p, q) around q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' In carrying out this expansion, one employs the following two key relations, C1(0, r, −r) = 0 , C(r, −r, 0) = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (39) The first one follows directly from the Bose symmetry of the three-gluon vertex, which implies that C1(q, r, p) = −C1(q, p, r);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' as we will see in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 10, it may also be derived in a completely independent way from the fundamental STIs satisfied by the three-gluon vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The justi- fication of the second relation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (39) is less straightforward;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' its derivation, presented in Appendix A, relies on the BQI [14,212] linking the conventional ghost-gluon vertex, IΓα(r, p, q), with its background counterpart, �IΓα(r, p, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Particles 2023, 1 13 = + ν, n r p µ, m q α, a ν, n r p µ, m q α, a α, a + r µ, m p ν, n r µ, m p ν, n + · · · k k + q k k + q K11 K12 q α, a q (a) (b) = + n r p m q α, a n r p m q α, a α, a + r m p n r m p n + · · · k k + q k k + q q α, a q (c) (d) K21 K22 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The coupled system of Schwinger-Dyson equations (SDEs) for the three-gluon and ghost-gluon vertices, IΓαµν(q, r, p) and IΓα(r, p, q), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The orange ellipses represent four-point scattering kernels, denoted by Kij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' We omit diagrams containing five-point scattering kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Thus, after taking Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (39) into account, the Taylor expansion of C1(q, r, p) and C(r, p, q) around q = 0 yields lim q→0 C1(q, r, p) = 2(q · r)C(r) + · · · , lim q→0 C(r, p, q) = 2(q · r)C(r) + · · · , (40) with C(r) := �∂C1(q, r, p) ∂p2 � q=0 , C(r) := �∂C(r, p, q) ∂p2 � q=0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (41) The functions C(r) and C(r) are of central importance for the rest of this review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' In particular, there are three key points related to them that will be elucidated in detail in what follows: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' C(r) and C(r) are the BS amplitudes describing the formation of gluon-gluon and ghost- antighost colored composite bound states, respectively, see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The gluon mass is determined by certain integrals that involve C(r) and C(r), given explicitly in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' C(r) and C(r) lead to smoking-gun displacements of the WIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' In fact, the displacement induced by C(r), has been confirmed by lattice QCD, by combining judiciously the results of several lattice simulations, see subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' We end this section by emphasizing that the BFM vertices develop poles in exactly the same way as their conventional counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' In particular, the main relations Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (33), (34), (39) and (41) remain valid, with the only modification that all quantities carry hats or tildes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' these BFM vertices will be used extensively in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Note that the conventional and background vertices, including their pole content, are related through appropriate BQIs, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=', Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (A3) and (A6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Dynamical formation of massless poles One crucial aspect of the implementation of the Schwinger mechanism in a Yang-Mills context is that the poles that comprise the components Vαµν(q, r, p) and Vα(q, r, p) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (34) are not introduced by hand;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' rather, they are generated dynamically, as massless composite excitations that carry color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' In fact, this subtle process is controlled by a system of coupled linear BSEs for the functions C(r) and C(r), which play the role of the BS amplitudes for generating composite massless scalars out of two gluons and a ghost-antighost pair, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Particles 2023, 1 14 The starting point for the derivations of the aforementioned BSEs are the SDEs for IΓαµν(q, r, p) and IΓα(r, p, q), shown diagrammatically in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 4, and given by [125] IΓαµν = Γαµν 0 − λ � k IΓαβγ∆βρ∆γσKµνσρ 11 + 2λ � k IΓαDDKµν 12 , IΓα = Γα 0 − λ � k IΓαβγ∆βρ∆γσKσρ 21 − λ � k IΓαDDK22 , (42) where λ := ig2CA/2 , (43) and the tree-level expressions for the vertices IΓαµν and IΓα are given by Γαµν 0 (q, r, p) = (q − r)νgαµ + (r − p)αgµν + (p − q)µgνα , Γα 0 (r, p, q) = rα .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (44) Note that, for compactness, all momentum arguments have been suppressed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' they may be easily restored by appealing to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The following steps are subsequently implemented: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Substitute into both sides of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (42) the expressions for the fully-dressed vertices given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' In order to exploit Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (38), multiply the first equation by the factor Pµ′µ(r)Pµ′ ν (p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Take the limit of the system as q → 0: this activates Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (40) and introduces the functions C(r) and C(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Isolate the tensorial structures proportional to qα, and match the terms on both sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Employ the “one-particle exchange” approximation for the kernels Kij, to be denoted by K0 ij, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Thus, we arrive at a system of homogeneous equations involving C(r) and C(r), C(r) = −λ 3 � k C(k)∆2(k)Pρσ(k)Pµν(r) �Kµνσρ 11 + 2λ 3 � k C(k)D2(k)Pµν(r) �Kµν 12 , C(r) = −λ � k C(k)∆2(k)Pσρ(k) �Kσρ 21 − λ � k C(k)D2(k) �K22 , (45) where �Kij := (r · k/r2) K0 ij(r, −r, k, −k);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' the system is diagrammatically depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Before turning to the numerical analysis, the BSE system must be passed to the Euclidean space, following standard conversion rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' In doing so we note that the integral measure is modified according to d4k → id4kE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' this extra factor of i combines with the λ defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (43) to give real expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' As announced, the system of coupled equations given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (45) represents the BSEs that govern the formation of massless colored bound states out of two gluons and a ghost-antighost pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The functions C(r) and C(r) are the corresponding BS amplitudes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' finding nontrivial solutions for them, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=', something other than C(r) = C(r) = 0 identically, is crucial for the implementation of the Schwinger mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The equations in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (45) are linear and homogeneous in the unknown functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' There are two main consequences arising from this fact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' First, the numerical solution of the system will be reduced to an eigenvalue problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Second, the overall scale of the solutions is undetermined, since the multiplication of a given solution by an arbitrary real constant produces another solution 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 2 The ambiguity originates from considering only leading terms in the expansion around q = 0, and may be resolved if further orders in q are kept, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=', [220,264,265].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Particles 2023, 1 15 = r r k k K0 11 r k k r r − k = r r k k K0 12 k r k r r − k = k r r k K0 21 k r k r r − k = k r k r r − k k r r k K0 22 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The one-particle exchange approximations, K0 ij, of the kernels Kij appearing in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' It turns out that the condition for obtaining nontrivial solutions, when expressed in terms of the strong coupling, αs := g2/4π, states that they exist for αs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='63, when the renormalization point µ = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='3 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The solutions obtained when αs acquires this special value are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' they have undergone scale fixing3, and are denoted by C⋆(r) and C⋆(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Observe that C⋆(r) is significantly larger in magnitude than C⋆(r), implying that the three-gluon vertex accounts for the bulk of the gluon mass, as originally claimed in [216].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' It is important to compare the value of αs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='63, imposed by the BSE eigenvalue, with the expected value for αs for the renormalization scheme employed: within the asymmetric momentum subtraction (MOM) scheme (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 8), we have that αs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='27 [72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' This numerical discrepancy in the values of αs is clearly an artifact of the truncation employed, and concretely of the approximation of the kernels Kij by their one-particle exchange diagrams, K0 ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' A preliminary analysis reveals that mild modifications of the kernels Kij lead to considerable variations in the value of αs, but leave the form of the solutions for C⋆(r) and C⋆(r) practically unaltered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' This observation suggests that, while a more complete knowledge of the BSE kernels is required in order to bring αs closer to its MOM value, the solutions obtained with the present approximations should be considered as particularly stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Generation of the gluon mass We next demonstrate in detail how the presence of the massless poles in the vertices that enter in the SDE of the gluon propagator generate a gluon mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' We start by pointing out that, since the fundamental STIs of the theory remain intact under the action of the Schwinger mechanism, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (7) and (8) remain valid, and the mass term m2 = ∆−1(0) will appear in the transverse combination ∆−1(0)Pµν(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' However, the determination of the mass proportional to gµν exposes an entirely different array of principles compared to the corresponding computation for the qµqν/q2 component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The calculation with respect to the qµqν/q2 component is rather direct;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' since the massless poles in the vertices are themselves longitudinally coupled, their contribution to the qµqν/q2 component of Πµν(q) is easily worked out, as will be illustrated in Subsec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' In contrast, the emergence of a mass proportional to gµν is intimately connected with a powerful relation, 3 The scale was fixed by requiring the best possible matching with the result obtained for C(r) from the WI displacement, see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Particles 2023, 1 16 k r − 2 q = 0 α, a µ, m ν, n k r r − k r r − k q = 0 α, a r µ, m ν, n k k C ν, n = r r µ, m q = 0 α, a C C q = 0 n m α, a C r r k r + q = 0 α, a m n k r r − k = r − k m n k r r k α, a q = 0 C C Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The diagrammatic representation of the coupled system of Bethe-Salpeter equations (BSEs) that governs the evolution of the functions C(r2) and C(r2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='1 Displacement functions C⋆(r) C⋆(r) Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The solutions for C⋆(r) (purple dot-dashed) and C⋆(r) (red dashed) obtained from the coupled BSE system of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' known as seagull identity [114,167], which in the absence of the Schwinger mechanism would enforce the masslessness of the propagator, as will be discussed in Subsec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' In fact, one main conceptual difference between the two approaches is that in the gµν case, the use of the PT-BFM-based version of the SDE given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (19) is crucial for the emergence of the correct result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' In order to simplify the technical aspects of the calculation without compromising its conceptual content, we will determine the contribution to the gluon mass due the pole in the ghost-gluon vertex, namely Vα(r, p, q) in the case of IΓα(r, p, q), and �Vα(r, p, q) in the case of �IΓα(r, p, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' To that end, we will focus on the subset of self-energy graphs containing only ghost loops, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=', graph (d3) in the case of Πµν(q), and graphs (a3) and (a4) in the case of �Πµν(q), shown in the upper and lower row of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Particles 2023, 1 17 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Gluon mass from the qµqν component Let us calculate the contribution to the gluon mass stemming from the ghost loop, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=', the diagram (d3) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 1, which, for general values of q, reads (d3)µν(q) = g2CA � k(k + q)µD(k + q)D(k)IΓν(−k, k + q, −q) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (46) To isolate the qµqν/q2 component of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (46) at the origin, we first decompose the full vertex IΓν(−k, k + q, −q) as in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (33) and (34), and drop directly the pole-free part, since it does not contribute at q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Then, denoting by (dV 3 )µν(q) the contribution of Vν(−k, k + q, −q) to (d3)µν(q), we obtain (dV 3 )µν(q) = −g2CA qν q2 � k(k + q)µD(k + q)D(k)C(−k, k + q, −q) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (47) Next, a Taylor expansion around q = 0, using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (39) and (40), yields (dV 3 )µν(q) = −2g2CA qνqρ q2 � k kµkρD2(k)C(k) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (48) Evidently, the integral above can only be proportional to gµρ, such that (dV 3 )µν(q) = −2g2CA d �qµqν q2 � � k k2D2(k)C(k) , (49) where the tensor structure qµqν/q2 is already isolated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Then, let us denote by ∆−1 gh (0) the contribution to the mass originating in the qµqν/q2 of the ghost loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Noting that the contribution of (dV 3 )µν(q) to the propagator is i times the negative of its qµqν/q2 form factor, we obtain that ∆−1 gh (0) = 4λ d � k k2D2(k)C(k) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (50) At this point, we set d = 4 and renormalize Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' This leads to the appearance of the finite renormalization constant of the ghost-gluon vertex, Z1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Next, we express the result in terms of the ghost dressing function F, pass to Euclidean space, and employ hyperspherical coordinates, to obtain the final expression ∆−1 gh (0) = ˆλ Z1 � ∞ 0 dy F2(y) C(y) , (51) where ˆλ := CAαs/8π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The derivation of the contributions from the diagrams (d1) and (d4) proceeds in a com- pletely analogous way, but is algebraically more involved, see [167] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' It is instructive to consider how the result of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (51) emerges in the context of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' To this end, we consider the ghost block �Π(2) µν (q) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 1, whose diagrams have the expressions given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (27);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' clearly, only diagram (a3)µν(q) can contribute to the qµqν component of �Π(2) µν (q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Then, we decompose �IΓα(r, p, q) in complete analogy with Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (33) and (34), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=', �IΓα(r, p, q) = �Γα(r, p, q) + qα q2 �C(r, p, q) , (52) Particles 2023, 1 18 and expand the (a3)µν(q) of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (27) around q = 0, isolating its qµqν/q2 component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' These steps eventually lead to �∆−1 gh (0) = 4λ d � k k2D2(k) �C(k) , (53) where �C(q) is defined in the exact same way as C(q), namely through Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (41) but with tildes over all relevant quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' It is now easy to establish that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (53) is completely equivalent to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (50), simply by multiplying both of its sides by Z1F(0), and then using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (A4) on the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' and Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (19) and (18) on the l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Hence, when the mass is computed through the qµqν/q2 component of the self-energy, the contributions originating from the ghost diagrams of either the BQ or the QQ propagator furnish the same result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The same is not true for the calculation through the gµν component, since the ghost diagram (d3)µν of the QQ propagator is not by itself transverse, and a meaningful analysis is preferably carried out within the BFM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Gluon mass from the gµν component: seagull identity and Ward identity displacement The fact that the activation of the Schwinger mechanism is crucial for the self-consistent generation of a gluon mass may be best appreciated in conjunction with the so-called seagull identity [114,167].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The content of this identity is that � k k2 ∂ f (k) ∂k2 + d 2 � k f (k) = 0 , (54) for functions f (k) that satisfy Wilson’s criterion [266];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' the cases of physical interest are f (k) = ∆(k), D(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The general demonstration of the validity of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (54) has been given in [167];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' for a detailed discussion of how Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (54) prevents the photon from acquiring a mass in scalar electrodynamics, see [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' What is so special about Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (54) is that, within the PT-BFM formalism, the l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (54) coincides with the contributions of loop diagrams to the gµν component of the gluon mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Therefore, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (54) enforces the nonperturbative masslessness of the gluon in the absence of the Schwinger mechanism: even if a massive gluon propagator (made “massive” through a procedure other than the Schwinger mechanism) were to be substituted inside Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (54), one would obtain zero as contribution to the gluon mass!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' For example, the simple choice f = (k2 − m2)−1, reduces the l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='s of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (54) to (dimensionally regularized) text-book integrals, which add up to give precisely zero [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' In order to appreciate in some detail how the seagull identity prevents the gµν component of the propagator from acquiring a mass in the absence of the Schwinger mechanism, let us consider once again the ghost block �Π(2) µν (q) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' now both graphs, (a3) and (a4), contribute to the gµν component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Let us assume that the Schwinger mechanism is turned off;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' at the level of the Bcc vertex this means that �Vα(r, p, q) vanishes identically, and �IΓα(r, p, q) = �Γα(r, p, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Consequently, �Γα(r, p, q) saturates the STI of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (21), qα�Γα(r, p, q) = D−1(p) − D−1(r) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (55) Since the form-factors of the vertex �Γα(r, p, q) do not contain any poles, the derivation from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (55) of the corresponding WI proceeds in the standard text-book way: both sides of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (55) undergo a Taylor expansion around q = 0, and terms at most linear in q are retained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Thus, one arrives at the simple QED-like WI �Γα(r, −r, 0) = ∂D−1(r) ∂rα =⇒ D2(r)�Γα(r, −r, 0) = −2rα ∂D(r) ∂r2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (56) Particles 2023, 1 19 We now compute the gµν component of �Π(2) µν (q) at q = 0, or, equivalently, �∆−1 gh (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (27), we see that (a4)µν is proportional to gµν in its entirety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' On the other hand, (a3)µν(q) contains both gµν and qµqν components;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' however, the latter vanishes in the limit q → 0 if the vertex is pole-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Then, it is straightforward to show that, as q → 0, �∆−1 gh (0) = 2λ d �� k kµD2(k)�Γµ(−k, k, 0) + d � k D(k) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (57) At this point, employing the WI of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (56) (with r → −k), we get �∆−1 gh (0) = 4λ d �� k k2 ∂D−1(k) ∂k2 + d 2 � k D(k) � � �� � seagull identity = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (58) Hence, the WI satisfied by the vertex in the absence of the Schwinger mechanism triggers the seagull identity, which, in turn, enforces the masslessness of the propagator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' When the Schwinger mechanism gets activated, the STIs satisfied by the vertices of the theory retain their original form, but are resolved through the nontrivial participation of the terms containing the massless poles [97,113,160–162,167,263,267].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' In particular, the full vertex �IΓα(r, p, q) satisfies precisely Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (21), namely qα �IΓα(r, p, q) = qα�Γα(r, p, q) + �C(r, p, q) = D−1(p) − D−1(r) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (59) Notice in particular that the contraction of �IΓα(r, p, q) by qα cancels the massless pole in q2, leading to a completely pole-free result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Therefore, the WI obeyed by �Γα(r, p, q) may be derived as before, through a standard Taylor expansion, leading to qα�Γα(r, −r, 0) = − �C(r, −r, 0) + qα � � � ∂D−1(r) ∂rα − � ∂ �C(r, p, q) ∂qα � q=0 � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (60) Evidently, the unique zeroth-order contribution appearing in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (60), namely �C(r, −r, 0), must vanish, �C(r, −r, 0) = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (61) Note that this particular property may be independently derived from the antisymmetry of �C(r, p, q) under r ↔ p, �C(r, p, q) = − �C(p, r, q), which is a consequence imposed by the ghost- antighost symmetry of the B(q)¯c(r)c(p) vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The above result, together with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (A3), is used to prove Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (39) in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Thus, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (60) becomes qα�Γα(r, −r, 0) = qα �∂D−1(r) ∂rα − 2rα �C(r) � , �C(r) := � ∂ �C(r, p, q) ∂p2 � q=0 , (62) and the matching of the terms linear in q yields the WI �Γα(r, −r, 0) = ∂D−1(r) ∂rα − 2rα �C(r) � �� � WI displacement .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (63) Particles 2023, 1 20 Comparing Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (56) and (63), it becomes clear that the Schwinger mechanism induces a char- acteristic displacement to the WIs that are satisfied by the pole-free parts of the vertices [167].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Returning to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (57), but now substituting in it the displaced version of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (56), namely D2(k)�Γµ(−k, k, 0) = 2kµ �∂D(k) ∂k2 + D2(k) �C(k) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (64) When Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (64) is substituted into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (57), the first term of its r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' triggers the seagull identity and vanishes, exactly as before;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' however, the second term survives, furnishing precisely the result given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (53).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Completely analogous procedures may be applied to the remaining two blocks, �Π(1) µν (q) and �Π(3) µν (q), by exploiting the Abelian STIs of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (20) and (22), respectively [162].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Renormalization group invariant interaction strength The PT-BFM formalism provides the natural framework for the construction of the RGI version of the naive one-gluon exchange interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' To fix the ideas, recall that in QED, the one-photon exchange interaction, defined as α∆A(q), where α := e2/4π is the hyper-fine structure constant and ∆A(q) the photon propagator, is an RGI combination, by virtue of the relation Ze = Z−1/2 A ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' see comments following Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Moreover, this particular combination is universal (process-independent) because it may be identified within any two-to-two scattering process, regardless of the nature of the initial and final states (electrons, muons, taus, etc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Instead, in QCD, the corresponding combination αs∆(q) is (trivially) universal but not RGI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' When the vertices that connect the gluon to the external particles are “dressed” (Γ0 → Γ), the combination Γ αs∆ Γ becomes RGI;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' however, it is no longer process-independent, because the vertices Γ contain information on the characteristics of the external particles, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=', the Γ is not the same if the external particles are quarks or gluons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' This apparent conundrum may be resolved by resorting to the PT, which reconciles harmoniously the notions of RGI and process-independence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Within the PT framework, the starting point of the construction are “on-shell” pro- cesses [14,97,101,194,195], such as those depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The fundamental observation is that the dressed vertices appearing there contain propagator-like contributions, which may be unambiguously identified by means of a well-defined diagrammatic procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' After dis- carding terms that vanish on shell, the contributions extracted from a vertex have a two-fold effect: (i) the genuine vertex contributions left behind form a new vertex, �Γ, which satisfies Abelian STIs, and (ii) when the propagator-like pieces from both vertices are allotted to the conventional propagator, ∆µν(q), the resulting effective propagator, �∆µν(q), captures all RG logarithms associated with the running of the coupling;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' for example, at one loop and for large q2, one has �∆−1(q) ≈ q2� 1 + bg2 ln(q2/µ2) � , (65) where b = 11CA/48π2 is the first coefficient of the Yang-Mills β function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' We emphasize that the PT construction goes through to all orders in perturbation theory, as well as nonperturbatively, and all key properties of the PT Green’s function persist unaltered [195,196].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The correspondence between the PT and the BFM may be summarized by stating that the PT rearrangement outlined above amounts effectively to replacing the Q-type gluon that is being exchanged (carrying momentum q) by a B-type gluon [194,268–270];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' external (on-shell) fields are always of the Q-type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Thus, the notation used above for the PT effective Green’s functions (“tildes” and “hats”) corresponds precisely to the BFM notation introduced in Sec- tion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Note that the formal expression of all PT rearrangements implemented diagrammatically are the BQIs that relate conventional Green’s functions to their BFM counterparts [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' For Particles 2023, 1 21 example, in the case of the quark-gluon vertex, we have that the vertices Γµ(q, k1, −k2) [with external fields Qa µ(q)qb(k1) ¯qc(−k2)] and �Γµ(q, k1, −k2) [Ba µ(q)qb(k1) ¯qc(−k2)] are related by the BQI [271] �Γµ(q, k1, −k2) = [1 + G(q)]Γµ(q, k1, −k2) + · · · , (66) where the ellipsis denotes terms that vanish on shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Similarly, the BQI of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (A5), when evaluated on-shell, yields a completely analogous result, to wit, �IΓµαρ(q, k1, −k2) = [1 + G(q)]IΓµαρ(q, k1, −k2) + · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (67) It is now clear how the PT gives rise to a process-independent propagator-like component: regardless of the process (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=', the type of vertex connecting the internal gluon to the external states), each vertex contributes to the conventional ∆(q) a factor of [1 + G(q)]−1, finally leading to the BQI of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (11) [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The culmination of the above sequence of ideas is reached by noting that, by virtue of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (15), the combination �d(q) := αs�∆(q) = αs∆(q) [1 + G(q)]2 , (68) is RGI: it retains exactly the same form before and after renormalization, and, consequently, does not depend on the renormalization point µ [97].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The quantity �d(q) has mass dimension of −2, and is known in the literature as the “RGI running interaction strength” [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' PT ==⇒ ∆ g2 �∆ gΓν �Γν �Γµ gΓµ µ ν µ ν q k1 k2 k3 k4 q k1 k2 k3 k4 PT ==⇒ ∆ g2 �∆ �Γβνσ �Γαµρ gΓβνσ gΓαµρ µ ν q k1 k2 k3 k4 q k1 k2 k3 k4 β σ ρ α β σ ρ α µ ν Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Diagrammatic representation of the basic PT rearrangement in the case of quark-antiquark scattering, corresponding to the S-matrix element Tq ¯q→q ¯q of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (69) (left), and gluon-gluon scattering, corresponding to Tgg→gg of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (70) (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The steps leading to the natural appearance of �d(q) within any given process may be summarized in the case of quark-antiquark, or gluon-gluon scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Consider the S-matrix elements Tq ¯q→q ¯q, for the scattering of a quark and an antiquark, and Tgg→gg, for the scattering of two gluons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The quark-antiquark scattering is depicted in the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Using the BQI of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (11) we obtain Tq ¯q→q ¯q = � gΓµ(q, k1, −k2) � ∆(q)Pµν(q)[gΓν(−q, k3, −k4)] PT = � g[1 + G(q)]−1�Γµ(q, k1, −k2) � ∆(q)Pµν(q) � g[1 + G(q)]−1�Γν(−q, k3, −k4) � PT = �Γµ(q, k1, −k2) � g2[1 + G(q)]−2∆(q) � Pµν(q)�Γν(−q, k3, −k4) PT = �Γµ(q, k1, −k2) � g2�∆(q) � � �� � 4π �d(q) Pµν(q)�Γν(−q, k3, −k4) , (69) Particles 2023, 1 22 where we omit color structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Similarly, the scattering of two gluons, depicted in the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 8, yields Tgg→gg = � gΓαµρ(k1, q, −k2) � ∆(q)Pµν(q) � gΓβνσ(k3, −q, −k4) � PT = � g[1 + G(q)]−1�Γαµρ(k1, q, −k2) � ∆(q)Pµν(q) � g[1 + G(q)]−1�Γβνσ(k3, −q, −k4) � PT = �Γαµρ(k1, q, −k2) � g2[1 + G(q)]−2∆(q) � Pµν(q)�Γβνσ(k3, −q, −k4) PT = �Γαµρ(k1, q, −k2) � g2�∆(q) � � �� � 4π �d(k) Pµν(q)�Γβνσ(k3, −q, −k4) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (70) Evidently, the same �d(q), defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (68), appears naturally in both Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (69) and (70): it is, in that sense, a process-independent RGI interaction capturing faithfully the one-gluon exchange dynamics [3,16,20,80,97,130–132].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The actual determination of �d(q) proceeds by means of the second equality in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (68), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=', by combining the standard gluon propagator, ∆(q), together with the function 1 + G(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' In the top left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 9 we show lattice data for the conventional gluon propagator from [86] (points) and a physically motivated fit (blue continuous), given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (C11) of [125].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' In the top right panel of the same figure we show the 1 + G(q) auxiliary function, which can be computed by contracting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (12) with Pµν(q)/3 (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=', [132]), using the results of [229] for the ghost-gluon kernel, Hνµ(r, p, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Then, in the bottom left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 9 we show the �d(q) that results from combining the fit for ∆(q) and the 1 + G(q) shown in the top panels of the same figure and using αs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='27 [72] and Z1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='9333 [see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' From the �d(q) of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (68) one may define the dimensionless RGI interaction [16], I(q), I(q) := q2 �d(q) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (71) As explained in [16], this quantity provides the strength required in order to describe ground- state hadron observables using SDEs in the matter sector of the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' In that sense, I(q) bridges a longstanding gap that has existed between nonperturbative continuum QCD and ab initio predictions of basic hadron properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Three-gluon vertex and its planar degeneracy The three-gluon vertex, IΓαµν(q, r, p), plays a pivotal role in the dynamics of QCD [235], manifesting its non-Abelian nature through the gluon self-interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' In fact, the most cel- ebrated perturbative feature of QCD, namely asymptotic freedom, hinges on the properties of this particular interaction vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Its importance in the nonperturbative domain has led to an intense effort for unveiling its elaborate features [21,28,33–36,41,50,69,70,72,72,79,82,87, 88,123,173–180,180,181,272].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Indeed, as we have seen in Secs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 3 and 4, the pole structure of the three-gluon vertex is crucial for the onset of the Schwinger mechanism and the dy- namical generation of a gluon mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Moreover, its pole-free part provides highly nontrivial contributions to the SDEs of several Green’s functions, most notably the gluon propagator (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 1), as well as in the Bethe-Salpeter and Faddeev equations that determine the properties of glueballs [236,237,239–241] and hybrid mesons [238], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' For general momenta, IΓαµν(q, r, p) is a particularly complicated function, comprised by 14 tensor structures and their associated form factors [252].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Fortunately, in the Landau gauge, considerable simplifications take place, making the treatment of the three-gluon vertex less Particles 2023, 1 23 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='0 0 1 2 3 4 5 6 7 8 9 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='0 2.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='5 Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Top left: Gluon propagator, ∆(q), from lattice simulations of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' [86] (points) and a fit given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (C11) of [125] (blue continuous).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Top right: The auxiliary function 1 + G(q), defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Bottom left: The renormalization group invariant (RGI) running interaction strength �d(q) defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (68), computed using the ∆(q) and 1 + G(q) shown in the top panels, with αs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='27 [72] and Z1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='9333 [see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Bottom right: The corresponding dimensionless RGI interaction I(q), defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (71).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' cumbersome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Indeed, in the latter gauge, quantities of interest require only the knowledge of the transversely projected three-gluon vertex [127,175,176,224], Γαµν(q, r, p), defined as Γαµν(q, r, p) = IΓα′µ′ν′(q, r, p)Pα′α(q)Pµ′µ(r)Pν′ν(p) = Γα′µ′ν′(q, r, p)Pα′α(q)Pµ′µ(r)Pν′ν(p) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (72) Note that Γαµν(q, r, p) does not contain massless poles, by virtue of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Furthermore, Γαµν(q, r, p) can be parametrized in terms of only 4 independent tensor structures, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=', Γαµν(q, r, p) = 4 ∑ i=1 �Γi(q2, r2, p2) �λαµν i (q, r, p) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (73) Due to the Bose symmetry of Γαµν(q, r, p), the �λαµν i (q, r, p) can be chosen to be individually Bose symmetric, such that its form factors �Γi(q2, r2, p2) are symmetric under the exchange of any two arguments [87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' In fact, they can only depend on three totally symmetric combinations of momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Quite remarkably, lattice [87–89] and continuum [175,176,224] studies alike, have demon- strated that, to a very good level of accuracy, the �Γi depend exclusively on a single judiciously Particles 2023, 1 24 chosen variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Specifically, the �Γi computed on the lattice in [87–89] can be parametrized in terms of the special Bose symmetric combination s2 = 1 2 � q2 + r2 + p2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (74) Thus, the �Γi are the same for any combination of q2, r2, and p2 that fulfils Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (74) for a given value of s2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' This property has been denominated planar degeneracy, because Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (74) with fixed s defines a plane, normal to the vector (1, 1, 1), in the first octant of the coordinate system (q2, r2, p2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' In particular, the form factor �Γ1(q2, r2, p2) of the classical tensor structure is rather accu- rately approximated by �Γ1(q2, r2, p2) ≈ �Γ1(s2, s2, 0) ≈ Lsg(s) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (75) In the above equation, Lsg is the single transverse form factor of the three-gluon vertex in the soft gluon limit [125], and is obtained in lattice simulations as the q = 0 limit of the following totally transverse projection [85] Lsg(r) = Γαµν 0 (q, r, p)Pαα′(q)Pµµ′(r)Pνν′(p)IΓα′µ′ν′(q, r, p) Γαµν 0 (q, r, p)Pαα′(q)Pµµ′(r)Pνν′(p)Γα′µ′ν′ 0 (q, r, p) ������ q→0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (76) A particular realization of the planar degeneracy property is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 10, where we show the classical form factor �Γ1(q2, r2, p2), obtained from the lattice simulation of [87];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' we consider three different kinematic configurations, characterized by a single momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Specifically, the orange stars correspond to the soft-gluon limit, q = 0, which implies p2 = r2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' the green diamonds denote the symmetric limit, where all of the momenta have the same magnitude, q2 = p2 = r2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' and the purple circles represent points with p2 = r2 and q2 = 2r2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' When plotted against the momentum r, the three configurations of �Γ1(q2, r2, p2) produce three clearly distinct curves;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' however, when plotted in terms of the Bose symmetric variable s of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (74), they become statistically indistinguishable, manifesting the validity of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (75).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 0 1 2 3 4 5 6 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='5 0 1 2 3 4 5 6 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='5 Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Lattice data from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' [87] for the classical form factor, �Γ1(q2, r2, p2), of the transversely projected three-gluon vertex in three different kinematic configurations: the soft-gluon (q = 0, p2 = r2, orange stars), the symmetric limit (q2 = p2 = r2, green diamonds), and the case p2 = r2 with q2 = 2r2 (purple circles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' In the left panel �Γ1(q2, r2, p2) is plotted as a function of r, while in the right it is plotted as a function of the Bose symmetric variable s defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (74).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Particles 2023, 1 25 In addition to the planar degeneracy property, lattice [85,87–89] and continuum [175,176, 180,224] results show a clear dominance of the classical form factor �Γ1 over the remaining ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Based on these considerations, the special approximation Γαµν(q, r, p) ≈ Lsg(s)Γαµν 0 (q, r, p) , (77) has been put forth, where Γαµν 0 (q, r, p) is the tree-level value of Γαµν(q, r, p), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=', Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (72) with Γα′µ′ν′(q, r, p) → Γα′µ′ν′ 0 (q, r, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (77) provides an accurate and exceptionally compact approximation for Γαµν(q, r, p) in general kinematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' We emphasize that the shape of Lsg(r) has been very precisely determined through dedicated lattice studies with large-volume simulations [69,72,85,86].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The outcome of this exploration is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 11, where we plot the lattice data of [85] for Lsg(r), together with a physically motivated fit given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (C12) of [125] (blue continuous curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The approximation given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (77), with the fit for Lsg shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 11, will be used explicitly in Secs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 8 and 11, where the Γαµν(q, r, p) in general kinematics will be needed as input for the determination of other physically important quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='2 Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Lattice data from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' [85] for Lsg(q), compared to the fit for it given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (C12) of [125] (blue continuous curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Ghost dynamics from Schwinger-Dyson equations We next turn our attention to the ghost sector of the theory, whose scrutiny is important for several reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' First, it has been connected to particular scenarios of color confinement [273, 274].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Second, the Green’s functions associated with the ghost sector appear as ingredients in the SDEs of several key functions, such as the gluon propagator and the three-gluon vertex [41, 50,69,70,72,82,123,173–180,275], affecting their nonperturbative behavior in nontrivial ways, as will be discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Third, the SDEs governing the ghost sector are simpler than their gluonic counterparts, because they are comprised by fewer diagrams;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' in fact, the SDE of the ghost propagator contains a single diagram, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Fourth, in the Landau gauge, the validity of Taylor’s theorem [208] facilitates considerably the task of renormalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Consequently, the SDEs of the ghost sector are an excellent testing ground for (a) probing the impact of the gluonic Green’s functions that contribute to them [86];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (b) assessing the reliability of truncation schemes [276,277];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' and (c) testing the agreement between lattice and continuum approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' One of the central results of numerous studies in the continuum [21,63,86,113,179,226,228– 234] as well as a variety of lattice simulations [42,47,49,51,56,64,74,80] may be summarized by stating that the ghost propagator, D(q), remains massless, while the corresponding dressing function, F(q), saturates at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' As we will discuss in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 9, the nonperturbative Particles 2023, 1 26 masslessness of the ghost has important implications for the infrared behavior of the gluon propagator and the three-gluon vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' In what follows we provide a concrete example of the state-of-the-art SDE analysis of the ghost sector, by solving the coupled system of equations that governs the ghost-dressing function and the ghost-gluon vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' In order to obtain a closed system of equations, we use lattice results for the gluon propagator, the three-gluon vertex, and the value of the coupling constant in the particular renormalization scheme employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The main points of this analysis may be summarized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (i) We begin by considering the coupled system of SDEs given by Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 12, which determines the ghost propagator and ghost-gluon vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The treatment will be simplified by neglecting diagram (dν 3) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 12, thus eliminating the dependence on the ghost-ghost-gluon-gluon vertex, Γµσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' This is a particularly robust truncation, because the impact of the neglected diagram on the ghost-gluon vertex has been shown to be less than 2% [276].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (ii) Note that due to the fully transverse nature of the gluon propagators in the Landau gauge, in conjunction with the fact that various projections need to be implemented during the treatment of this system, the pole parts V of all fully dressed vertices appearing in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 12 will be annihilated;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' thus, we will have throughout the replacement IΓ → Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (iii) We proceed by decomposing the pole-free part, Γν(r, q, p), of the ghost-gluon vertex into its most general Lorentz structure, namely Γν(r, q, p) = rνB1(r, q, p) + pνB2(r, q, p) , (78) whose scalar form factors reduce to B0 1 = 1 and B0 2 = 0 at tree level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Evidently, due to the transversality of the gluon propagator, only the classical tensor rν, accompanied by the form factor B1, will survive in all SDE diagrams of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' + k q q k + q = ( −1 q ) ) q −1 ( Γµσ = rν + + + k − p k k + r (g ν 1 ) k + r k − p p k ν, a q (g ν 2 ) c q k ν, a p (g ν 3 ) k + r c b r b r p ν, a q c b r b q c r p ν, a Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Top: SDE governing the momentum evolution of the ghost propagator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Bottom: SDE for the ghost-gluon vertex, IΓν(r, q, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (iv) The SDE of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 12 is given by F−1(r) = 1 + 2λ � k f (k, r)B1(−r, k + r, −k)∆(k)D(k + r) , (79) Particles 2023, 1 27 where λ is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (43), and we define f (k, r) := 1 − (r · k)2 r2k2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (80) (v) Next, we note that the form factor B1(r, q, p) can be extracted from Γν(r, q, p) through the projection B1(r, q, p) = ενΓν(r, q, p) , εν := p2rν − (r · p)pν r2p2 − (r · p)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (81) Hence, acting with εν on the diagrams in the second line of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 12, we obtain B1(r, q, p) = 1 − λ[a(r, q, p) − b(r, q, p)] , (82) where a(r, q, p) = qαrµεν � k D(k)D(k − p)∆(k + r)B1(p − k, q, k + r)B1(−k, k − p, p)Pαµ(k + r)kν , b(r, q, p) = qαrµεν � k ∆(k)∆(k − p)D(k + r)B1(k + r, q, p − k)Γνµα(p, −k, k − p) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (83) (vi) At this point, we invoke the property of the planar degeneracy of Γαµν(q, r, p), dis- cussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Employing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (77) into the SDE for B1, the term b(r, q, p) of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (83) becomes b(r, q, p) = qαrµεν � k ∆(k)∆(k − p)D(k + r)B1(k + r, q, p − k)Γ0 νµα(p, −k, k − p)Lsg(¯s) , (84) with ¯s2 = p2 + k2 − 2(k · p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' We emphasize that although Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (77) constitutes in general an approximation, there is one particular kinematic limit in which the expression for b(r, q, p) given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (84) becomes exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Specifically, in the soft gluon limit (p = 0), it can be shown exactly that [86] Pµ′ µ (k)Pν′ ν (k)Γαµ′ν′(0, k, −k) = 2Lsg(k)kαPµν(k) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (85) Then, starting from either the general expression for b(r, q, p) of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (83) and using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (85), or from the approximate version given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (84), it can easily be shown that the p = 0 limit is the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' As such, the use of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (77) yields not only an excellent approximation in general kinematics, but also the exact soft gluon limit for the contribution of the three-gluon vertex to the form factor B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (vii) Now we consider the renormalization of the coupled system of equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Since the ghost-gluon vertex is finite in the Landau gauge [208], most SDE treatments [86,225–229] of the ghost sector employ the so-called Taylor renormalization scheme, defined in such a way that the finite renormalization constant of the ghost-gluon vertex has the exact value Z1 = 1 [54,60,81,86,208].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' However, in order to employ Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (77) most expeditiously, it is more convenient to renor- malize in the so-called asymmetric MOM scheme, because this is precisely the scheme employed in the lattice calculations of Lsg [69,72,85,86].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Specifically, this scheme is defined by imposing the normalization conditions [85,86] ∆−1 R (µ) = µ2 , FR(µ) = 1 , LR sg(µ) = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (86) Past this point, we denote by �Z1 the finite value of the ghost-gluon renormalization constant in the asymmetric MOM scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Evidently, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (14) and (78) imply that BR 1 = �Z1B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Particles 2023, 1 28 The renormalization of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (79) and (82) proceeds by substitution of the unrenormalized quantities by their renormalized counterparts, following Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (14), and imposing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (86) for F(µ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Note that, in principle, �Z1, may be determined from the relation �Z1 = Z3ZcZ−1 A , imposed by the corresponding STI [278];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' however, these renormalization constants are not available to us, given that the associated Green’s functions have been obtained from the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Therefore, �Z1 is treated as an adjustable parameter, whose value is determined by requiring that the solution of the SDE for F(q) reproduces the corresponding lattice data of [74,86] as well as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (viii) Finally, we transform Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (79) and (82) from Minkowski to Euclidean space, using standard conversion rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Note that, once in Euclidean space, we will express the functional dependence of B1(r, q, p) in terms of the squared momenta of the antighost and gluon legs, r2 and p2, and the angle, θ, between them, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=', B1(r, q, p) ≡ B1(r2, p2, θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The result of these manipulations is that Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (79) and (82) become F−1(r) = 1 − αsCA �Z1 2π2 � ∞ 0 dk2k2∆(k) � π 0 dφ s4 φ × � B1(r2, k2, φ) F(√z) z − B1(µ2, k2, φ) F(√u) u � , (87) and B1(r2, p2, θ) = �Z1 − αsCA �Z1 8π2 � a + 2b � , (88) respectively, with a = 1 sθ � ∞ 0 dk2k2F(k) � π 0 dφs3 φ ∆(√z) z � π 0 dωsω F(√v) v B1(k2, p2, α)B1(v, z, β)Ka , (89) b = 1 sθ � ∞ 0 dk2k2∆(k) � π 0 dφs3 φ F(√z) z � π 0 dωsω ∆(√v) v B1(z, v, β)Lsg(s)Kb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' In the above equations we employ the notation cx := cos x and sx := sin x, and define the following variables r · k := rkcφ , p · k := pk(cθcφ + sθsφcω) , z := r2 + k2 + 2rkcφ , u := µ2 + k2 + 2µkcφ , s2 := (p2 + k2 + v)/2 , v := p2 + k2 − 2pk(cθcφ + sθsφcω) , α := π − cos−1� cθcφ + sθsφcω � , β := cos−1 � k(pcθcφ + psθsφcω − rcφ) + prcθ − k2 √vz � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Particles 2023,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 1 29 Finally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' the kernels Ka and Kb are given by Ka =(cθcωsφ − cφsθ) � ksφ(pcθ + r) − pcθcω(kcφ + r) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Kb =cω � k2pcφ � cθp � s2 θ(s2 φs2 ω − 4s2 φ + 1) + s2 φ � + r � s2 φ − s2 θ(2s2 φ + 1) �� − k3� s2 φ � rcθ − 2ps2 θ + p � + ps2 θ � + kp2� s2 φ � 2s2 θ(p − rcθ) − rcθ − p � + s2 θ(rcθ − p) � −cφp3rs2 θ � + sθsφ � cθp � r � p2 − k2(s2 ω + s2 φs2 ω − 2s2 φ) � − cφk(s2 ω − 2)(k2 + p2) � + k � cφk2r − cφp2r � s2 θ(s2 ω − 2) + s2 ω � + kp2� 3s2 θs2 φs2 ω − 2s2 θs2 ω − 4s2 θs2 φ + 3s2 θ +(3 − 2s2 ω)s2 φ − 2 ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' We are now in position to solve Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (87) and (88) numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' We choose the renormal- ization point at µ = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='3 GeV, and employ for ∆(q) and Lsg(q) the fits to lattice data shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 9 and 11, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Note that for large momenta these fits recover the behavior dictated by the corresponding anomalous dimensions [125].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' For the strong coupling, we use the value αs(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='3 GeV) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='27, determined from the lattice simulations of [72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Below we discuss the main results of this analysis: The value of �Z1 was obtained by solving the SDE system for various values of this constant until the χ2 of the comparison between the solution for F(q) and the lattice data of [74,86] was minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' This procedure yields �Z1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='9333 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='0075.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' In the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 13 we show as a blue continuous line the SDE result for F(q), with the above value of �Z1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The result is compared to the lattice data of [74,86], which have been cured from discretization artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' As it turns out, the SDE and lattice results for F agree within 1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' We next consider the form factor B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' In the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 13 we show B1(r2, p2, θ) as a surface, for arbitrary values of the magnitudes of the momenta r and p, and for the angle θ formed between them at θ = 2π/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' In the same panel, we highlight as a red dot-dashed curve the soft gluon limit4 B1(r2, 0, 2π/3) of the general kinematics B1(r2, p2, 2π/3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The only available SU(3) lattice data for B1 have been obtained in the soft gluon limit [42, 43], and have sizable error bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Furthermore, they have been computed within the Taylor scheme, while in the present work we used the asymmetric MOM scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Nevertheless, we can meaningfully compare our SDE results with those of the lattice, and perform a statistical analysis to assess their agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Specifically, denoting by BT 1 the Taylor scheme value of the form factor B1, it can easily be shown that B1(r2, p2, θ) = �Z1BT 1(r2, p2, θ) , (90) which allows us to carry out the desired comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Then, we use Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (90) to compute BT 1(r2, 0, θ) from the B1(r2, 0, 2π/3) slice (red dot-dashed curve) in the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 13, and compare the result to the lattice data of [42,43] (points) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Evidently, the SDE determination agrees with the lattice results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' In order to quantify this agreement, we next conduct a χ2 analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' To this end, we consider only the 22 lattice points ri in the interval ri ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='3, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='5] GeV, where the signal is most pronounced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Then, we compute the χ2 of the data through χ2 j = ∑ i [Blat 1 (r2 i , 0, θ) − gj(ri)]2 ϵB1(r2 i , 0, θ) , (91) 4 The soft gluon limit is approached by taking p → 0 in B1(r2, p2, θ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' in the nonperturbative case, this limit is independent of the value of θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Particles 2023, 1 30 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='0 Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Left: ghost dressing function F(q) obtained from the coupled system of SDEs of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (79) and (82) (blue continuous line) compared to the lattice data of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' [74,86].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Right: The corresponding result for B1(r2, p2, θ) for arbitrary magnitudes of the antighost and gluon momenta, r and p, respectively, and a representative value of θ = 2π/3 for the angle between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The red dot-dashed curve highlights the soft gluon limit (p = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 0 1 2 3 4 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='3 Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Soft gluon limit, BT 1(r2, 0, θ), of the classical form factor of the ghost-gluon vertex in Taylor scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The points correspond to the lattice data of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' [42,43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The red dot-dashed line shows the SDE solution with the three-gluon vertex dressed according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (77), while the green dashed represents the SDE solution with tree-level three-gluon vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' where Blat 1 (r2 i , 0, θ) are the lattice points shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 14, ϵB1(r2 i , 0, θ) their respective errors, and gj(ri) are three hypotheses which we will compare to the lattice data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Specifically, for the gj we consider the three cases gj(ri) = � � � � � 1 if j = 1 , SDE with Γαµν = Γαµν 0 Lsg(s) if j = 2 , SDE with Γαµν = Γαµν 0 if j = 3 , (92) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=', g1 is the tree-level value of B1, g2 the solution of the SDE using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (77) for dressing the three-gluon vertex, corresponding to the red dot-dashed curve of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 14, and g3 is the solution of the SDE obtained setting the three-gluon vertex to tree-level, which amounts to the substitution Lsg → 1 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (88), and is represented by a green dashed curve in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' [Vo] b [GeA] 0 T 士 8 3 5 于 1 G Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content="0 B(3) T I'S 1'3Particles 2023, 1 31 Then, for each χ2 j we compute the probability Pj that normally distributed errors would yield a χ2 at least as large as χ2 j , through Pj = � ∞ χ2 j χ2 PDF(22, x)dx = Γ(nr/2, χ2/2) Γ(nr/2) ���� χ2=χ2 j nr=22 ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (93) In the above equation, χ2 PDF(n, x) = xn/2−1e−x/2/[2n/2Γ(n/2)] denotes the χ2 probability distribution function with n degrees of freedom, while Γ(z, x) is the incomplete Γ function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The results of the above analyses are collected in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' We note that the case g1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=', the tree-level value of B1, is discarded at 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='1σ confidence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' As for case g3, it is discarded at the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='4σ level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' On the other hand, the SDE result with dressed three-gluon vertex, g2, is statistically indistinguishable from the lattice data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Statistical results of the χ2 analysis for the three hypotheses given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (92) for the form factor B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' For each case (first column), we give the corresponding χ2 j computed from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (91) (second column), probability Pj computed from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (93) (third row), and the same Pj expressed in terms of confidence levels σ (fourth row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Case (j) χ2 j Pj Confidence level in σ 1 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='37 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='0 × 10−7 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='1 2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='399 1 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='8 × 10−6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='2 × 10−6 3 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='03 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='8 × 10−4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='4 Lastly, we point out that for both F and B1 we find a good qualitative agreement with various related studies [21,29,179,180,225,227–229,279,280], including kinematics other than the soft gluon limit considered in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Divergent ghost loops and their impact on the QCD Green’s functions The masslessness of the ghost propagator, discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 8, has important implications for the infrared behavior of other Green’s functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Specifically, while the saturation of the gluon propagator renders gluon loops infrared finite, ghost loops furnish infrared divergent contributions [173], akin to those encountered in perturbation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' In this section, we highlight with two characteristic examples how the effects of ghost loops manifest themselves at the level of the two- and three-point functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Specifically, the ghost loops induce the appearance of a moderate maximum in the gluon propagator and are responsible for the zero-crossing and the logarithmic divergence at the origin displayed by the dominant form factors of the three-gluon vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The basic observation at the level of the gluon SDE shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 1 is that, the ghost loop of (d3), due to the masslessness of its ingredients, furnishes “unprotected” logarithms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=', terms of the type ln q2, which diverge as q2 → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Instead, gluonic loops contain infrared finite gluon propagators, and, therefore, give rise to contributions that remain finite as q2 → 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=', they may be described in terms of “protected” logarithms of the type ln(q2 + m2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The circumstances described above may be modeled by ∆−1(q) = q2 + m2 + c1q2 ln �q2 + ρm2 Λ2 � � �� � f (q) +c2q2 ln � q2 Λ2 � , (94) Particles 2023, 1 32 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='0 Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Lattice data for the gluon propagator in the deep infrared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The data displayed correspond to the two lattice setups with the largest volumes of [49], namely, V = 724 (left) and V = 804 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The red dashed lines are smooth fits from which the position of the maximum can be estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' where m is the gluon mass, Λ the mass scale of QCD, and c1, c2 and ρ are constants;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' note that ∆−1(0) = f (0) = m2 Differentiating Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (94) with respect to q2 we obtain d∆−1(q) dq2 = d f (q) dq2 + c2 � 1 + ln � q2 Λ2 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (95) The second term on the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (95) is infrared divergent, and necessarily dominates the behavior of the derivative of the propagator for sufficiently small q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Moreover, the value of the coefficient c2 can be computed explicitly by expanding the ghost block �Π(2) µν (q) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 1 around q = 0 and using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (19), which yields c2 = αsCA �Z2 1F2(0) 48π .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (96) Therefore, d∆−1(q)/dq2 has the asymptotic behavior lim q→0 d∆−1(q) dq2 = � αsCA �Z2 1F2(0) 48π � ln � q2 Λ2 � , (97) which diverges to −∞ as q → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Now, since the gluon propagator is a decreasing function in the ultraviolet, we have that d∆−1(q)/dq2 is positive for large momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Therefore, there must exist a special momentum, denoted by q⋆, such that [d∆(q)/dq2]q=q⋆ = 0, which corresponds to a maximum5 of ∆(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The maximum of ∆(q), predicted by means of the simple arguments presented above, is observed in lattice simulations of the gluon propagator [49,56,86].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' In particular, it is clearly visible in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 15, where the data from the two largest volume lattice setups of [49] are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The red dashed lines represent smooth functions, fitted to each of the data sets, in the window q ∈ [0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='5] GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' For each of the volumes considered, V = 724 (left panel) and V = 804 (right panel), the estimate obtained for q⋆ is q⋆ = 140 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 5 Note that d∆−1(q)/dq2 is an increasing function, since it is negative in the infrared and positive in the ultraviolet, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=', d2∆−1(q)/d(q2)2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Therefore, assuming that d∆−1(q)/dq2 only crosses zero once, q = q⋆ must be a maximum of ∆(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Particles 2023, 1 33 It is interesting to observe in passing that the existence of a maximum of ∆(q) has an interesting implication on the form of the spectral function of the gluon propagator [281–286].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' In particular, the standard Källén-Lehmann representation [287,288] states that ∆(q) = � ∞ 0 dλ2 ρ(λ2) q2 + λ2 , (98) where ρ(λ2) is the gluon spectral function (with a factor 1/π absorbed in it).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Thus, the differentiation of both sides of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (98) with respect to q2 yields d∆(q) dq2 = − � ∞ 0 dλ2 ρ(λ2) (q2 + λ2)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (99) Then, from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (99) follows that the existence of a maximum for ∆(q) at q = q⋆ leads necessarily to the violation of reflection positivity [11,168,169,172], because the condition � ∞ 0 dλ2 ρ(λ2) (q2⋆ + λ2)2 = 0 , (100) may be fulfilled only if ρ(λ2) reverses its sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Note that an analogous argument based on the existence of an inflection point has been presented recently in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Turning to the three-gluon vertex, it is well known that the corresponding ghost loops induce characteristic features to the form factors associated with its classical (tree-level) tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' There are two complementary continuum descriptions of the dynamics that determine the behavior of these form factors: (i) the SDE of the three-gluon vertex [175–177,227], depicted diagrammatically in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 16, and (ii) the STI of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (23) [173], which, in the limit of vanishing gluon momentum, and when the displacement function and the ghost sector are neglected, yields the approximate WI IΓαµν(0, r, −r) ≈ ∂∆−1 µν (r) ∂rα , (101) which transmits the properties of the propagator derivative to the vertex form factors, as shown schematically in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' In the simplified kinematic circumstances where only a single representative momentum is considered, to be denoted by r, the conclusions drawn by either method may be qualitatively described in terms of a simple model, namely L(r) = b0 + bgl ln �r2 + m2 Λ2 � + bgh ln � r2 Λ2 � , (102) where L(r) denotes the particular combination of form factors, such that, at tree level, L0(r) = 1, and b0, bgl, and bgh are positive constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The model in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (102) encompasses two important cases studied on the lattice [69,70,72,82], namely (i) the soft gluon limit, L(r) → Lsg(r), corre- sponding to the kinematic choice q → 0 , p = −r , θ := �pr = π, defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (76), and (ii) totally symmetric limit, L(r) → Lsym(r), corresponding to q2 = p2 = r2 , θ := �qr = � qp = �rp = 2π/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Upon inspection of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (102) we note that, as r → 0, the term with the unprotected logarithm will eventually dominate, forcing L(r) to reverse its sign (zero crossing), and finally display a logarithmic divergence, L(0) → −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Given that, in practice, bgl is considerably larger than bgh, the unprotected logarithm overtakes the protected one rather deep in the infrared: the location of the zero-crossing is at about 160 MeV [72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Consequently, in the intermediate region of momenta, which is considered relevant for the onset of nonperturbative dynamics, we have Particles 2023, 1 34 L(r) < 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' this effect is known in the literature as the infrared suppression of the three-gluon vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' + + · · · (e1) (e2) µ, b α, a p = + α, a ν, c q p r ν, c µ, b q r Figure 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The SDE of the three-gluon vertex at the one-loop dressed level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The diagrams (e1) and (e2) are the gluon and the ghost triangle contributions entering in the skeleton expansion of the three-gluon vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Most importantly, the special features of infrared suppression, zero-crossing, and log- arithmic divergence at the origin have been corroborated through a variety of lattice re- sults [50,69,70,72,73,82,85], as shown, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=', in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The central curve of this figure is presented as the blue line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 17, where the aforementioned characteristics have been explicitly marked for the benefit of the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Note the close proximity of the blue curve to the d∆−1(r)/dr2 (red dashed line), especially below 1 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='2 Figure 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Comparison of Lsg(r) (blue continuous) from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 11 and d∆−1(r)/dr2 (red dashed) resulting from the fit for ∆(r) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Note that both display the characteristic features of infrared suppression with respect to their tree-level values (which is 1 for both quantities), zero-crossing, and logarithmic divergence at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' We end this section by pointing out that, in the case of Yang-Mills in d = 3 [28,173,224,289– 303], the situation is qualitatively similar to the one described above, but the divergences induced due to the masslessness of the ghost are stronger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Specifically, as may be already established at the level of a simple one-loop calculation [303], the first derivative of the gluon propagator diverges at the origin as 1/q rather than a ln q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Consequently, the corresponding effects are significantly enhanced;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' in particular, the maximum of the gluon propagator is considerably more pronounced, becoming plainly visible on the lattice [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Similarly, an abrupt negative divergence is observed in the corresponding vertex form factors [41,83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Ward identity displacement of the three-gluon vertex In complete analogy to the case of the ghost-gluon vertex discussed in subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='2, the WI satisfied by the pole-free part of the three-gluon vertex is also displaced in the presence of Particles 2023, 1 35 ⊃ ⊃ STI Figure 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The ghost triangle present in the three-gluon vertex SDE (top) and the ghost loop contributing to the gluon propagator in the corresponding equation (middle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The infrared divergences arising from these diagrams are connected through the Slavnov-Taylor identity (STI) of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (23), as shown schematically in the bottom panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' longitudinally coupled massless poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Quite importantly, the associated displacement function, C(r), coincides with the BS amplitude that controls the formation of a (colored) scalar bound state with vanishing mass out of a gluon pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The displacement of the WI circumvents the seagull cancellation involving the gluon propagator [i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=', f = ∆ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (54)], furnishing to the gµν component the mass originating from graphs (d1) and (d4) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' In addition, it permits the indirect determination of the displacement function C(r) from the lattice;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' this is particularly important, given that, by virtue of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (35), the lattice “observables” do not perceive directly the presence of the massless poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The starting point of the analysis is the STI satisfied by the three-gluon vertex, IΓαµν(q, r, p), given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' In order to eliminate the poles in r and p, thus isolating the displacement of the WI originating from the pole in the channel q, we contract that equation with Pµ µ′(r)Pν ν′(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Note that this procedure also eliminates any longitudinal pole terms in the Hσµ(p, q, r) and Hσν(r, q, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Then, we decompose IΓαµν(q, r, p) into pole-free and longitudinally coupled massless pole parts, as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (33), and use Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (38), to obtain Pµ µ′(r)Pν ν′(p) � qαΓαµν(q, r, p) + gµνC1(q, r, p) + qµqνC5(q, r, p) � = Pµ µ′(r)Pν ν′(p)Rνµ(p, q, r) , (103) where Rνµ(p, q, r) := F(q) � ∆−1(p)Pσ ν (p)Hσµ(p, q, r) − ∆−1(r)Pσ µ (r)Hσν(r, q, p) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (104) At this point, we expand Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (103) around q = 0 and match coefficients of equal orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' At zeroth order in this expansion we obtain immediately that C1(0, r, −r) = 0 , (105) in exact analogy to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (61).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Note that we have arrived once again at the result of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (39), but through an entirely different path: while Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (39) is enforced by the Bose symmetry of the three-gluon vertex, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (105) is a direct consequence of the STI that this vertex satisfies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Particles 2023, 1 36 We next gather the terms in the expansion of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (103) that are of first order in q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Evidently, the term C5 does not contribute to this order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Then, the expansion leads to the appearance of derivatives of the gluon propagator, in analogy to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (63), but also of the ghost-gluon kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Specifically, we obtain for the WI of the three-gluon vertex and its displacement the expression Lsg(r) = F(0) � �Z1 d∆−1(r) dr2 + W(r) r2 ∆−1(r) � − C(r) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (106) In the above equation, Lsg(r) is the form factor of the three-gluon vertex defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (76) and with lattice results shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 11, while W(r) is a particular derivative of the ghost-gluon kernel, namely [125,242] W(r) = − 1 3r2 Pµν(r) �∂Hνµ(p, q, r) ∂qα � q=0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (107) For the detailed derivation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (106), we refer to [94,125].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' In the following section, we will use Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (106) to determine the displacement amplitude C(r) from lattice inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' To this end, we must first pass to Euclidean space, where we note that CE(r2 E) = −C(r)|r2=−r2 E , (108) with the extra sign originating from the fact that C is a derivative [see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (41)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Then, suppress- ing the indices “E” and solving for C(r2), we obtain the central relation C(r) = Lsg(r) − F(0) �W(r) r2 ∆−1(r) + �Z1 d∆−1(r) dr2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (109) For the determination of C(r), we use lattice inputs for all the quantities that appear on the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (109), with the exception of the function W(r), which will be computed from the SDE satisfied by the ghost-gluon kernel, derived and analyzed in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The ghost-gluon kernel contribution to the Ward identity In this section, we derive the SDE that determines the key function W(r);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' the resulting SDE will be solved using lattice inputs for the various quantities entering in it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' In addition, the infrared behavior of W(r) will be analyzed in detail, following an analytic procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Our discussion starts with the SDE of the ghost-gluon kernel, Hµν(r, q, p), shown diagram- matically in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 19, from which W(r) can be obtained using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (107).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Note that the similarity between the diagrams shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 19 and those in the bottom panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 12, depicting the SDE of the ghost-gluon vertex, is a simple reflection of the fundamental STI relating the ghost-gluon kernel with the ghost-gluon vertex, Γν(r, q, p) = rµHµν(r, q, p) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (110) Specifically, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (110) is preserved by the SDEs of Γν(r, q, p) and Hµν(r, q, p);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' indeed, contraction of each diagram (hµν i ) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 19 by rµ yields the corresponding diagram (gν i ) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 12 (up to a shift of k → −k − r for i = 1, introduced to simplify certain expressions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Note that, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 19, the diagram corresponding to the (g3) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 12 has been omitted, for the reason explained in the item (i) of Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' It is well known that, in the Landau gauge, the momentum q of the ghost field in Hµν(r, q, p) factors out of its quantum corrections [1], allowing us to write [125,229,242] Hµν(r, q, p) = gµν + qαKµνα(r, q, p) , (111) Particles 2023, 1 37 = gµν + + k − q q p k + r r k ν, a µ, b (h µν 1 ) k + r k − p p k r ν, a q µ, b (h µν 2 ) c c µ, b k q ν, a p r k + r c Figure 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' SDE for the ghost-gluon scattering kernel, Hµν(r, q, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' We omit a diagram containing a 1PI four-point function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' where no particular assumptions are made about the structure of the function Kµνα(r, q, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Following Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (107), we differentiate the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (111) with respect to q and subsequently set q = 0, to obtain W(r) = −1 3rαPµν(r)Kµνα(r, 0, −r) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (112) Lastly, the finite renormalization of W proceeds through the use of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (14) and (86), which leads to the appearance of an overall factor of �Z1 in the equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The outcome of the above steps is that W(r) can be written as W(r) = W1(r) + W2(r) , (113) where the Wi(r) are the contributions originating from the diagrams (hµν i ) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 19, respec- tively, and read W1(r) = λ�Z1 3 � k ∆(k)D(k)D(k + r)(r · k) f (k, r)B1(k + r, −k, −r)B1(k, 0, −k) , W2(r) = λ�Z1 3 � k ∆(k)∆(k + r)D(k + r)B1(k + r, 0, −k − r)IW(−r, −k, k + r) , (114) where f (k, r) is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (80), and we define the specific contribution of the three-gluon vertex to the kernel of W(r2) as IW(q, r, p) := 1 2(q − r)νΓα αν(q, r, p) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (115) Note that, from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (115) and the Bose symmetry of the Γαµν(q, r, p) under the exchange {q, α} ↔ {r, µ}, it follows that IW(q, r, p) = IW(r, q, p) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (116) At this point, by capitalizing on the planar degeneracy of Γαµν(q, r, p) discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 7, we obtain a compact, and yet accurate, approximation for IW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Specifically, using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (77), we find IW(q, r, p) ≈ I0 W(q, r, p)Lsg(s) , (117) Particles 2023, 1 38 where I0 W(q, r, p) is the tree-level value of IW, given by I0 W(q, r, p) := 2f (q, r) p2 � 2q2r2 − (q2 + r2)(q · r) − (q · r)2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (118) We remark that the approximation given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (117) becomes exact in the limit p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Using the above approximation for IW, the contribution W2(r) reads W2(r) =2λ�Z1 3 � k ∆(k)∆(k + r)D(k + r) (k + r)2 B1(k + r, 0, −k − r) f (k, r) × � 2r2k2 − (r2 + k2)(r · k) − (r · k)2� Lsg(ˆs) , (119) where we now have ˆs2 = r2 + k2 + (r · k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Lastly, we transform W1 of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (114) and W2 of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (119) to Euclidean space to obtain the final expression to be used for the numerical determination of W, W1(r) = −rαsCA �Z1 12π2 � ∞ 0 dk2k∆(k)F(k)B1(k2, k2, π) � π 0 dφs4 φcφ F(√z) z B1(z, r2, χ) , W2(r) = −rαsCA �Z1 6π2 � ∞ 0 dk2 k3∆(k) � π 0 dφ s4 φ∆(√ z)B1(z, z, π) F(√z) z2 � kr(2 + c2 φ) − zcφ � × Lsg � r2 + k2 + rkcφ � , (120) where z has been defined below Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (89) and χ := cos−1 � −(r + kcφ) √z � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (121) We emphasize that we have used into the SDEs of both B1 and W, given by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (88) and (120), respectively, the same approximation for the three-gluon vertex, namely Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (77).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Therefore, our analyses of B1 and W are self-consistent, in the sense that the STI in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (110) is strictly preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Before embarking on the numerical determination of W(r) for the entire range of Euclidean momenta, we discuss the infrared behavior of this function, and demonstrate an important self-consistency proof involving C(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Specifically, as discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 9, the Lsg(r) and d∆−1(r)/dr2 that appear in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (109) are infrared divergent, due to massless ghost loops present in their SDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Nevertheless, the BSE solutions for the amplitude C(r) are all found to be finite at r = 0, (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 7) [118,122,125,216].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Therefore, in order for the WI displacement of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (109) to be consistent with the finite C(0) obtained from BSE solutions, the infrared divergences of the ingredients appearing in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (109) must cancel against each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Indeed, a careful analysis of diagram (e2) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 16 yields lim r→0 Lsg(r) = � αsCA �Z3 1F3(0) 96π � ln � r2 µ2 � , (122) up to infrared finite terms6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Then, assuming that only Lsg(r) and d∆−1/dr2 diverge and using the asymptotic form of d∆(r)/dr2 given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (97) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (109), we find that the divergences 6 We note that results identical to Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (97) and (122) for the infrared divergences of d∆−1(r)/dr2 and Lsg(r), respectively, have been previously derived within the Curci-Ferrari model [181].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Particles 2023, 1 39 do not fully cancel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Therefore, the finiteness of C(0) demands that the term W(r)/r2 appearing in the WI must be infrared divergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Now, it is evident from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (120) that W(r) vanishes as r → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Nevertheless, the ratio W(r)/r2 is found to diverge at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Specifically, expanding Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (120) around r = 0, it can be shown that W(r)/r2 has the asymptotic behavior lim r→0 W(r) r2 = − � αsCA �Z3 1∆(0)F2(0) 96π � ln � r2 µ2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (123) Then, combining Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (123), (122) and (123) we find that the infrared divergences in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (109) cancel out exactly, leaving a finite C(0), in full agreement with the BSE results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' r r 0 µ ν ρ − F(0) �Z1 d dr2 r r 0 µ ν ρ − F(0) ∆(0) lim r2→0 = IR finite r µ ν � �� � Kµνρ(r, 0, −r) Figure 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Diagrammatic representation of the cancellation of the infrared divergences originating from massless ghost loops in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (109) to yield a finite C(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The red cross indicates that the overall ghost momentum is factored out before being set to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' We finish the discussion of the infrared finiteness of C(0) with a remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' In the absence of the Schwinger mechanism, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=', for an identically zero C(r), the infared divergences of Lsg(r), W(r)/r2 and d∆−1(r)/dr2 must also cancel in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (109).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' For instance, this cancellation can be explicitly verified at the one loop level7, where, evidently, C(r) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' In that case, however, the gluon propagator is also massless, causing the gluonic loops contributing to the functions entering Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (109) to also diverge, such that the cancellation occurs among all radiative diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' In contrast, in the presence of a gluon mass, the cancellation of the remaining infrared divergences takes place at the level of the ghost loops only, as illustrated diagrammatically in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Figure 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' W(r) obtained using the approximation Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (117) based on the observed planar degeneracy of the three-gluon vertex in its kernel (blue solid curve) together with uncertainty estimate (blue band).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 7 In the perturbative realization of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (109) F(0) also diverges, participating in the overall cancellation of infrared divergences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='2-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='0 3 r[GeV].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='0Particles 2023, 1 40 We now return to the numerical determination of W(r) from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (120).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' To this end, we employ the fits to lattice data of [85] for ∆(q) and Lsg(q) shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 9 and 11, respectively, and the SDE solution for F(q) shown in the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' All of the fits employed are constructed so as to reproduce the correct ultraviolet behavior of the respective Green’s functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' For the value of the coupling in the asymmetric MOM scheme we employ g2 = 4παs, with αs(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='3 GeV) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='27, as determined in the lattice study of [72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Lastly, for B1 we use the SDE result of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 8, shown in the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 13, which reproduces accurately the available lattice data for the ghost-gluon vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Using the above ingredients into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (120) we obtain the W(r) shown as the blue solid curve in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The blue band in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 21 represents the error estimate on our results;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' the procedures followed to obtain it are described in detail in [127].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Displacement function from lattice inputs In this section we determine C(r) from the main relation given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (109).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' For W(r) we use the result shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 21, together with the curves for Lsg(r) from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 11, ∆(r) and d∆−1(r)/dr2 from Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 9 and 17, respectively, and the F(r) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The C(r) obtained is shown as a black solid curve in the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' In the same panel, we show as points the estimates of C(r) obtained by using into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (109) the lattice data points of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' [85] directly, rather than a fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' To estimate the uncertainty in the resulting C(r), we combine the error estimate of W(r), represented by the blue band in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 21, with the statistical error of the lattice data points for Lsg(r) of [85], and neglect the error in the gluon propagator, which is much smaller than the errors in Lsg and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Then, a conservative error propagation analysis produces the error bars shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='2 C(r) C(r) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='2 C(r) C(r) C⋆(r) Figure 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Left: Result for C(r) (black continuous line) obtained from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (109) using the W(r) shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 21, the fits to lattice data for ∆(r) and Lsg(r) shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 9 and 17, respectively, and the SDE solution for F(r) shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The points are obtained using for Lsg(r) the data in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' [85], with error bars denoting the error propagated from Lsg and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The green band is obtained by fitting the upper and lower bounds of the points and guide the eye to the typical error associated with C(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Right: The C(r) of the left panel is compared to the BSE prediction C⋆(r) (purple dot-dashed and error band) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' At this point, we quantify the significance of the C(r) obtained above, in comparison to the null hypothesis result;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' evidently, in the absence of the Schwinger mechanism, this latter quantity, to be denoted by C0 in what follows, vanishes identically, namely C0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' To this end, we first compute the χ2 of our points as χ2 = ∑ i [C(ri) − C0(ri)]2 ϵ2 C(ri) , (124) Particles 2023, 1 41 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=', the null hypothesis is taken as the estimator for our data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The sum runs over the nr = 515 indices i such that ri ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='3, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='3] GeV, the interval of momenta for which the systematic error in our calculation of W(r) is best known, and ϵC(ri) denotes the error estimate of C(ri).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Then we obtain χ2 = 2 630, corresponding to χ2 d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The probability PC0 that our result for C is consistent with the null hypothesis is vanishingly small, given by the formula PC0 = � ∞ χ2=2 630 χ2 PDF(515, x)dx = Γ(nr/2, χ2/2) Γ(nr/2) ���� χ2=2 630 nr=515 = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='3 × 10−280 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (125) In fact, even if the errors were 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='95% larger, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=', nearly doubled, we could still discard C0 at the 5σ confidence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' In the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 22 we compare C(r) to the BSE prediction, C⋆(r), of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 7, shown as a purple dot-dashed curve and corresponding error band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' In that panel, we observe an excellent qualitative agreement between the two results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The most noticeable quantitative difference is in the position of the minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Specifically, C reaches the minimum value of −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='36 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='11 at r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='93+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='09 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='06 GeV, while the minimum of C⋆ is −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='341 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='003 at r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Nevertheless, it is clear in the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 22 that the BSE prediction lies within the error estimate of the lattice-derived C(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' In fact, defining a χ2 measure for the discrepancy between C and C⋆ as χ2 ⋆ = ∑ i [C(ri) − C⋆(ri)]2 ϵ2 C(ri) , (126) we obtain χ2⋆ = 258.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='5, which is smaller than the number of degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Then, this value of χ2⋆ amounts to a probability of PC⋆ = Γ(nr/2, χ2⋆/2) Γ(nr/2) ���� χ2⋆=258.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='5 nr=515 = 1 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='0 × 10−23 , (127) showing that C⋆ is statistically compatible with the lattice derived C, with probability extremely near unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Conclusions The gauge sector of QCD is host to a wide array of subtle mechanisms that are of vital importance for the self-consistency and infrared stability of the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' In the present work, we have offered a comprehensive review of the intricate dynamics that account for some of the most prominent infrared phenomena, such as the generation of a gluon mass through the action of the Schwinger mechanism, the nonperturbative masslessness of the ghost, and the characteristic features induced by this particular mass pattern to the form factors of the three-gluon vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The SDEs, supplemented by the judicious use of certain key results from lattice QCD, provide a robust continuum framework for carrying out such a demanding investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' In fact, the results obtained from the SDEs are increasingly reliable, passing successfully all sorts of tests imposed on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' A particularly impressive, and certainly not isolated, case of such a success has been outlined in detail in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Symmetry and dynamics are tightly interwoven;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' therefore, the information encoded in the STIs and WIs of the theory is particularly decisive for unraveling basic dynamical patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' A striking manifestation of the profound connection between symmetry and dynamics is provided by the dual role played by the function C(r): it is both the BS amplitude of the massless states composed by a pair of gluons, and the quantity that embodies the displacement induced to the WIs by the presence of these states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Particles 2023, 1 42 In our opinion, the determination of C(r) described in Section 12 represents a major success of the entire set of concepts and techniques surrounding the generation of a gluon mass through the action of the Schwinger mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Thus, fifty years after the genesis of QCD, we seem to be closing in on the mechanism that the theory uses for curing the infrared instabilities endemic to perturbation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' We hope to be able to report further progress in this direction in the near future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Funding: The authors are supported by the Spanish MICINN grant PID2020-113334GB-I00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' acknowledges financial support from Generalitat Valenciana through contract CIAPOS/2021/74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' also acknowledges funding from the regional Prometeo/2019/087 from the Generalitat Valenciana.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Data Availability Statement: Not applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Acknowledgments: The authors thank A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Aguilar, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Binosi, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Ibáñez, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Pawlowski, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Roberts, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Rodríguez-Quintero for several collaborations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Conflicts of Interest: The authors declare no conflict of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='Abbreviations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='The following abbreviations are used in this work: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='BFM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='background field method ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='BQI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='background-quantum identity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='BRST ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='Becchi-Rouet-Stora-Tyutin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='BS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='Bethe-Salpeter ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='BSE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='Bethe-Salpeter equation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='EHM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='emergent hadron mass ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='MOM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='momentum subtraction (renormalization schemes) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='PT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='pinch technique ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='QCD ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='Quantum Chromodynamics ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='QED ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='Quantum Electrodynamics ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='RGI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='renormalization group invariant ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='SDE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='Schwinger-Dyson equation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='STI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='Slavnov-Taylor identity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='WI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='Ward identity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' BQIs for the BSE amplitudes In this Appendix, we use two particular BQIs in order to relate the displacement functions C and C with their BFM counterparts �C and �C, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The ghost-gluon vertices IΓµ(r, p, q) and �IΓµ(r, p, q) are related by a BQI [14], which reads �IΓµ(r, p, q) = � [1 + G(q)]gν µ + L(q)qµqν q2 � IΓν(r, p, q) +F−1(p)pνKµν(r, q, p) + r2F−1(r)Kµ(r, q, p) , (A1) where Kµ and Kµν are two auxiliary functions, shown diagrammatically in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' A1, while G(q) and L(q) are the form factors of Λµν(q), defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Next, we decompose the �IΓµ(r, p, q) and IΓµ(r, p, q) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (A1) into their regular and pole parts, using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (33) and (52), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Note that the second and third terms in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (A1) do not contain poles in q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' this is so because Kµν(r, q, p) can contain (longitudinally coupled) poles only in the pν channel, whereas Kµ(r, q, p) has no external gluon legs, and hence no poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Particles 2023, 1 43 −gf amnKµ(r, q, p) = n µ, a p m q r −gf anmKµν(r, q, p) = gf amngµν + p ν, m n µ, a q r Figure A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The auxiliary functions Kµ(q, r, p) and Kµν(q, r, p), appearing in the BQI of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (A1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Then, multiplying Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (A1) by q2 we obtain qµ �C(r, p, q) = qµ[1 + G(q) + L(q)]C(r, p, q) + O(q2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (A2) Setting q = 0 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (A2) and using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (18), we find C(r, −r, 0) = Z1F(0) �C(r, −r, 0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (A3) Hence, using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (61), we obtain the result in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Then, expanding Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (A2) to first order in q, using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (41) for C(r, p, q) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (62) for �C(r, p, q), entails C(r) = Z1F(0) �C(r) , (A4) which is one of the main results of this Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' A relation identical to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (A4) can be obtained for C(r) and its BFM analog, �C(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The starting point of the derivation is the BQI [14] �IΓαµν(q, r, p) = � [1 + G(q)]gρ α + L(q)qαqρ q2 � IΓρµν(q, r, p) (A5) +Kρνα(r, q, p)Pρ µ(r)∆−1(r) − Kρµα(p, q, r)Pρ ν (p)∆−1(p) , where Kµνα(r, q, p) is the function defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (111).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Then, we note that the only longitudinal poles at q = 0 present in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (A5) are those contained in the IΓαµν(q, r, p) and �IΓαµν(q, r, p) vertices, with the auxiliary functions Kανρ(q, p, r) having poles only in the rµ and pν channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' As such, isolating the qαgµν/q2 pole and expanding around q = 0, one eventually finds �C1(0, r, −r) = Z−1 1 F−1(0)C1(0, r, −r) = 0 , (A6) and C(r) = Z1F(0)�C(r) , (A7) where �C1(q, r, p) and �C(r2) are defined in analogy to the Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (37) and (41), and we used Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' (39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Marciano, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' ;' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Empirical Consequences of Emergent Mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Symmetry 2020, 12, 1468, [arXiv:hep- ph/2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='04011].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='3390/sym12091468.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Kaon and pion parton distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' C 2020, 80, 1064.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' https: //doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Binosi, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Chang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' De Soto, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Roberts, C.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Kizilersu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Oliveira, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Silva, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' ;' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Rodríguez-Quintero, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Zafeiropoulos, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Discretization effects on renormalized gauge-field Green’s functions, scale setting, and the gluon mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Rev.' metadata={'source': 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Quantum electrodynamics at small distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 1954, 95, 1300– 1312.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='1103/PhysRev.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Quantum Field Theory;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' International Series in Pure and Applied Physics, New York, USA: Mcgraw-Hill (1980) 705 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=', 1980.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 135.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Nambu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Jona-Lasinio, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Dynamical model of elementary particles based on an analogy with superconductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 1961, 122, 345–358.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Chiral Symmetry Breaking in QCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Running Coupling Constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} 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200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Kallosh, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The Renormalization in Nonabelian Gauge Theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' B 1974, 78, 293–312.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Renormalization of Nonabelian Gauge Theories in a Background Field Gauge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Green Functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' D 1975, 12, 482–488.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='12 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='482.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Arefeva, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Faddeev, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Slavnov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Generating Functional for the s Matrix in Gauge Theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Teor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Fiz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 1974, 21, 311–321.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='1007/BF01038094.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Abbott, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The Background Field Method Beyond One Loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' B 1981, 185, 189–203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='1016/0550-3213(81)90371-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Weinberg, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Effective Gauge Theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' B 1980, 91, 51–55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='1016/0370 2693(80)90660-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Abbott, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Introduction to the Background Field Method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Acta Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Polon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 1982, B13, 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Shore, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Symmetry Restoration and the Background Field Method in Gauge Theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Annals Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 1981, 137, 262.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='1016/0003-4916(81)90198-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Abbott, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Grisaru, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Schaefer, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' The Background Field Method and the S Matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' B 1983, 229, 372–380.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='1016/0550-3213(83)90337-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Taylor, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Ward Identities and Charge Renormalization of the Yang-Mills Field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' B 1971, 33, 436–444.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='1016/0550-3213(71)90297-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Slavnov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Ward Identities in Gauge Theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' 1972, 10, 99–107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content=' https: //doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE0T4oBgHgl3EQfZAA9/content/2301.02314v1.pdf'} +page_content='org/10.' metadata={'source': 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El Bourakadi1,∗ M. Koussour1,† G. Otalora2,‡ M. Bennai1,§ and Taoufik Ouali3¶ +1Quantum Physics and Magnetism Team, LPMC, +Faculty of Science Ben M’sik, +Casablanca Hassan II University, Morocco. +2Departamento de F´ısica, Facultad de Ciencias, +Universidad de Tarapac´a, Casilla 7-D, Arica, Chile. +and +3Laboratory of Physics of Matter and Radiations, +Mohammed first University, BP 717, Oujda, Morocco. +(Dated: January 11, 2023) +In this study, we investigate the consequence of the constant-roll condition and examine the role +of f(Q, T ) gravity in the cosmological inflation process. We analyze the inflationary scenario by +calculating modified Friedmann equations, and giving an alternative technique that enables relating +modified slow-roll parameters to the constant roll parameter β. +Considering both chaotic and +hilltop models, we calculate the spectral index and the tensor-to-scalar ratio and compare their +compatibility with Planck’s data for different choices of the constant roll parameter β. We examine +the evolution of primordial black holes in our chosen modified gravity model taking into account +the accretion process and the evaporation due to Hawking radiation. We compute the evaporation +and accretion masses rate and provide an analytic estimation of the primordial black holes masse +and of the radiation in the f(Q, T ) gravity model. +I. +INTRODUCTION +Inflationary cosmology, or simply inflation, is the current widely accepted paradigm for explaining the physics +of very early Universe. In this theory a quasi-exponential accelerating phase before the radiation decelerating era +is proposed to solve the several long-standing puzzles of the Hot Big-Bang standard cosmological model [1–4]. In +addition, the most fascinating feature of inflation is that it provides a mechanism to explain the origin of the Cosmic +Microwave Background (CMB) temperature anisotropies and the Large-Scale Structure (LSS) [6], and the subsequent +phases at which matter appeared for the first time in the Universe [5]. The mechanism of inflation is based on the +generation of small quantum fluctuations in the inflaton field, which is the field driving the accelerated expansion. +These quantum fluctuations are amplified in physical scale during inflation, leading to a Gaussian, scale-invariant +and adiabatic primordial density perturbations [6, 7]. This information is encoded into the primordial scalar power +spectrum Ps and its scale dependence is characterized by the scalar spectral index ns, which is tightly constrained +by the latest Planck data [8] to be ns = 0.9649 ± 0.0042 (at 68% C.L.). Furthermore, inflation also predicts the +generation of tensor perturbations as a background of primordial gravitational waves (PGWs) [9–11]. In this case +the amplitude of the primordial tensor power spectrum Pt can be parameterized in terms of the tensor-to-scalar ratio +r ≡ Pt/Ps [12]. Although no primordial gravitational waves have been detected so far, current observations give us +an upper bound on r. New data from BICEP/Keck 2021 [13] have been published, leading to a considerably stronger +upper bound on r: r0.05 = 0.014+0.010 +−0.011 (r0.05 < 0.036 (at 95% C.L.)), in comparison to Planck 2018 data [8]. +In general, the dynamics of inflation is based on the slow-roll approximation. The scalar potential is chosen to be +flat such that the scalar field slowly rolls down the potential [6]. So, the equation of the mode functions associated to +quantum fluctuations can be written as a Bessel equation of order ν, and ν is approximated to the first order of the +Hubble slow-roll parameters ǫH ≪ 1 and ηH ≪ 1. Thus, the mode functions and the corresponding power spectra are +obtained by applying the so-called Bessel function approximation [14]. However, another alternative that has already +been considered in the literature is the case of ultra-slow roll inflation [15–17]. The scalar potential is assumed to +be extremely flat, and then the Klein-Gordon equation of motion of the homogeneous inflaton field is simplified by +neglecting the slope term. Therefore, the friction term is now locked with the acceleration term. Furthermore, one +has that ηH ≈ 3 and then the slow-roll approximation is no longer valid. For instance, the mechanism to generate +∗ k.elbourakadi@yahoo.com +† pr.mouhssine@gmail.com +‡ giovanni.otalora@academicos.uta.cl +§ mdbennai@yahoo.fr +¶ t.ouali@ump.ac.ma + +2 +primordial black holes within the framework of ultra-slow roll inflation has been investigated in Refs. [100, 101]. Ultra- +slow roll inflation has also been generalized to constant-roll inflation such that the slow-roll parameter ηH becomes a +constant [113, 114] (see also Refs. [18–23]). Interestingly enough, constant-roll inflation differs from slow-roll inflation +due to its new dynamical features that result in richer physics. It gives rise to a large local non-Gaussianity and the +curvature perturbation may grow on the super-horizon scales [24, 113, 114]. +Primordial Black Holes (PBHs) [25] started to be considered seriously after Hawking proposed that PBHs could +be formed with Planck-order masses in the early Universe [26]. +In this direction one would consider that PBHs +formation could be a result of inflation [27] inhomogeneities from the early Universe [28, 29] or the phase transition +[30]. Furthermore, it was first proposed that PBHs could be a candidate of dark matter (DM) in [31] and reconsidered +in [32, 33] when black holes mergers were first detected by LIGO [34]. Hawking found that black holes release thermal +radiation [35]. As a result, depending on their mass, black holes evaporate. The faster the PBHs evaporate, the +smaller their masses. A black hole’s density, on the other hand, varies inversely with its mass. Such high densities +are required for the formation of lighter black holes, and such densities are only possible in the early Universe. +Primordial black holes are thus the only black holes with masses small enough to have vanished by now. A number +of observations restrict the mass ranges of PBHs, for instance, PBHs masses bounded as ≲ 10M⊙ are excluded +from null microlensing searches [36, 37], while masses ≳ 102M⊙ are excluded from wide binary surveys [38, 39]. +The essential assumption behind these constraints is that in the early Universe, PBHs accrete primordial gas and +subsequently convert a percentage of the accreted mass to radiation. The ensuing infusion of energy into the primordial +plasma then influences its thermal and ionization processes, causing anomalies in the CMB’s frequency spectrum and +polarization power spectra [40, 41]. +In General Relativity (GR), the gravitational interaction is described in terms of the curvature associated with +the Levi-Civita Connection [42]. Moreover, it is well-known that gravity can also be described in terms of torsion +in the context of the so-called Teleparallel Equivalent of GR, or simply Teleparallel Gravity (TG) [43–50]. TG is +a gauge theory for the translation group in which the dynamical variable is the tetrad field rather than the metric +tensor and the torsion is associated to the Weitzenb¨ock connection that substitutes the Levi-Civita connection [51– +54]. Additionally, the Lagrangian density of TG is proportional to the torsion scalar T , which is equivalent to the +curvature scalar R up to a total derivative term. Therefore, GR and TG are equivalent at the level of field equations +[53, 55]. In all the cases, the equivalence with GR is only guaranteed for a gravitational action linear in torsion or +decoupled from other fields. On the other hand, as a natural extension to modified gravity, and similarly to f(R) +gravity [62–66], the Lagrangian of TG may be promoted to an arbitrary function f(T ) [56–58] (see also Ref [59] and +references therein). Also, following the perspective of scalar-tensor gravity theories [62, 63, 67], we can further extend +the above theory to f(T, φ) gravity [60, 61]. These novel modified gravity theories have shown a rich structure by +explaining the dynamics of dark energy [68–71, 74] and inflation [61, 72, 73]. +Thus, in the context of GR, without considering any modification to gravity, one can see that there exist at +least two equivalent descriptions. The curvature representation, in which the torsion and nonmetricity vanish, and +the teleparallel representation, in which the curvature and nonmetricity vanish. +But, recently a third equivalent +representation has also been proposed. In this approach, called Symmetry Teleparallel Gravity (STG), the curvature +and torsion vanish, and then the gravitational interaction is described by the nonmetricity Q of the metric [75]. +STG has also been further developed to tackle modify gravity in the context of f(Q) gravity theory, also known as +nonmetric gravity [76]. Recently, interest in this theory has increased rapidly due to its novel geometrical and physical +features [77–87], f(Q) gravity is also used to explain cosmic acceleration [88–92]. Moreover, another proposed theory +of modified gravity based on nonmetricity is f(Q, T ) gravity [93], where T is the trace of the matter energy-momentum +tensor. Thus, it is an extension of f(Q) gravity where a direct coupling between nonmetricity and matter has been +assumed. In fact, one can verify that the divergence of the matter energy-momentum tensor does not vanish by +taking the covariant derivative of the modified field equations. Several different aspects of f(Q, T ) gravity have been +studied, among these we have observational constraints [94], energy conditions [95, 96], cosmological solutions [97], +phase-space dynamics [98], and cosmological inflation [99]. +The plan of the paper is the following: In Section II, we present an introduction to f (Q, T ) gravity. In Section III, +we study the dynamics of constant-roll inflation within the context of f(Q, T ) gravity. In Section IV, we investigate +the generation of primordial black holes in f(Q, T ) gravity. Finally, in Section V, we summarize the results obtained. +II. +BASIC FORMALISM IN f (Q, T ) GRAVITY +In the so-called Weyl-Cartan geometry, gravitational effects are caused not only by a variation of the direction of a +vector in parallel transport, but also by a variation in its length. The variation in the length of a vector is described +geometrically by what is called non-metricity, and it is mathematically described as the covariant derivative of the +metric tensor which is a generalization of the gravitational potential. In this case, general connexion Σα +µν is written + +3 +in terms of all contributions as curvature, torsion and non-metricity of spacetime [102], +Σα +µν = Γα +µν + Cα +µν + Lα +µν, +(2.1) +where Γα +µν is the famous Levi-Civita connection of the metric tensor gµν in GR given by, +Γα +µν ≡ 1 +2gαλ(gµλ,ν + gλν,µ − gµν,λ), +(2.2) +here Cα +µν represents the Contortion tensor expressed as, +Cα +µν ≡ 1 +2( ˜T α +µν + ˜T α +µ ν + ˜T α +ν µ) = −Cα +νµ, +(2.3) +and Lα +µν is the Disformation tensor given by, +Lα +µν ≡ 1 +2(Qα +µν − Q α +µ ν − Q α +ν µ) = Lα +νµ, +(2.4) +where ˜T α +µν and Qαµν in Eqs. (2.3) and (2.4) are the torsion tensor and the non-metricity tensor, respectively. In +non-flat space-time, the geodesic structure is designated by the connection form. +In addition, in Einstein’s GR, +the presumption of a non-torsion and metric compatible connection leads to the so-called Levi-Civita connection, +connected to the metric tensor and its first derivatives. It is possible to define two tensors of order 3 related to the +antisymmetric part of Σα +µν and the covariant derivative of the metric tensor as, +˜T α +µν ≡ 2Σα +[µν], +(2.5) +and +Qαµν ≡ ∇αgµν = ∂αgµν − gνσΣσµα − gσµΣσνα ̸= 0. +(2.6) +Depending on the proposed connection, different theories of gravity that can be extracted. In this work, we will +consider the modified Einstein-Hilbert action in f(Q, T ) extended symmetric teleparallel gravity expressed as [93, 103], +S = +ˆ √−gd4x +� 1 +2κf(Q, T ) + Lm +� +, +(2.7) +where f(Q, T ) is a general function of the non-metricity scalar Q and the trace of the energy-momentum tensor T , +κ = 1/M 2 +p, g is the determinant of the metric tensor gµν i.e. g = det (gµν), and Lm is the Lagrangian function for the +matter fields. The non-metricity scalar Q is defined as, +Q ≡ −gµν(Lβ +αµLα +νβ − Lβ +αβLα +µν). +(2.8) +The trace of the non-metricity tensor is obtained as, +Qβ = gµνQβµν +�Qβ = gµνQµβν. +(2.9) +Further, the superpotential tensor P β +µν (or non-metricity conjugate) is related to the non-metricity scalar Q as, +P β +µν = −1 +2Lβ +µν + 1 +4(Qβ − �Qβ)gµν − 1 +4δβ +(µQν). +(2.10) +By using the definition above, the non-metricity scalar in terms of the superpotential tensor is given by, +Q = −QβµνP βµν. +(2.11) +As usual, we define of the energy-momentum tensor of the matter fields by +Tµν = − +2 +√−g +δ(√−gLm) +δgµν +, +(2.12) +and +Θµν = gαβ δTαβ +δgµν . +(2.13) + +4 +So that the variation of energy-momentum tensor with respect to the metric tensor gµν read as, +δ (g µν T µν) +δ g α β += T αβ + Θ α β. +(2.14) +Now, varying the gravitational action (2.7) with respect to metric tensor gµν, the fleld equations of f(Q, T ) gravity +can be derived as, +− +2 +√−g ∇β +� +fQ +√−gP β +µν +� +− 1 +2fgµν + fT (Tµν + Θµν) − fQ(PµβαQ βα +ν +− 2Qβα +µ Pβαν) = κTµν. +(2.15) +Here, fQ = df(Q,T ) +dQ +, fT = df(Q,T ) +dT +, and ∇β denotes the covariant derivative. Now, we consider a homogeneous and +spatially flat Friedmann-Lemaitre-Robertson-Walker (FLRW) metric described by the line element +ds2 = −N 2 (t) dt2 + a2(t) +� +dx2 + dy2 + dz2� +, +(2.16) +where a (t) is the scale factor of the Universe, depending only on the cosmic time t (which is scaled to be unity at +the present time, i.e. a0 = 1) and N (t) is the lapse function regarded to be 1 in the standard model. The rates of +expansion and dilation are fixed as H ≡ +.a +a and D ≡ +. +N +N , respectively. Hence, the corresponding non-metricity scalar +is given by Q = 6 H2 +N 2 . In our current analysis, we suppose that the content of the Universe as a perfect non-viscosity +fluid for which the energy-momentum tensor is given by +T µ +ν = diag (−ρ, p, p, p), +(2.17) +where p is the perfect non-viscosity fluid pressure and ρ is the energy density of the Universe. Hence, for the tensor +Θµ +ν the expression is obtained as Θµ +ν = diag (2ρ + p, −p, −p, −p). Taking into account the case as N = 1, the Einstein +field equations using the line element (2.16) are given by, +κρ = f +2 − 6FH2 − +2E +1 + E +� . +FH + F +. +H +� +(2.18) +and +κp = −f +2 + 6FH2 + 2 +� . +FH + F +. +H +� +. +(2.19) +where we used Q = 6H2 and (·) represents a derivative with respect to cosmic time (t). In this case, F ≡ fQ and +κE ≡ fT represent differentiation with respect to Q and T respectively. The evolution equation for the Hubble +parameter H can be derived by combining Eqs. (2.18) and (2.19) as, +. +H + +. +F +F H = κ +2F (1 + E) (ρ + p) . +(2.20) +Einstein’s field equations (2.18) and (2.19) can be regarded as extended symmetric teleparallel equivalents to +standard Friedmann’s equations with supplementary terms from the non-metricity of space-time and the trace of the +energy-momentum tensor T which behaves as an effective component. Therefore, the effective energy density ρeff +and effective pressure peff are determined by, +3H2 = κρeff = f +4F − κ +2F [(1 + E) ρ + Ep] , +(2.21) +2 ˙H + 3H2 = −κpeff = f +4F − 2 ˙FH +F ++ κ +2F [(1 + E) ρ + (2 + E) p] . +(2.22) +Taking into account Eqs. (2.20) and (2.21) one gets +ρ = +f − 12H2F +2κ [(1 + ω) E + 1]. +(2.23) + +5 +III. +INFLATIONARY SCENARIO IN f(Q, T ) GRAVITY +A. +Slow-roll and constant-roll in the standard cosmology +The simple inflation model is described with an isotropic and homogeneous scalar field known as the inflaton. The +dynamics of such inflaton field can be defined via the Lagrangian +Lm = −1 +2gµν∂µφ∂νφ − V (φ). +(3.1) +From the Lagrangian of the scalar field φ the energy-momentum tensor is defined as +Tµν ≡ − +2 +√−g +δ (√−gLm) +δgµν += gµνLm − 2 ∂Lm +∂gµν . +(3.2) +Considering the inflaton field as a perfect fluid with an equation of state ω = pφ/ρφ, knowing that ρφ and pφ are the +energy density and the pressure of the inflaton field, the energy-momentum tensor in Eq. 3.2 gives +ρφ = 1 +2 +˙φ2 + V (φ), +(3.3) +and +pφ = 1 +2 +˙φ2 − V (φ). +(3.4) +Here the dot represents the derivative with respect to the cosmic time. The equation of state parameter is then +written as +ω = pφ +ρφ += +1 +2 ˙φ2 − V (φ) +1 +2 ˙φ2 + V (φ) +. +(3.5) +Taking into account the Einstein field equations in addition to Eq. (3.3) and (3.4), the Friedmann equations are easily +obtained as +H2 = κ +3 +� ˙φ2 +2 + V (φ) +� +, +(3.6) +H2 + ˙H = −κ +3 +� +˙φ2 − V (φ) +� +, +(3.7) +˙H = −κ +2 +˙φ2, +(3.8) +here H ≡ ˙a/a is the Hubble parameter. The so called Klein-Gordon (KG) equation, by taking the time derivative of +Eq. (3.6) and by taking into considering Eq. (3.8), can be obtained +¨φ + 3H ˙φ + V ′ = 0, +(3.9) +here the prime denotes the derivative with respect to the φ-field. The accelerated expansion of the inflationary phase +must last for enough time for a successful period where the Hubble radius decreased over time. For this, the slow-roll +parameters are defined as [105] +ǫ = − +˙H +H2 , +(3.10) +η = +˙ǫ +Hǫ ≈ − +¨H +2 ˙HH +. +(3.11) +An additional parameter used to study the period of inflation is the e-folding number which describes the rate of the +Univers expansion during this phase [107, 108] +N ≡ ln +�aend +a +� += +ˆ tend +t +Hdt, +(3.12) + +6 +the index ”end” denotes the time when inflation ended. Inflation will continue as long as ǫ < 1 to solve the standard +cosmological problems. However, at the end of inflation, the slow-roll parameter must reach ǫ = 1. The slow roll +parameters and the curvature perturbations are related by the spectral index ns, and the ratio of tensor to scalar +perturbations r, in the following way [109] +ns − 1 = −6ǫ + 2η, +(3.13) +r = 16ǫ. +(3.14) +The Planck data provides strong constraints on ns and r parameters. In fact, any inflationary model predict such +parameters can be tested to decide whether it can be ruled out or not [110, 111]. In addition to the slow roll conditions, +one may consider a type of constant roll of the scalar field, which is expressed as follows +¨φ +˙φ += βH. +(3.15) +In Ref. [112] they considered two slow-roll stages, separated by a constant-roll stage that imprints an alternative +dynamic to the scalar field [113], where the constant β determines the deviation from a flat potential. When β ≃ 0, +the slow roll inflation is recovered, whereas β = 0 corresponds to the “ultra slow roll” inflation [114, 115]. +In order to construct the idea of constant roll inflation this paper aims to use the condition ǫ ≪ 1 that leads to +˙φ2 ≪ V (φ). +(3.16) +which simplifies Eq. (3.11) as +η ≈ − +���¨φ +��� +H +��� ˙φ +��� +. +However, for a successful constant roll scenario, we consider the expression ¨φ = ˙φβH that simplifies the KG equation +as +˙φH (3 + β) + V ′ = 0. +(3.17) +In the next section, we will investigate the effect of the constant roll stage on the inflationary parameters by deriving +H, ˙H, ˙φ and ¨φ from an f(Q, T ) gravity perspective. +B. +Inflation in f(Q, T ) Gravity +In the context of cosmic inflation, we consider the model f(Q, T ) = αQ+σT [93], with α = F ̸= 0, σ is an arbitrary +constant, and T = −ρ + 3p. It is crucial to keep in mind that σ = 0 and α = −1 is equivalent to the case of the GR +theory. Furthermore, the theory is reduced to f(Q, T ) = −Q for α = −1 at σ ̸= 0 and T = 0 which is equivalent to +GR as well. By taking Eqs. (2.20), (2.21), (2.22) and (2.23), we obtain the following equations +˙H = (κ + σ) ρ (1 + ω) +2α +, +(3.18) +3H2 = κρeff = [(ω − 3) σ − 2κ] ρ +2α +, +(3.19) +2 ˙H + 3H2 = −κpeff = ρ [(3ω − 1) σ + 2ωκ] +2α +, +(3.20) +ρ = +−6αH2 +σ (3 − ω) + 2κ. +(3.21) +Aside from the equation of state that corresponds to the inflationary phase, we can derive an effective equation of +state for this model as +ωeff = peff +ρeff += −1 − 2 (κ + σ) (1 + ω) +(ω − 3) σ − 2κ . +(3.22) + +7 +It is clear that from the effective equation of state above, we can have an accelerated de Sitter expansion as long +as ((ω − 3) σ − 2κ ̸= 0) and (ω = −1) . Considering the trace of the energy-momentum tensor as T = ˙φ2 − 4V , the +cosmological inflation in the context of f(Q, T ) gravity, from Eqs. (3.1), Eqs.(2.21) and (2.22) one obtains +ρeff = −(κ + σ) ˙φ2 + 2V (φ) (κ + 2σ) +2κα +, +(3.23) +peff = −(κ + σ) ˙φ2 − 2V (φ) (κ + 2σ) +2κα +. +(3.24) +On the other hand, the modified Friedmann equations can be given as +3 +2H2 = −(κ + σ) ˙φ2 + 2V (φ) (κ + 2σ) +4α +, +(3.25) +2 ˙H + 3 +2H2 = 3 (κ + σ) ˙φ2 − 2V (φ) (κ + 2σ) +4α +. +(3.26) +Taking into account the previous two equations one gets +˙H = +˙φ2 +2α (κ + σ) , +(3.27) +the effective equation of state parameter can be obtained as +ωeff = −(κ + σ) ˙φ2 − 2V (φ) (κ + 2σ) +(κ + σ) ˙φ2 + 2V (φ) (κ + 2σ) +. +(3.28) +The derivation of Eq.(3.25) and using Eq. (3.27), we obtain the modified equation +¨φ (κ + σ) + 3H ˙φ (κ + σ) + V ′ (κ + 2σ) = 0. +(3.29) +From the constant roll condition ¨φ = βH ˙φ, we obtain +˙φ = − +V ′ (κ + 2σ) +H (β + 3) (κ + σ). +(3.30) +However, one may consider the case where the bound (κ + σ) ˙φ2 ≪ V (φ) (κ + 2σ) takes place and from Eq.(3.25), +Eq.(3.10) and Eq.(3.27) to obtain the following slow roll parameter +ǫ ≈ 3 (κ + σ) ˙φ2 +2 (κ + 2σ) V = +−9α +2 (β + 3)2 (κ + σ) +�V ′ +V +�2 +, +(3.31) +deriving Eq.(3.27) and using Eq.(3.11), the second slow roll parameter can simply be given as +η ≈ − +¨H +2 ˙HH += − +���¨φ +��� +H +��� ˙φ +��� += +−3α +(β + 3) (κ + σ) +�V ′′ +V +� +. +(3.32) +Eqs. (3.31) and (3.32) are obtained by assuming that the constant roll stage succeeds the slow roll one. +C. +Chaotic potential +Let us consider the simplest model known as chaotic potential, given by the form +V (φ) = 1 +2m2φ2, +(3.33) +the parameters ǫ and η are expressed as a function of the constant roll parameter β in the following way +ǫ = 3 +2 +1 − ns +6 − β , +(3.34) +η = +�3 + β +6 − β +� 1 − ns +2 +, +(3.35) + +8 +here we should note that (6 − β) ̸= 0. To obtain Eqs. (3.31) and (3.32), we should develop a little bit the transition +from slow roll stage to a constant roll one, for the chaotic potential we see that the behavior of the inflationary phase +is determined through the constant roll parameter β. +FIG. 1. r as a function of ns for a chaotic potential in modified symmetric teleparallel gravity, light and dark blue regions are +constraints in combination with CMB lensing reconstruction and BAO from Planck data. on the right panel we choose three +different values of β, the black line for β = 0, the red line represents β = 0.5, and the blue line for β = 1. While on the left +one, We choose three different values of β, the black line for β = −0.5, the red line represents β = −1.5, and the blue line for +β = −2.5. +Fig. 1 show the decreasing behavior of the tensor-to-scalar ratio, r, with respect to the spectral index, ns for the +chaotic potential in our chosen f(Q, T ) gravity model. The results show a good consistency for a specific range of the +constant roll parameter β with the latest observations from Planck data. Furthermore, the case with β > 0 provides +inconsistent results with Planck’s data. However, the constant roll parameter must be bounded as β ≤ 0 in order to +produce consistent observational parameters with recent results. +D. +Hilltop inflation +Hilltop inflation is classified among small-field models. This potential naturally involves eternal inflation, that raises +the question of initial conditions, which is a problem in most inflation models [116]. Hilltop potential is given by [117] +V (φ) = M 4 +� +1 − +�φ +µ +�p� +, +(3.36) +this model of inflation has two free parameters M and µ plus the parameter p that will be given specific values at the +end of this section. The e-folds number during inflation using Eqs. (3.30) and (3.25) is given by +N ≡ +ˆ φend +φk +H +˙φ +dφ ≈ −(β + 3) (κ + σ) +3α +ˆ φk +φend +V +V ′ dφ, +(3.37) +considering Φ = φ/µ and p > 2, the expression of inflationary e-folds number is given by +N = −µ2 +2p +(β + 3) (κ + σ) +3α +� +Φ2 +k − Φ2 +end + +2 +p − 2Φ2−p +k +− +2 +p − 2Φ2−p +end +� +, + +rn +STT,TE,EE+IOWE ++ lensing+BK15 ++BAO +TT,TE,EE+lowE ++ lensing +Nj= 50 +N; = 609 +here the subscripts ”k” and ”end” mean the time the pivot scale crossed outside the horizon and the end of inflation +respectively. Since the slow roll parameters are defined by the value of the field at the horizon crossing Φk, for this +model ǫ and η are calculated as follows +ǫ = − +9α +2(β + 3)2 (κ + σ) +p2 +µ2 +Φ2(p−1) +k +(1 − Φp +k)2 , +(3.38) +η = +3α +(β + 3) (κ + σ) +p (p − 1) +µ2 +Φp−2 +k +1 − Φp +k +. +(3.39) +Furthermore, we can compute the tensor-to-scalar ratio as a function of the inflationary e-folds N, the constant-roll +parameter β for p > 2 in the following way, +r = +4p +(β + 3) +1 +N . +(3.40) +Taking into consideration Eq. (3.14), we can study the behavior of r as a function of the spectral index ns, β and p +parameters using the first slow roll parameter ǫ given as +ǫ = +3p (1 − ns) +18p − 4 (p − 1) (β + 3), +(3.41) +η = 2 (p − 1) (β + 3) (1 − ns) +−18p + 4 (p − 1) (β + 3) . +(3.42) +FIG. 2. r as a function of ns for the hilltop potential in modified symmetric teleparallel gravity, light and dark blue regions +are constraints in combination with CMB lensing reconstruction and BAO from Planck data. We choose three different values +of p, the black line for p = 3, the red line represents p = 4 and the blue line for p = 5. +In Fig. 2, we consider the hilltop inflation, where the inflationary parameters are linked to f(Q, T ) gravity. It is +apparent from the calculations that the constant roll parameter has negligible effects on the (r, ns) behavior, the plot +shows that increasing the value of p makes the decreasing function obtained from our model consistent with the latest +observations provided by Planck’s results. +Based on the previously established equation Eq.(3.40), we provide an examination of the tensor-to-scalar ratio +taking into account different inflation durations that varies from 50 to 65 e-folds. In addition to an increasing values +of the constant roll β parameter, the tensor-to-scalar ratio r can be compatible with the observational bound for +specific values of p and N. Furthermore, the constant roll parameter and the inflationary e-folds provide compatible +results only for higher values since they are inversely proportional to r. +Now as we construct a constant roll model for the f(Q, T ) gravity in the context of chaotic and hilltop inflation, we +will study the evolution of primordial black holes that were supposed to occur just after the inflation period taking +into account our f(Q, T ) gravity model. + +rn +STT,TE,EE+IOWE ++ lensing+BK15 ++BAO +TT,TE,EE+lowE ++ lensing +Nj= 50 +N; = 6010 +β +p N +r +−1.5 6 50 0.32 +−1.1 5 55 0.19 +−0.5 4 60 0.10 +0.7 +3 65 0.049 +TABLE I. Testing the tensor-to-scalar ratio for different parameters for the hilltop inflation. +IV. +PRIMORDIAL BLACK HOLES EVOLUTION IN f(Q, T ) GRAVITY +A. +Rotating and non-rotating Primordial Black Holes +Primordial black holes abundance is determined by the primordial power spectrum. In fact, PBHs were formed as a +result of amplification in the primordial power spectrum on small scales at the time when primordial inhomogeneities +re-enter the Hubble horizon in a radiation-dominated Universe era. Some regions with a significant positive curvature +which are considered equivalent to a closed Universe will collapse into a black hole [119]. At first, black holes were +considered to be eternal and evolved with an increasing mass by absorbing more matter or even other stars and +BHs. However, studying their quantum properties shows the possibility of emitting particles with a thermal spectrum +related to BHs surface gravity [35, 120]. In this process of emitting particles, BHs lose mass and angular momentum +with different properties depending on the specific characteristics of BHs. In this direction, we will discuss the thermal +properties of evaporating PBHs for two cases namely the Schwarzschild and Kerr PBHs. +As an analogy to non-rotating Schwarzschild black holes, we consider a PBH with mass MBH, the thermal spectrum +of emitted particles from the evaporation process has the following expression [121] +TBH = +1 +8πGMBH +∼ 1.06 +�1010kg +MBH +� +GeV, +(4.1) +Another option is that the evaporating BHs have some angular momentum, such spinning BHs, also known as Kerr +BHs, might have originated with an initial spin or acquired their angular momenta by a variety of events, such as +mergers [122–124]. A PBH temperature for the Kerr scenario can be modified due to their spinning and it’s given as +[125] +TBH = +1 +4πGMBH +� +1 − a2∗ +1 + +� +1 − a2∗ +, +(4.2) +where a∗ is a dimensionless parameter bounded by the interval 0 ≤ a∗ ≤ 1. The primary distinction between Kerr +(a∗ ̸= 0) and Schwarzschild (a∗ = 0) BHs is that Kerr BHs are axially symmetric rather than spherically symmetric. +When a∗ grows, the emission of particles with angular momentum spinning in the same direction as the BH is +enhanced [126]. For the case a∗ = 1 a BH would have TBH = 0, this is traditionally prohibited in any statistical +system. Furthermore, its horizon would have vanished revealing a bare space-time singularity and contradicting the +Cosmic Censorship Conjecture [126–128]. +B. +Primordial Black Holes evolution +Since we are interested in studying the rate of mass loss from PBH, we recall the process which reduces the mass +of the black hole due to Hawking evaporation which is defined by [129, 130] +�dM +dt +� +eva += −ℏc4 +G2 +α′ +M 2 +BH +. +(4.3) +where G is Newton’s gravitational constant, ℏ is the Planck constant, c is light speed, α′ is the spin parameter of +the emitting particles, and the black hole radius is given by rBH = 2GM. After integration Eq. (4.3) we obtain the +evolution of PBH mass as +Meva = Mi +� +1 − +t +teva +� 1 +3 +, +(4.4) + +11 +we can define the Hawking evaporation time scale teva as +teva = G2 +ℏc4 +M 3 +i +3α′ , +(4.5) +according to Ref. [131] fine-tuning the initial PBH masse Mi along with the α′ parameter would be a good method to +probe the early Universe. On the other hand, additional contributions suggest that the evaporation of PBHs would +have interesting implications on the CMB and the standard cosmological nucleosynthesis scenario [132, 133]. +FIG. 3. On the left, the evolution of the primordial black hole reduced the mass ratio to the initial mass Meva/Mi as functions +of time ratio t/teva is plotted. On the right, the evaporation time of primordial black holes as a function of the initial mass is +presented, and the plot in the left corner shows the variation of teva for higher values of Mi. +In Fig. 3 we test the evolution of the primordial black holes evaporation and initial mass ratio Meva/Mi with +respect to the ratio t/teva, and we plotted the variation of the evaporation time parameter teva as functions of the +initial mass of primordial black holes Mi. Our results show that the evaporation mass can simply decrease as we move +forward in time. However, one must study the effect of the initial primordial black holes masses on the time that +must take in order to completely evaporate. In this direction, the second plot indicates that as we increase the initial +mass of PBHs we need more time to achieve a complete evaporation process. In fact, an initial mass in the order of +Mi ≳ 1013kg needs time more than the current age of our Universe to evaporate, as a result, we must consider lower +bounds on PBHs initial mass for fine consistency with cosmic history. +The accretion of cosmic fluid surrounding the black hole will prolong the evaporation of the primordial black holes. +Therefore, it is necessary to add a mass accretion rate which for a cosmic fluid with ρeff and peff will be given as +�dM +dt +� +accr += 16πG2 +c5 +M 2 (ρeff + peff) , +(4.6) +which can be integrated to be in the final form [134] +Maccr = Mi +� +1 − +t +taccr +�−1 +, +(4.7) +where from Eqs. (3.19) and (3.20) the accretion time scale taccr is defined as +taccr = +�16πG2 +c5 +Mi (ρeff + peff) +�−1 +. +(4.8) +Additionally, we can write the following equation + +t +tevaMeva +MiMi12 +ρeff + peff = 3 +κH2 +�2 (σ + κ) (1 + ω) +2κ − (ω − 3) σ +� +(4.9) +Here, at a radiation-dominated era, and using the fact that the Hubble parameter is given by H = ˙a +a, we can simply +choose a(t) ∝ t1/2 and finally obtain H = 1/ (2t)2 to estimate the time at which the radiation surrounded primordial +black-holes. +FIG. 4. The accretion time taccr variation as a function of the equation of state in the interval ω ∈ [−0.3, 0.3] considering +different values of the f(Q, T ) gravity model parameter σ. The left panel provides the case for Mi ∝ 1012kg, while the right +shows the case of Mi ∝ 1014kg. +Fig. 4 illustrates the evolution of the accretion time as a function of a specific interval of the equation of state +parameter. In our study, the accretion time is the moment at which primordial black holes started the accretion +process, the variation of the accretion time shows to be minimally dependent on the choice of σ parameter for our +chosen f(Q, T ) gravity model. In fact, the behavior of accretion time increases for higher values of σ. Considering +the fact that in order for inflation to initiate the EoS must be bounded by ω > −1/3. Moreover, ω evolves toward +1/3 for inflation to be ended and the subsequent periods to take place [9, 135]. In Eq.(4.9) we choose a fixed value of +time using the Hubble parameter which represents the time of PBHs formation, following the results of [129] which +suggests that PBHs formed at t ∝ 10−23s, our results provide precise values of the time of accretion which decreases +as we consider higher values of the EoS parameter. On the other hand, for lower initial PBHs masses, the accretion +initiates sooner than in the case of higher Mi values. + +13 +FIG. 5. On the left panel, the evolution of the primordial black holes accretion mass ratio to the initial mass Maccr/Mi as +functions of the time ratio t/taccr is shown. On the right one, the accretion time of primordial black holes as a function of the +initial mass is presented for specific values on the equation of state ω parameter. +In Fig. 5 we provide the variation of the primordial black holes accretion mass ratio to the initial mass Maccr/Mi +as a function of the ratio t/taccr. From this figure we conclude that when t −→ taccr the resulting primordial black +hole mass due to the accretion process became in the order of 100Mi. On the other hand, accretion occurs faster when +we consider more significant initial PBH masses for several values of ω parameter. +The total PBHs mass evolution may now be rewritten in the following form MBH = Meva + Maccr +σ +ω +t(s) +Mi(kg) +MBH(kg) +TBH(GeV ) +1.5 κ −0.3 10−15 +106 +∼ 2 × 106 +∼ 5.3 × 103 +2 κ +−0.2 10−5 +107 +∼ 1.9 × 107 +∼ 5.3 × 102 +2.5 κ −0.1 +10 +108 +∼ 1.9 × 108 +∼ 53 +3 κ +0 +105 +109 +∼ 1.9 × 109 +∼ 5.3 +3.5 κ +0.1 +109 +1010 +∼ 1.9 × 1010 ∼ 53 × 10−2 +4 κ +0.2 +1013 +1011 +∼ 1.9 × 1011 ∼ 53 × 10−3 +4.5 κ +0.3 +1017 +1012 +∼ 1.9 × 1012 ∼ 53 × 10−4 +TABLE II. Testing the evolution of the total non-rotating PBHs masses and temperature in f(Q, T ) gravity. +σ +ω +t(s) +Mi(kg) +a∗ +MBH(kg) +TBH(GeV ) +1.5 κ −0.3 10−15 +106 +0.1 +∼ 2 × 106 +∼ 5.28 × 103 +2 κ +−0.2 10−5 +107 +0.3 +∼ 1.9 × 107 +∼ 5.17 × 102 +2.5 κ −0.1 +10 +108 +0.5 +∼ 1.9 × 108 +∼ 49.1 +3 κ +0 +105 +109 +0.7 +∼ 1.9 × 109 +∼ 4.41 +3.5 κ +0.1 +109 +1010 +0.9 ∼ 1.9 × 1010 ∼ 32.1 × 10−2 +4 κ +0.2 +1013 +1011 +0.96 ∼ 1.9 × 1011 ∼ 23.1 × 10−3 +4.5 κ +0.3 +1017 +1012 +0.99 ∼ 1.9 × 1012 ∼ 13.1 × 10−4 +TABLE III. Testing the evolution of the total rotating PBHs masses and temperature in f(Q, T ) gravity. + +t +taccrM +accr +MiMi14 +From tables (II) and (III) we study the evolution of rotating and non-rotating PBHs masses and temperature as +functions of different parameters. According to these results, we conclude that PBHs masses increase with higher +initial masses Mi. On the other hand, PBHs temperature decreases as we move forward in time. Moreover, TBH is +inversely proportional to PBHs masses which makes the temperature evolve from T eV to MeV taking into account +masses in the order of 1012kg. Finally, when we compare rotating and non-rotating black holes, we find that higher +values of the parameter a∗ can simply lead to a lower temperature for the case of rotating PBHs. +V. +CONCLUDING REMARKS +Over the past few decades, numerous studies have been conducted to examine the early and late evolution of the +Universe. The standard model of cosmology, based on general relativity (GR), has proven to be a reliable model for +describing the dynamics of the Universe. However, there are still some unresolved issues, such as the flatness and +horizon problems, that need to be addressed through further research on cosmological inflation. Additionally, while +GR has been successful in predicting the behavior of the Universe, it is unable to fully explain the influence of dark +sectors on its dynamics in a way that is consistent with observed data. As a result, it may be worthwhile to consider +alternative models of gravity such f(Q, T ) model to address these challenges. +In this study, we examined constant-roll inflation in the context of f(Q, T ) gravity. To do this, we started by +explaining the basic theory of cosmological inflation using the isotropic and homogeneous inflaton scalar field. We +then assumed a flat FLRW spacetime and an equation of state ω, we provided a new technique for studying the +constant-roll process and correlating the slow-roll equations to the constant-roll β parameter based on the modified +Friedmann equations obtained through f(Q, T ) gravity. Furthermore, we have calculated the inflationary observables, +the spectral index ns, and the tensor-to-scalar ratio r for two cases of inflationary potentials, namely the chaotic and +hiltop models. We showed that for each model, a bound on the constant-roll parameter is preferred. In the case of +chaotic inflation, for a consistent value of r and ns, β must be bounded as β ≤ 0. While for the Hiltop Inflation, +several parameters are involved to reproduce compatible values of r and ns, for instance, we must consider the bounds +p ≥ 3, N ≥ 60, and β ≥ −0.5 for a good index spectral and tensor-to-scalar ratio consistencies. +Finally, for the PBHs evolution in the context of f(Q, T ) gravity, we analyzed the accretion process and the +evaporation through Hawking radiation. From the obtained results, we conclude that both the evaporated mass and +the evaporation time are directly related to the initial mass Mi that must be bounded as Mi < 1013kg to be able to +completely evaporate currently or much earlier in the cosmic time. For the accretion process, we can summarise that +the accretion of matter and radiation are model dependent in the context of the f(Q, T ) gravity, which motivates us +to explore the PBHs evolution in the framework of modified gravity. According to the results, if we supposed that +PBHs formed at t ∝ 10−23s the PBHs mass due to the accretion can reach 100Mi taking into account that for lower +initial PBHs masses the accretion can occur faster. Lastly, we studied the PBHs evaporation taking into account +the effect of several parameters, and concluded that the Hawking temperature can simply decrease for higher initial +masses through the cosmic time for both rotating and non-rotating black holes. +[1] A. A. 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Lett. +119.6, 061301 (2017). + diff --git a/oNE2T4oBgHgl3EQfJwbV/content/tmp_files/load_file.txt b/oNE2T4oBgHgl3EQfJwbV/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..95dfdd8fc777a136eb8543a2b6265e6fc26b735f --- /dev/null +++ b/oNE2T4oBgHgl3EQfJwbV/content/tmp_files/load_file.txt @@ -0,0 +1,1509 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf,len=1508 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='03696v1 [gr-qc] 9 Jan 2023 Constant-roll and primordial black holes in f(Q, T) gravity K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' El Bourakadi1,∗ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' Koussour1,† G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' Otalora2,‡ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' Bennai1,§ and Taoufik Ouali3¶ 1Quantum Physics and Magnetism Team, LPMC, Faculty of Science Ben M’sik, Casablanca Hassan II University, Morocco.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' 2Departamento de F´ısica, Facultad de Ciencias, Universidad de Tarapac´a, Casilla 7-D, Arica, Chile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' and 3Laboratory of Physics of Matter and Radiations, Mohammed first University, BP 717, Oujda, Morocco.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' (Dated: January 11, 2023) In this study, we investigate the consequence of the constant-roll condition and examine the role of f(Q, T ) gravity in the cosmological inflation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' We analyze the inflationary scenario by calculating modified Friedmann equations, and giving an alternative technique that enables relating modified slow-roll parameters to the constant roll parameter β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' Considering both chaotic and hilltop models, we calculate the spectral index and the tensor-to-scalar ratio and compare their compatibility with Planck’s data for different choices of the constant roll parameter β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' We examine the evolution of primordial black holes in our chosen modified gravity model taking into account the accretion process and the evaporation due to Hawking radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' We compute the evaporation and accretion masses rate and provide an analytic estimation of the primordial black holes masse and of the radiation in the f(Q, T ) gravity model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' INTRODUCTION Inflationary cosmology, or simply inflation, is the current widely accepted paradigm for explaining the physics of very early Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' In this theory a quasi-exponential accelerating phase before the radiation decelerating era is proposed to solve the several long-standing puzzles of the Hot Big-Bang standard cosmological model [1–4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' In addition, the most fascinating feature of inflation is that it provides a mechanism to explain the origin of the Cosmic Microwave Background (CMB) temperature anisotropies and the Large-Scale Structure (LSS) [6], and the subsequent phases at which matter appeared for the first time in the Universe [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' The mechanism of inflation is based on the generation of small quantum fluctuations in the inflaton field, which is the field driving the accelerated expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' These quantum fluctuations are amplified in physical scale during inflation, leading to a Gaussian, scale-invariant and adiabatic primordial density perturbations [6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' This information is encoded into the primordial scalar power spectrum Ps and its scale dependence is characterized by the scalar spectral index ns, which is tightly constrained by the latest Planck data [8] to be ns = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='9649 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='0042 (at 68% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' Furthermore, inflation also predicts the generation of tensor perturbations as a background of primordial gravitational waves (PGWs) [9–11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' In this case the amplitude of the primordial tensor power spectrum Pt can be parameterized in terms of the tensor-to-scalar ratio r ≡ Pt/Ps [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' Although no primordial gravitational waves have been detected so far, current observations give us an upper bound on r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' New data from BICEP/Keck 2021 [13] have been published, leading to a considerably stronger upper bound on r: r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='05 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='014+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='010 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='011 (r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='05 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='036 (at 95% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' )), in comparison to Planck 2018 data [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' In general, the dynamics of inflation is based on the slow-roll approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' The scalar potential is chosen to be flat such that the scalar field slowly rolls down the potential [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' So, the equation of the mode functions associated to quantum fluctuations can be written as a Bessel equation of order ν, and ν is approximated to the first order of the Hubble slow-roll parameters ǫH ≪ 1 and ηH ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' Thus, the mode functions and the corresponding power spectra are obtained by applying the so-called Bessel function approximation [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' However, another alternative that has already been considered in the literature is the case of ultra-slow roll inflation [15–17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' The scalar potential is assumed to be extremely flat, and then the Klein-Gordon equation of motion of the homogeneous inflaton field is simplified by neglecting the slope term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' Therefore, the friction term is now locked with the acceleration term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' Furthermore, one has that ηH ≈ 3 and then the slow-roll approximation is no longer valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' For instance, the mechanism to generate ∗ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='elbourakadi@yahoo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='com † pr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='mouhssine@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='com ‡ giovanni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='otalora@academicos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='uta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='cl § mdbennai@yahoo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='fr ¶ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='ouali@ump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='ma 2 primordial black holes within the framework of ultra-slow roll inflation has been investigated in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' [100, 101].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' Ultra- slow roll inflation has also been generalized to constant-roll inflation such that the slow-roll parameter ηH becomes a constant [113, 114] (see also Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' [18–23]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' Interestingly enough, constant-roll inflation differs from slow-roll inflation due to its new dynamical features that result in richer physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' It gives rise to a large local non-Gaussianity and the curvature perturbation may grow on the super-horizon scales [24, 113, 114].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' Primordial Black Holes (PBHs) [25] started to be considered seriously after Hawking proposed that PBHs could be formed with Planck-order masses in the early Universe [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' In this direction one would consider that PBHs formation could be a result of inflation [27] inhomogeneities from the early Universe [28, 29] or the phase transition [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' Furthermore, it was first proposed that PBHs could be a candidate of dark matter (DM) in [31] and reconsidered in [32, 33] when black holes mergers were first detected by LIGO [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' Hawking found that black holes release thermal radiation [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' As a result, depending on their mass, black holes evaporate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' The faster the PBHs evaporate, the smaller their masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' A black hole’s density, on the other hand, varies inversely with its mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' Such high densities are required for the formation of lighter black holes, and such densities are only possible in the early Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' Primordial black holes are thus the only black holes with masses small enough to have vanished by now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' A number of observations restrict the mass ranges of PBHs, for instance, PBHs masses bounded as ≲ 10M⊙ are excluded from null microlensing searches [36, 37], while masses ≳ 102M⊙ are excluded from wide binary surveys [38, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' The essential assumption behind these constraints is that in the early Universe, PBHs accrete primordial gas and subsequently convert a percentage of the accreted mass to radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' The ensuing infusion of energy into the primordial plasma then influences its thermal and ionization processes, causing anomalies in the CMB’s frequency spectrum and polarization power spectra [40, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' In General Relativity (GR), the gravitational interaction is described in terms of the curvature associated with the Levi-Civita Connection [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' Moreover, it is well-known that gravity can also be described in terms of torsion in the context of the so-called Teleparallel Equivalent of GR, or simply Teleparallel Gravity (TG) [43–50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' TG is a gauge theory for the translation group in which the dynamical variable is the tetrad field rather than the metric tensor and the torsion is associated to the Weitzenb¨ock connection that substitutes the Levi-Civita connection [51– 54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' Additionally, the Lagrangian density of TG is proportional to the torsion scalar T , which is equivalent to the curvature scalar R up to a total derivative term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' Therefore, GR and TG are equivalent at the level of field equations [53, 55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' In all the cases, the equivalence with GR is only guaranteed for a gravitational action linear in torsion or decoupled from other fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' On the other hand, as a natural extension to modified gravity, and similarly to f(R) gravity [62–66], the Lagrangian of TG may be promoted to an arbitrary function f(T ) [56–58] (see also Ref [59] and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' Also, following the perspective of scalar-tensor gravity theories [62, 63, 67], we can further extend the above theory to f(T, φ) gravity [60, 61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' These novel modified gravity theories have shown a rich structure by explaining the dynamics of dark energy [68–71, 74] and inflation [61, 72, 73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' Thus, in the context of GR, without considering any modification to gravity, one can see that there exist at least two equivalent descriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' The curvature representation, in which the torsion and nonmetricity vanish, and the teleparallel representation, in which the curvature and nonmetricity vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' But, recently a third equivalent representation has also been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' In this approach, called Symmetry Teleparallel Gravity (STG), the curvature and torsion vanish, and then the gravitational interaction is described by the nonmetricity Q of the metric [75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' STG has also been further developed to tackle modify gravity in the context of f(Q) gravity theory, also known as nonmetric gravity [76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' Recently, interest in this theory has increased rapidly due to its novel geometrical and physical features [77–87], f(Q) gravity is also used to explain cosmic acceleration [88–92].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' Moreover, another proposed theory of modified gravity based on nonmetricity is f(Q, T ) gravity [93], where T is the trace of the matter energy-momentum tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' Thus, it is an extension of f(Q) gravity where a direct coupling between nonmetricity and matter has been assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' In fact, one can verify that the divergence of the matter energy-momentum tensor does not vanish by taking the covariant derivative of the modified field equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' Several different aspects of f(Q, T ) gravity have been studied, among these we have observational constraints [94], energy conditions [95, 96], cosmological solutions [97], phase-space dynamics [98], and cosmological inflation [99].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' The plan of the paper is the following: In Section II, we present an introduction to f (Q, T ) gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' In Section III, we study the dynamics of constant-roll inflation within the context of f(Q, T ) gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' In Section IV, we investigate the generation of primordial black holes in f(Q, T ) gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' Finally, in Section V, we summarize the results obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' BASIC FORMALISM IN f (Q, T ) GRAVITY In the so-called Weyl-Cartan geometry, gravitational effects are caused not only by a variation of the direction of a vector in parallel transport, but also by a variation in its length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' The variation in the length of a vector is described geometrically by what is called non-metricity, and it is mathematically described as the covariant derivative of the metric tensor which is a generalization of the gravitational potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' In this case, general connexion Σα µν is written 3 in terms of all contributions as curvature, torsion and non-metricity of spacetime [102], Σα µν = Γα µν + Cα µν + Lα µν, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='1) where Γα µν is the famous Levi-Civita connection of the metric tensor gµν in GR given by, Γα µν ≡ 1 2gαλ(gµλ,ν + gλν,µ − gµν,λ), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='2) here Cα µν represents the Contortion tensor expressed as, Cα µν ≡ 1 2( ˜T α µν + ˜T α µ ν + ˜T α ν µ) = −Cα νµ, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='3) and Lα µν is the Disformation tensor given by, Lα µν ≡ 1 2(Qα µν − Q α µ ν − Q α ν µ) = Lα νµ, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='4) where ˜T α µν and Qαµν in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='3) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='4) are the torsion tensor and the non-metricity tensor, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' In non-flat space-time, the geodesic structure is designated by the connection form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' In addition, in Einstein’s GR, the presumption of a non-torsion and metric compatible connection leads to the so-called Levi-Civita connection, connected to the metric tensor and its first derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' It is possible to define two tensors of order 3 related to the antisymmetric part of Σα µν and the covariant derivative of the metric tensor as, ˜T α µν ≡ 2Σα [µν], (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='5) and Qαµν ≡ ∇αgµν = ∂αgµν − gνσΣσµα − gσµΣσνα ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='6) Depending on the proposed connection, different theories of gravity that can be extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' In this work, we will consider the modified Einstein-Hilbert action in f(Q, T ) extended symmetric teleparallel gravity expressed as [93, 103], S = ˆ √−gd4x � 1 2κf(Q, T ) + Lm � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='7) where f(Q, T ) is a general function of the non-metricity scalar Q and the trace of the energy-momentum tensor T , κ = 1/M 2 p, g is the determinant of the metric tensor gµν i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' g = det (gµν), and Lm is the Lagrangian function for the matter fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' The non-metricity scalar Q is defined as, Q ≡ −gµν(Lβ αµLα νβ − Lβ αβLα µν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='8) The trace of the non-metricity tensor is obtained as, Qβ = gµνQβµν �Qβ = gµνQµβν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='9) Further, the superpotential tensor P β µν (or non-metricity conjugate) is related to the non-metricity scalar Q as, P β µν = −1 2Lβ µν + 1 4(Qβ − �Qβ)gµν − 1 4δβ (µQν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='10) By using the definition above, the non-metricity scalar in terms of the superpotential tensor is given by, Q = −QβµνP βµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='11) As usual, we define of the energy-momentum tensor of the matter fields by Tµν = − 2 √−g δ(√−gLm) δgµν , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='12) and Θµν = gαβ δTαβ δgµν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='13) 4 So that the variation of energy-momentum tensor with respect to the metric tensor gµν read as, δ (g µν T µν) δ g α β = T αβ + Θ α β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='14) Now, varying the gravitational action (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='7) with respect to metric tensor gµν, the fleld equations of f(Q, T ) gravity can be derived as, − 2 √−g ∇β � fQ √−gP β µν � − 1 2fgµν + fT (Tµν + Θµν) − fQ(PµβαQ βα ν − 2Qβα µ Pβαν) = κTµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='15) Here, fQ = df(Q,T ) dQ , fT = df(Q,T ) dT , and ∇β denotes the covariant derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' Now, we consider a homogeneous and spatially flat Friedmann-Lemaitre-Robertson-Walker (FLRW) metric described by the line element ds2 = −N 2 (t) dt2 + a2(t) � dx2 + dy2 + dz2� , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='16) where a (t) is the scale factor of the Universe, depending only on the cosmic time t (which is scaled to be unity at the present time, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' a0 = 1) and N (t) is the lapse function regarded to be 1 in the standard model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' The rates of expansion and dilation are fixed as H ≡ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='a a and D ≡ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' N N , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' Hence, the corresponding non-metricity scalar is given by Q = 6 H2 N 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' In our current analysis, we suppose that the content of the Universe as a perfect non-viscosity fluid for which the energy-momentum tensor is given by T µ ν = diag (−ρ, p, p, p), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='17) where p is the perfect non-viscosity fluid pressure and ρ is the energy density of the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' Hence, for the tensor Θµ ν the expression is obtained as Θµ ν = diag (2ρ + p, −p, −p, −p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' Taking into account the case as N = 1, the Einstein field equations using the line element (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='16) are given by, κρ = f 2 − 6FH2 − 2E 1 + E � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' FH + F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' H � (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='18) and κp = −f 2 + 6FH2 + 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' FH + F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' H � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='19) where we used Q = 6H2 and (·) represents a derivative with respect to cosmic time (t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' In this case, F ≡ fQ and κE ≡ fT represent differentiation with respect to Q and T respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' The evolution equation for the Hubble parameter H can be derived by combining Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='18) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='19) as, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' H + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' F F H = κ 2F (1 + E) (ρ + p) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='20) Einstein’s field equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='18) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='19) can be regarded as extended symmetric teleparallel equivalents to standard Friedmann’s equations with supplementary terms from the non-metricity of space-time and the trace of the energy-momentum tensor T which behaves as an effective component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' Therefore, the effective energy density ρeff and effective pressure peff are determined by, 3H2 = κρeff = f 4F − κ 2F [(1 + E) ρ + Ep] , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='21) 2 ˙H + 3H2 = −κpeff = f 4F − 2 ˙FH F + κ 2F [(1 + E) ρ + (2 + E) p] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='22) Taking into account Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='20) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='21) one gets ρ = f − 12H2F 2κ [(1 + ω) E + 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='23) 5 III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' INFLATIONARY SCENARIO IN f(Q, T ) GRAVITY A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' Slow-roll and constant-roll in the standard cosmology The simple inflation model is described with an isotropic and homogeneous scalar field known as the inflaton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' The dynamics of such inflaton field can be defined via the Lagrangian Lm = −1 2gµν∂µφ∂νφ − V (φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='1) From the Lagrangian of the scalar field φ the energy-momentum tensor is defined as Tµν ≡ − 2 √−g δ (√−gLm) δgµν = gµνLm − 2 ∂Lm ∂gµν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='2) Considering the inflaton field as a perfect fluid with an equation of state ω = pφ/ρφ, knowing that ρφ and pφ are the energy density and the pressure of the inflaton field, the energy-momentum tensor in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='2 gives ρφ = 1 2 ˙φ2 + V (φ), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='3) and pφ = 1 2 ˙φ2 − V (φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='4) Here the dot represents the derivative with respect to the cosmic time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' The equation of state parameter is then written as ω = pφ ρφ = 1 2 ˙φ2 − V (φ) 1 2 ˙φ2 + V (φ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='5) Taking into account the Einstein field equations in addition to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='3) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='4), the Friedmann equations are easily obtained as H2 = κ 3 � ˙φ2 2 + V (φ) � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='6) H2 + ˙H = −κ 3 � ˙φ2 − V (φ) � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='7) ˙H = −κ 2 ˙φ2, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='8) here H ≡ ˙a/a is the Hubble parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' The so called Klein-Gordon (KG) equation, by taking the time derivative of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='6) and by taking into considering Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='8), can be obtained ¨φ + 3H ˙φ + V ′ = 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='9) here the prime denotes the derivative with respect to the φ-field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' The accelerated expansion of the inflationary phase must last for enough time for a successful period where the Hubble radius decreased over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' For this, the slow-roll parameters are defined as [105] ǫ = − ˙H H2 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='10) η = ˙ǫ Hǫ ≈ − ¨H 2 ˙HH .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='11) An additional parameter used to study the period of inflation is the e-folding number which describes the rate of the Univers expansion during this phase [107, 108] N ≡ ln �aend a � = ˆ tend t Hdt, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='12) 6 the index ”end” denotes the time when inflation ended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' Inflation will continue as long as ǫ < 1 to solve the standard cosmological problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' However, at the end of inflation, the slow-roll parameter must reach ǫ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' The slow roll parameters and the curvature perturbations are related by the spectral index ns, and the ratio of tensor to scalar perturbations r, in the following way [109] ns − 1 = −6ǫ + 2η, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='13) r = 16ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='14) The Planck data provides strong constraints on ns and r parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' In fact, any inflationary model predict such parameters can be tested to decide whether it can be ruled out or not [110, 111].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' In addition to the slow roll conditions, one may consider a type of constant roll of the scalar field, which is expressed as follows ¨φ ˙φ = βH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='15) In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' [112] they considered two slow-roll stages, separated by a constant-roll stage that imprints an alternative dynamic to the scalar field [113], where the constant β determines the deviation from a flat potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' When β ≃ 0, the slow roll inflation is recovered, whereas β = 0 corresponds to the “ultra slow roll” inflation [114, 115].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' In order to construct the idea of constant roll inflation this paper aims to use the condition ǫ ≪ 1 that leads to ˙φ2 ≪ V (φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='16) which simplifies Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='11) as η ≈ − ���¨φ ��� H ��� ˙φ ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' However, for a successful constant roll scenario, we consider the expression ¨φ = ˙φβH that simplifies the KG equation as ˙φH (3 + β) + V ′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='17) In the next section, we will investigate the effect of the constant roll stage on the inflationary parameters by deriving H, ˙H, ˙φ and ¨φ from an f(Q, T ) gravity perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' Inflation in f(Q, T ) Gravity In the context of cosmic inflation, we consider the model f(Q, T ) = αQ+σT [93], with α = F ̸= 0, σ is an arbitrary constant, and T = −ρ + 3p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' It is crucial to keep in mind that σ = 0 and α = −1 is equivalent to the case of the GR theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' Furthermore, the theory is reduced to f(Q, T ) = −Q for α = −1 at σ ̸= 0 and T = 0 which is equivalent to GR as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' By taking Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='20), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='21), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='22) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='23), we obtain the following equations ˙H = (κ + σ) ρ (1 + ω) 2α , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='18) 3H2 = κρeff = [(ω − 3) σ − 2κ] ρ 2α , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='19) 2 ˙H + 3H2 = −κpeff = ρ [(3ω − 1) σ + 2ωκ] 2α , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='20) ρ = −6αH2 σ (3 − ω) + 2κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='21) Aside from the equation of state that corresponds to the inflationary phase, we can derive an effective equation of state for this model as ωeff = peff ρeff = −1 − 2 (κ + σ) (1 + ω) (ω − 3) σ − 2κ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='22) 7 It is clear that from the effective equation of state above, we can have an accelerated de Sitter expansion as long as ((ω − 3) σ − 2κ ̸= 0) and (ω = −1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' Considering the trace of the energy-momentum tensor as T = ˙φ2 − 4V , the cosmological inflation in the context of f(Q, T ) gravity, from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='1), Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='21) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='22) one obtains ρeff = −(κ + σ) ˙φ2 + 2V (φ) (κ + 2σ) 2κα , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='23) peff = −(κ + σ) ˙φ2 − 2V (φ) (κ + 2σ) 2κα .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='24) On the other hand, the modified Friedmann equations can be given as 3 2H2 = −(κ + σ) ˙φ2 + 2V (φ) (κ + 2σ) 4α , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='25) 2 ˙H + 3 2H2 = 3 (κ + σ) ˙φ2 − 2V (φ) (κ + 2σ) 4α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='26) Taking into account the previous two equations one gets ˙H = ˙φ2 2α (κ + σ) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='27) the effective equation of state parameter can be obtained as ωeff = −(κ + σ) ˙φ2 − 2V (φ) (κ + 2σ) (κ + σ) ˙φ2 + 2V (φ) (κ + 2σ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='28) The derivation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='25) and using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='27), we obtain the modified equation ¨φ (κ + σ) + 3H ˙φ (κ + σ) + V ′ (κ + 2σ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='29) From the constant roll condition ¨φ = βH ˙φ, we obtain ˙φ = − V ′ (κ + 2σ) H (β + 3) (κ + σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='30) However, one may consider the case where the bound (κ + σ) ˙φ2 ≪ V (φ) (κ + 2σ) takes place and from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='25), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='10) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='27) to obtain the following slow roll parameter ǫ ≈ 3 (κ + σ) ˙φ2 2 (κ + 2σ) V = −9α 2 (β + 3)2 (κ + σ) �V ′ V �2 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='31) deriving Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='27) and using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='11), the second slow roll parameter can simply be given as η ≈ − ¨H 2 ˙HH = − ���¨φ ��� H ��� ˙φ ��� = −3α (β + 3) (κ + σ) �V ′′ V � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='32) Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='31) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='32) are obtained by assuming that the constant roll stage succeeds the slow roll one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' Chaotic potential Let us consider the simplest model known as chaotic potential, given by the form V (φ) = 1 2m2φ2, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='33) the parameters ǫ and η are expressed as a function of the constant roll parameter β in the following way ǫ = 3 2 1 − ns 6 − β , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='34) η = �3 + β 6 − β � 1 − ns 2 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='35) 8 here we should note that (6 − β) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' To obtain Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='31) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='32), we should develop a little bit the transition from slow roll stage to a constant roll one, for the chaotic potential we see that the behavior of the inflationary phase is determined through the constant roll parameter β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' r as a function of ns for a chaotic potential in modified symmetric teleparallel gravity, light and dark blue regions are constraints in combination with CMB lensing reconstruction and BAO from Planck data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' on the right panel we choose three different values of β, the black line for β = 0, the red line represents β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='5, and the blue line for β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' While on the left one, We choose three different values of β, the black line for β = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='5, the red line represents β = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='5, and the blue line for β = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' 1 show the decreasing behavior of the tensor-to-scalar ratio, r, with respect to the spectral index, ns for the chaotic potential in our chosen f(Q, T ) gravity model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' The results show a good consistency for a specific range of the constant roll parameter β with the latest observations from Planck data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' Furthermore, the case with β > 0 provides inconsistent results with Planck’s data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' However, the constant roll parameter must be bounded as β ≤ 0 in order to produce consistent observational parameters with recent results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' Hilltop inflation Hilltop inflation is classified among small-field models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' This potential naturally involves eternal inflation, that raises the question of initial conditions, which is a problem in most inflation models [116].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' Hilltop potential is given by [117] V (φ) = M 4 � 1 − �φ µ �p� , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='36) this model of inflation has two free parameters M and µ plus the parameter p that will be given specific values at the end of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' The e-folds number during inflation using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='30) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='25) is given by N ≡ ˆ φend φk H ˙φ dφ ≈ −(β + 3) (κ + σ) 3α ˆ φk φend V V ′ dφ, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='37) considering Φ = φ/µ and p > 2, the expression of inflationary e-folds number is given by N = −µ2 2p (β + 3) (κ + σ) 3α � Φ2 k − Φ2 end + 2 p − 2Φ2−p k − 2 p − 2Φ2−p end � , rn STT,TE,EE+IOWE + lensing+BK15 +BAO TT,TE,EE+lowE + lensing Nj= 50 N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' = 609 here the subscripts ”k” and ”end” mean the time the pivot scale crossed outside the horizon and the end of inflation respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' Since the slow roll parameters are defined by the value of the field at the horizon crossing Φk, for this model ǫ and η are calculated as follows ǫ = − 9α 2(β + 3)2 (κ + σ) p2 µ2 Φ2(p−1) k (1 − Φp k)2 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='38) η = 3α (β + 3) (κ + σ) p (p − 1) µ2 Φp−2 k 1 − Φp k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='39) Furthermore, we can compute the tensor-to-scalar ratio as a function of the inflationary e-folds N, the constant-roll parameter β for p > 2 in the following way, r = 4p (β + 3) 1 N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='40) Taking into consideration Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='14), we can study the behavior of r as a function of the spectral index ns, β and p parameters using the first slow roll parameter ǫ given as ǫ = 3p (1 − ns) 18p − 4 (p − 1) (β + 3), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='41) η = 2 (p − 1) (β + 3) (1 − ns) −18p + 4 (p − 1) (β + 3) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='42) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' r as a function of ns for the hilltop potential in modified symmetric teleparallel gravity, light and dark blue regions are constraints in combination with CMB lensing reconstruction and BAO from Planck data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' We choose three different values of p, the black line for p = 3, the red line represents p = 4 and the blue line for p = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' 2, we consider the hilltop inflation, where the inflationary parameters are linked to f(Q, T ) gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' It is apparent from the calculations that the constant roll parameter has negligible effects on the (r, ns) behavior, the plot shows that increasing the value of p makes the decreasing function obtained from our model consistent with the latest observations provided by Planck’s results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' Based on the previously established equation Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='40), we provide an examination of the tensor-to-scalar ratio taking into account different inflation durations that varies from 50 to 65 e-folds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' In addition to an increasing values of the constant roll β parameter, the tensor-to-scalar ratio r can be compatible with the observational bound for specific values of p and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' Furthermore, the constant roll parameter and the inflationary e-folds provide compatible results only for higher values since they are inversely proportional to r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' Now as we construct a constant roll model for the f(Q, T ) gravity in the context of chaotic and hilltop inflation, we will study the evolution of primordial black holes that were supposed to occur just after the inflation period taking into account our f(Q, T ) gravity model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' rn STT,TE,EE+IOWE + lensing+BK15 +BAO TT,TE,EE+lowE + lensing Nj= 50 N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' = 6010 β p N r −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='5 6 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='32 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='1 5 55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='19 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='5 4 60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='7 3 65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='049 TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' Testing the tensor-to-scalar ratio for different parameters for the hilltop inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' PRIMORDIAL BLACK HOLES EVOLUTION IN f(Q, T ) GRAVITY A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' Rotating and non-rotating Primordial Black Holes Primordial black holes abundance is determined by the primordial power spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' In fact, PBHs were formed as a result of amplification in the primordial power spectrum on small scales at the time when primordial inhomogeneities re-enter the Hubble horizon in a radiation-dominated Universe era.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' Some regions with a significant positive curvature which are considered equivalent to a closed Universe will collapse into a black hole [119].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' At first, black holes were considered to be eternal and evolved with an increasing mass by absorbing more matter or even other stars and BHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' However, studying their quantum properties shows the possibility of emitting particles with a thermal spectrum related to BHs surface gravity [35, 120].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' In this process of emitting particles, BHs lose mass and angular momentum with different properties depending on the specific characteristics of BHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' In this direction, we will discuss the thermal properties of evaporating PBHs for two cases namely the Schwarzschild and Kerr PBHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' As an analogy to non-rotating Schwarzschild black holes, we consider a PBH with mass MBH, the thermal spectrum of emitted particles from the evaporation process has the following expression [121] TBH = 1 8πGMBH ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='06 �1010kg MBH � GeV, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='1) Another option is that the evaporating BHs have some angular momentum, such spinning BHs, also known as Kerr BHs, might have originated with an initial spin or acquired their angular momenta by a variety of events, such as mergers [122–124].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' A PBH temperature for the Kerr scenario can be modified due to their spinning and it’s given as [125] TBH = 1 4πGMBH � 1 − a2∗ 1 + � 1 − a2∗ , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='2) where a∗ is a dimensionless parameter bounded by the interval 0 ≤ a∗ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' The primary distinction between Kerr (a∗ ̸= 0) and Schwarzschild (a∗ = 0) BHs is that Kerr BHs are axially symmetric rather than spherically symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' When a∗ grows, the emission of particles with angular momentum spinning in the same direction as the BH is enhanced [126].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' For the case a∗ = 1 a BH would have TBH = 0, this is traditionally prohibited in any statistical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' Furthermore, its horizon would have vanished revealing a bare space-time singularity and contradicting the Cosmic Censorship Conjecture [126–128].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' Primordial Black Holes evolution Since we are interested in studying the rate of mass loss from PBH, we recall the process which reduces the mass of the black hole due to Hawking evaporation which is defined by [129, 130] �dM dt � eva = −ℏc4 G2 α′ M 2 BH .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='3) where G is Newton’s gravitational constant, ℏ is the Planck constant, c is light speed, α′ is the spin parameter of the emitting particles, and the black hole radius is given by rBH = 2GM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' After integration Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='3) we obtain the evolution of PBH mass as Meva = Mi � 1 − t teva � 1 3 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='4) 11 we can define the Hawking evaporation time scale teva as teva = G2 ℏc4 M 3 i 3α′ , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='5) according to Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' [131] fine-tuning the initial PBH masse Mi along with the α′ parameter would be a good method to probe the early Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' On the other hand, additional contributions suggest that the evaporation of PBHs would have interesting implications on the CMB and the standard cosmological nucleosynthesis scenario [132, 133].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' On the left, the evolution of the primordial black hole reduced the mass ratio to the initial mass Meva/Mi as functions of time ratio t/teva is plotted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' On the right, the evaporation time of primordial black holes as a function of the initial mass is presented, and the plot in the left corner shows the variation of teva for higher values of Mi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' 3 we test the evolution of the primordial black holes evaporation and initial mass ratio Meva/Mi with respect to the ratio t/teva, and we plotted the variation of the evaporation time parameter teva as functions of the initial mass of primordial black holes Mi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' Our results show that the evaporation mass can simply decrease as we move forward in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' However, one must study the effect of the initial primordial black holes masses on the time that must take in order to completely evaporate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' In this direction, the second plot indicates that as we increase the initial mass of PBHs we need more time to achieve a complete evaporation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' In fact, an initial mass in the order of Mi ≳ 1013kg needs time more than the current age of our Universe to evaporate, as a result, we must consider lower bounds on PBHs initial mass for fine consistency with cosmic history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' The accretion of cosmic fluid surrounding the black hole will prolong the evaporation of the primordial black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' Therefore, it is necessary to add a mass accretion rate which for a cosmic fluid with ρeff and peff will be given as �dM dt � accr = 16πG2 c5 M 2 (ρeff + peff) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='6) which can be integrated to be in the final form [134] Maccr = Mi � 1 − t taccr �−1 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='7) where from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='19) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='20) the accretion time scale taccr is defined as taccr = �16πG2 c5 Mi (ρeff + peff) �−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='8) Additionally, we can write the following equation t tevaMeva MiMi12 ρeff + peff = 3 κH2 �2 (σ + κ) (1 + ω) 2κ − (ω − 3) σ � (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='9) Here, at a radiation-dominated era, and using the fact that the Hubble parameter is given by H = ˙a a, we can simply choose a(t) ∝ t1/2 and finally obtain H = 1/ (2t)2 to estimate the time at which the radiation surrounded primordial black-holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' The accretion time taccr variation as a function of the equation of state in the interval ω ∈ [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='3] considering different values of the f(Q, T ) gravity model parameter σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' The left panel provides the case for Mi ∝ 1012kg, while the right shows the case of Mi ∝ 1014kg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' 4 illustrates the evolution of the accretion time as a function of a specific interval of the equation of state parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' In our study, the accretion time is the moment at which primordial black holes started the accretion process, the variation of the accretion time shows to be minimally dependent on the choice of σ parameter for our chosen f(Q, T ) gravity model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' In fact, the behavior of accretion time increases for higher values of σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' Considering the fact that in order for inflation to initiate the EoS must be bounded by ω > −1/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' Moreover, ω evolves toward 1/3 for inflation to be ended and the subsequent periods to take place [9, 135].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='9) we choose a fixed value of time using the Hubble parameter which represents the time of PBHs formation, following the results of [129] which suggests that PBHs formed at t ∝ 10−23s, our results provide precise values of the time of accretion which decreases as we consider higher values of the EoS parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' On the other hand, for lower initial PBHs masses, the accretion initiates sooner than in the case of higher Mi values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' 13 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' On the left panel, the evolution of the primordial black holes accretion mass ratio to the initial mass Maccr/Mi as functions of the time ratio t/taccr is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' On the right one, the accretion time of primordial black holes as a function of the initial mass is presented for specific values on the equation of state ω parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' 5 we provide the variation of the primordial black holes accretion mass ratio to the initial mass Maccr/Mi as a function of the ratio t/taccr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' From this figure we conclude that when t −→ taccr the resulting primordial black hole mass due to the accretion process became in the order of 100Mi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' On the other hand, accretion occurs faster when we consider more significant initial PBH masses for several values of ω parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' The total PBHs mass evolution may now be rewritten in the following form MBH = Meva + Maccr σ ω t(s) Mi(kg) MBH(kg) TBH(GeV ) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='5 κ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='3 10−15 106 ∼ 2 × 106 ∼ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='3 × 103 2 κ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='2 10−5 107 ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='9 × 107 ∼ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='3 × 102 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='5 κ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='1 10 108 ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='9 × 108 ∼ 53 3 κ 0 105 109 ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='9 × 109 ∼ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='5 κ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='1 109 1010 ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='9 × 1010 ∼ 53 × 10−2 4 κ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='2 1013 1011 ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='9 × 1011 ∼ 53 × 10−3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='5 κ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='3 1017 1012 ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='9 × 1012 ∼ 53 × 10−4 TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' Testing the evolution of the total non-rotating PBHs masses and temperature in f(Q, T ) gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' σ ω t(s) Mi(kg) a∗ MBH(kg) TBH(GeV ) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='5 κ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='3 10−15 106 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='1 ∼ 2 × 106 ∼ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='28 × 103 2 κ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='2 10−5 107 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='3 ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='9 × 107 ∼ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='17 × 102 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='5 κ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='1 10 108 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='5 ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='9 × 108 ∼ 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='1 3 κ 0 105 109 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='7 ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='9 × 109 ∼ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='41 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='5 κ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='1 109 1010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='9 ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='9 × 1010 ∼ 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='1 × 10−2 4 κ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='2 1013 1011 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='96 ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='9 × 1011 ∼ 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='1 × 10−3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='5 κ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='3 1017 1012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='99 ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='9 × 1012 ∼ 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='1 × 10−4 TABLE III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' Testing the evolution of the total rotating PBHs masses and temperature in f(Q, T ) gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' t taccrM accr MiMi14 From tables (II) and (III) we study the evolution of rotating and non-rotating PBHs masses and temperature as functions of different parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' According to these results, we conclude that PBHs masses increase with higher initial masses Mi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' On the other hand, PBHs temperature decreases as we move forward in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' Moreover, TBH is inversely proportional to PBHs masses which makes the temperature evolve from T eV to MeV taking into account masses in the order of 1012kg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' Finally, when we compare rotating and non-rotating black holes, we find that higher values of the parameter a∗ can simply lead to a lower temperature for the case of rotating PBHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' CONCLUDING REMARKS Over the past few decades, numerous studies have been conducted to examine the early and late evolution of the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' The standard model of cosmology, based on general relativity (GR), has proven to be a reliable model for describing the dynamics of the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' However, there are still some unresolved issues, such as the flatness and horizon problems, that need to be addressed through further research on cosmological inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' Additionally, while GR has been successful in predicting the behavior of the Universe, it is unable to fully explain the influence of dark sectors on its dynamics in a way that is consistent with observed data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' As a result, it may be worthwhile to consider alternative models of gravity such f(Q, T ) model to address these challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' In this study, we examined constant-roll inflation in the context of f(Q, T ) gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' To do this, we started by explaining the basic theory of cosmological inflation using the isotropic and homogeneous inflaton scalar field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' We then assumed a flat FLRW spacetime and an equation of state ω, we provided a new technique for studying the constant-roll process and correlating the slow-roll equations to the constant-roll β parameter based on the modified Friedmann equations obtained through f(Q, T ) gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' Furthermore, we have calculated the inflationary observables, the spectral index ns, and the tensor-to-scalar ratio r for two cases of inflationary potentials, namely the chaotic and hiltop models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' We showed that for each model, a bound on the constant-roll parameter is preferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' In the case of chaotic inflation, for a consistent value of r and ns, β must be bounded as β ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' While for the Hiltop Inflation, several parameters are involved to reproduce compatible values of r and ns, for instance, we must consider the bounds p ≥ 3, N ≥ 60, and β ≥ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content='5 for a good index spectral and tensor-to-scalar ratio consistencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' Finally, for the PBHs evolution in the context of f(Q, T ) gravity, we analyzed the accretion process and the evaporation through Hawking radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' From the obtained results, we conclude that both the evaporated mass and the evaporation time are directly related to the initial mass Mi that must be bounded as Mi < 1013kg to be able to completely evaporate currently or much earlier in the cosmic time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' For the accretion process, we can summarise that the accretion of matter and radiation are model dependent in the context of the f(Q, T ) gravity, which motivates us to explore the PBHs evolution in the framework of modified gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' According to the results, if we supposed that PBHs formed at t ∝ 10−23s the PBHs mass due to the accretion can reach 100Mi taking into account that for lower initial PBHs masses the accretion can occur faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' Lastly, we studied the PBHs evaporation taking into account the effect of several parameters, and concluded that the Hawking temperature can simply decrease for higher initial masses through the cosmic time for both rotating and non-rotating black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE2T4oBgHgl3EQfJwbV/content/2301.03696v1.pdf'} +page_content=' A.' 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RODRIGUEZ +Abstract. Recent works have shown that in contrast to classical linear elastic +fracture mechanics, endowing crack fronts in a brittle solid with Steigmann- +Ogden surface elasticity yields a model that predicts bounded strains at the +crack tips for plane-strain problems. +However, a logarithmic singularity is +still present in general for anti-plane shear (mode-III fracture) even when +Steigmann-Ogden surface elasticity is incorporated. +Motivated by obtaining a model of brittle fracture capable of predicting +bounded strains for all modes of loading, we formulate an exact general the- +ory of a bulk solid containing a boundary surface with strain-gradient surface +elasticity. For planar reference surfaces, the form of surface elasticity reduces +to that introduced by Hilgers and Pipkin, and when the surface energy is +independent of the surface gradient of the stretching, the theory reduces to +that of Steigmann and Ogden. We give a full discussion of material symmetry +using Murdoch and Cohen’s extension of Noll’s theory. We present a model +quadratic surface energy that incorporates resistance to geodesic distortion, +satisfies strong ellipticity, and requires the same material constants found in +the Steigmann-Ogden theory. +Finally, we derive and apply the linearized theory to mode-III fracture in +an infinite plate. We prove that there always exists a unique classical solu- +tion to the governing integro-differential equation, and in contrast to using +Steigmann-Ogden surface elasticity, our model is consistent with the lineariza- +tion assumption in predicting finite strains at the crack tips. +1. Introduction +1.1. Surface stressed solid bodies. The study of surface tension for solids was +initiated by Gibbs in 1857, and it is now widely accepted that surface tension and +more general surfaces stresses must be accounted for when modeling mechanical +structures at small length scales. In particular, interfaces1 between a bulk solid and +its environment can form due to various mechanisms including coating or atomic +rearrangement during fracture. A way to mathematically model such an interface is +by endowing part of the three-dimensional body’s two-dimensional boundary with +it’s own thermodynamic properties that are distinct from the bulk (such as energy). +Motivated by modeling the apparent compressive surface stresses in certain +cleaved crystals, Gurtin and Murdoch [4,5] developed a rigorous general theory of +material surface stresses that accounts for the surface’s resistance to stretching (but +not flexure). Their celebrated theory has been used to model a wide arrange of phe- +nomena over the past 45 years, especially recently due to advances in nanoscience +and nanotechnology. In the special case of a hyperelastic, three-dimensional bulk +1By an interface, we mean a thin region separating either two distinct materials or two distinct +phases of a material. +1 +arXiv:2301.13744v1 [math-ph] 31 Jan 2023 + +2 +C. RODRIGUEZ +solid with reference configuration B and material surface S ⊆ ∂B, the field equa- +tions for the current configuration χ : B → χ(B) are the Euler-Lagrange equations +associated to the total strain energy +Φ[χ] = +ˆ +B +W(C) dA + +ˆ +S +U(C) dA, +where we omit the possible explicit dependence of the functions W and U on points +in B and S (see Figure 1). Here C is the left Cauchy-Green stretch tensor and C +is the left Cauchy-Green surface stretch tensor, the pullbacks by χ of the metric +tensors on χ(B) and χ(S), respectively. In particular, the material surface stress +tensor is derived from the surface energy density U(C) in analogy with the bulk’s +Piola stress being derived from the bulk energy density W(C). +However, Steigmann and Ogden showed in their seminal works [22, 23] that +equilibrium states under compressive surface stresses obtained from the Gurtin- +Murdoch theory do not satisfy an associated Legendre-Hadamard condition, and +thus, these equilibrium states cannot be local energy minimizers.2 Steigmann and +Ogden [22, 23] rectified this inadequacy and incorporated the material surface’s +resistance to flexure by including curvature dependence in the surface energy: +Φ[χ] = +ˆ +B +W(C) dA + +ˆ +S +U(C, κ) dA, +where κ is the pullback of the second fundamental form on χ(S). Since κ depends +on the second derivatives of χ, the surface energy is of strain-gradient type. In +recent years, the Steigmann-Ogden theory has attracted considerable interest from +various perspectives including the study of contact problems [10–13, 31, 32, 37], +inclusion problems [2, 6, 14, 15, 27, 35]. The theory has also been used to model +fracture in brittle materials [30,33,34], the main phenomenon motivating this work. +One of the most successful and practical theories for modeling fracture in brittle +materials is classical linear elastic fracture mechanics. The governing linear par- +tial differential equations are derived from finite elasticity under the assumption +of infinitesimal strains (linearized elasticity), but the theory predicts unbounded +singular strains at the crack tips, a striking inconsistency and physically impossi- +ble prediction. There have been a vast number of suggestions for supplementing +2The fact that a pure membrane, a special type of Gurtin-Murdoch material surface, in equi- +lbrium and under compressive surface stresses cannot by locally energy minimizing is due to +Pipkin [20]. +S +B +Figure 1. The reference configuration of a bulk solid B with +strain-gradient elastic surface S ⊆ ∂B. + +STRAIN-GRADIENT ELASTIC SURFACES +3 +classical linear elastic fracture mechanics to correct this defect in the theory (see, +e.g., [1]). +A more recent approach aimed at eliminating these crack tip singularities is to +modify the boundary conditions of classical linear elastic fracture mechanics by +endowing the crack fronts with material surface stresses. Beginning with [19] and +developed further by Sendova and Walton [21], one line of thought has been to pre- +scribe the crack front’s surface stresses as a curvature dependent surface tension. +Although able to remove the singularities completely in a diverse range of settings +(see [3,26,28,29,36]), it is unclear if the surface stresses from the Sendova-Walton +theory can be derived from a surface energy density,3 a reasonable definition of +“elastic-like” behavior. +Deriving the material stresses for the crack fronts from +a Gurtin-Murdoch surface energy does not completely remove the crack tip sin- +gularities (see [9, 25]), but however, endowing the crack fronts with Steigmann- +Ogden surface energy does completely remove the singularities for plane-strain +problems [30, 33] and axisymmetric penny shaped cracks [34]. Unfortunately, for +anti-plane shear (mode-III loading), Steigmann-Ogden surface energy reduces to +Gurtin-Murdoch surface energy and crack tip singularities persist. The fact that +the Steigmann-Ogden theory reduces to the Gurtin-Murdoch theory for mode-III +loading is due to the fact that the linearized curvature vanishes for anti-plane shear, +i.e., for displacement fields tangent to the material surface. +1.2. Main results and outline. Inspired by [3] and motivated by obtaining a +model of brittle fracture capable of predicting bounded crack tip strains for all +modes of loading, this work proposes an augmentation of the Steigmann-Ogden +theory that is of strain-gradient type and includes the derivative of stretching in +the surface energy: +Φ[χ] = +ˆ +B +W(C) dA + +ˆ +S +U(C, κ, ∇C) dA, +(1.1) +where ∇ is the Levi-Cevita connection on S. For S contained in a plane, the surface +energy is equivalent to that introduced by Hilgers and Pipkin [7]. +In Section 2, the relevant kinematics of the boundary surface S convected by a +deformation of the bulk solid B are first summarized. In particular, we introduce +a third order tensor L that is obtained from certain transposes of ∇C and has +components in terms of the difference of Christoffel symbols on S and χ(S). This +tensor is used in a model surface energy proposed in Section 3. Physically, the tensor +L furnishes the rate of stretching of convected geodesics and locally characterizes +geodesic distortion, i.e., how convected geodesics fail to be geodesics on χ(S) (see +Proposition 2.1). The general form of (1.1) that we consider is then introduced +(see (2.4)), and inspired by Murdoch and Cohen’s extension to surfaces of Noll’s +classical theory of material symmetry, we introduce material symmetry for the +surface energy density U (see 2.2). In the final subsection of Section 2, we derive the +field equations (2.13) governing equilibrium states of the solid B with strain-gradient +elastic surface S ⊆ ∂B from a Lagrangian energy functional (2.10) including the +boundary relations between the boundary tractions and the surface stresses. +3Equivalently, it is unclear if the governing field equations from the Sendova-Walton model +can be derived from a Lagrangian energy functional. + +4 +C. RODRIGUEZ +In Section 3, we present a model hemitropic, quadratic surface energy that re- +quires the same number of material constants (with the same physical interpreta- +tions) as found in the Steigmann-Ogden theory (see (3.1)). In contradistinction, +however, the surface energy we propose satisfies the strong ellipticity condition +(see (3.6), (3.7)). We then derive the linearized field equations governing infinites- +imal displacements of B and S, including the boundary relations connecting the +linearized boundary tractions and the linearized surface stresses. +Finally, in Section 4, we apply the linearized theory (3.11) to the problem of +a brittle infinite solid with a straight, non-interfacial crack of finite length, under +mode-III loading. Using the explicit Dirichlet-to-Neumann map, the problem is +reduced to solving a fourth order integro-differential equation for the crack profile +along the boundary crack front (see (4.5)). Analytically, it is the surface energy +satisfying the strong ellipticity condition that implies that this integro-differential +equation is fourth order in the surface derivative of the displacement.4 The dimen- +sionless form of the equation implies that the behavior of the displacement depends +on the size of the crack, and for macro cracks, we expect the displacement to be +well-approximated by the solution given by classical linear elastic fracture mechan- +ics except in small regions near the crack tips (boundary layers). Finally, using the +Lax-Milgram theorem and regularity afforded by the presence of the fourth order +derivative, we prove that there always exist a unique classical solution to the gov- +erning integro-differential equation, and in contrast to using the Steigmann-Ogden +theory, our model predicts bounded strains up to the crack tips (see Theorem 4.4). +Acknowledgments. The author would like to thank Jay R. Walton for several +fruitful discussions related to fracture, surface elasticity and interfacial physics. +He would also like to thank Jeremy Marzuola for his help generating numerical +solutions to the mode-III fracture problem discussed in Section 4. +2. Kinematics and field equations +In this section formulate a general mathematical model for a hyperelastic bulk +solid containing a boundary surface with strain-gradient surface elasticity. +For +planar reference surfaces, the form of surface elasticity reduces to that introduced +by Hilgers and Pipkin [7], and when the surface energy is independent of the surface +gradient of the stretching, the theory reduces to that of Steigmann and Ogden +[22,23]. +2.1. Kinematics. Let E3 be three-dimensional Euclidean space with Cartesian +coordinates (X1, X2, X3) = X ∈ E3 and coordinate vector fields given by a fixed +orthonormal basis {ei}3 +i=1 of R3. We define the following operations for elementary +4Physically, it is the model incorporating resistance to geodesic distortion that implies that +this integro-differential equation is fourth order in the surface derivative of the displacement. + +STRAIN-GRADIENT ELASTIC SURFACES +5 +tensor products of vectors in R3, +(a1 ⊗ a2)b = (a2 · b)a1, +(a1 ⊗ a2 ⊗ a3)b = (a3 · b)a1 ⊗ a2, +(a1 ⊗ a2 ⊗ a3)(b1 ⊗ b2) = (a3 · b1)a1 ⊗ a2 ⊗ b2, +(b1 ⊗ b2)(a1 ⊗ a2 ⊗ a3) = (b2 · a1)b1 ⊗ a2 ⊗ a3, +(a1 ⊗ a2 ⊗ a3)[b1 ⊗ b2] = (a2 · b1)(a3 · b2)a1, +(a1 ⊗ a2)[b1 ⊗ b2] = (a1 · b1)(a2 · b2), +(a1 ⊗ a2 ⊗ a3)[b1 ⊗ b2 ⊗ b3] = (a1 · b1)(a2 · b2)(a3 · b3), +(a1 ⊗ a2)T = a2 ⊗ a1, +(a1 ⊗ a2 ⊗ a3)T = a1 ⊗ a3 ⊗ a2, +(a1 ⊗ a2 ⊗ a3)∼ = a2 ⊗ a1 ⊗ a3. +These operations are extended to general second and third order tensors on R3 by +linearity. +Let B ⊆ E3 be a domain with smooth boundary ∂B, the reference configuration +of a three-dimensional body, and let S ⊆ ∂B be a closed surface with smooth +boundary. Let χ : B → χ(B) ⊆ E3 be a smooth, invertible deformation of B. We +denote the current position of the reference particle X ∈ B by +x = χ(X) = (χ1(X), χ2(X), χ3(X)). +The deformation gradient F : R3 → R3 is the second order tensor field +F = ∂χi +∂Xa (X)ei ⊗ ea +where ea := ea, a = 1, 2, 3. We denote the left Cauchy-Green stretch tensor by +C = F T F . +Let Y = ˆY (θ1, θ2) be a local parameterization of the reference surface S ⊆ ∂B. +Then +y = ˆy(θ1, θ2) = χ( ˆY (θ1, θ2)) +is a local parameterization of the current surface χ(S) ⊆ χ(∂B). +The (local) +tangent vector fields on the reference and current surfaces are then given by +Y ,α ∈ TY S, +y,α = F Y ,α ∈ Tyχ(S), +where ·,α := +∂ +∂θα . +The dual tangent vector fields on the reference and current +surfaces are denoted by Y ,β and y,β respectively and satisfy +Y ,β · Y ,α = δβ +α, +y,β · y,α = δβ +α. +We remark that we may also write +y,α = FY ,α +where F = F k +β ek ⊗ Y ,β := yk +,βek ⊗ Y ,β = y,β ⊗ Y ,β is the surface deformation +gradient. The first fundamental forms for the reference and current surfaces are +then given by +G = GαβY ,α ⊗ Y ,β, +Gαβ = Y ,α · Y ,β, +g = gαβy,α ⊗ y,β, +gαβ = y,α · y,β, +and we note that +Y ,α = (G−1)αβY ,β, +y,α = (g−1)αβy,β + +6 +C. RODRIGUEZ +The Christoffel symbols of the second kind for the reference and current surfaces +are denoted by +Γα +βδ = Y ,α · Y ,βδ, +γα +βδ = y,α · y,βδ. +The left Cauchy-Green surface stretch tensor on S is the second order tensor +C := FT F = gαβY ,α ⊗ Y ,β, +and the surface Green-St. Venant tensor is the second order tensor E = 1 +2(C − G). +We assume that the reference and current surfaces are orientable with unit normals +to the reference and current surfaces (locally) given by +N = |Y ,1 × Y ,2|−1Y ,1 × Y ,2, +n = |y,1 × y,2|−1y,1 × y,2 +respectively. The second fundamental forms on the reference and current surfaces +are given by +B = BαβY ,α ⊗ Y ,β, +Bαβ := Y ,αβ · N, +b = bαβy,α ⊗ y,β, +bαβ := y,αβ · n. +The relative curvature tensor on S is the second order tensor K = KαβY ,α ⊗ Y ,β +given by +K = FT bF − B = [bαβ − Bαβ]Y ,α ⊗ Y ,β. +As discussed by Steigmann and Ogden [23], E and K furnish local differences in +length and scaled extrinsic normal curvature between a given curve on S and the +convected curve on χ(S), where ˙ := +d +ds. Indeed, if Z(s) is a a curve on S with +unit tangent vector field T and z(s) = χ(Z(s)) is the convected curve, then the +tangent to the convected curve is given by t = ˙z = F T = FT. The stretch ν of the +convected curve is +ν = |t|2 − 1 = FT · FT − 1 = CT · T − 1 = 2ET · T. +The extrinsic normal curvature of the convected curve is +κ = bt · t = |t|−2bFT · FT = (ν + 1)−1(FT bF)T, +and thus, the difference in scaled extrinsic normal curvature of the convected curve +and the original curve is given by +|t|2κ − |T|2BT · T = KT · T. +The Levi-Cevita connection on S is denoted by ∇, so +∇E = 1 +2∇C = 1 +2∇δgαβY ,α ⊗ Y ,β ⊗ Y ,δ, +where +∇δgαβ := gαβ,δ − Γµ +αδgµβ − Γµ +βδgµα. +For later use, we define the following third order tensor on S, +L = ∇E + (∇E)T − (∇E∼)T , +with components +Lαβδ = 1 +2(∇δgαβ + ∇βgαδ − ∇αgβδ) = (γµ +βδ − Γµ +βδ)gµα. + +STRAIN-GRADIENT ELASTIC SURFACES +7 +Physically, the tensor L (and thus ∇E) furnishes the rate of stretching of convected +geodesics, and more generally, it quantifies geodesic distortion, i.e. how convected +geodesics fail to be geodesics on χ(S). More precisely, we have the following. +Proposition 2.1. Let Z(s) : I → S be a geodesic on S with unit tangent vector field +T = ˙Z. Then L yields the rate of stretching of the convected curve z(s) = χ(Z(s)), +2L[T ⊗ T ⊗ T] = d +ds| ˙z|2. +(2.1) +Moreover, the convected curve z(s) is a geodesic on χ(S) if and only if for each +s0 ∈ I, +∀U ∈ TZ(s0)S, +L[U ⊗ T ⊗ T] +�� +s=s0 = 0. +(2.2) +Proof. To prove (2.1), we simply note that since Z is a geodesic, ∇TT = 0 so +d +ds| ˙z|2 = d +ds[CT · T] += ∇C[T ⊗ T ⊗ T] + 2C∇TT · T += ∇C[T ⊗ T ⊗ T] = 2L[T ⊗ T ⊗ T]. +We now prove (2.2). The convected curve z on χ(S) has tangent vector field +t = FT = tµy,µ and acceleration vector field +a = +�˙tµ + γµ +βδtβtδ� +y,µ. +The acceleration is zero and z is a geodesic if and only if for each s0 ∈ I, +∀u ∈ Tz(s0)χ(S), +u · a|s=s0 = 0. +(2.3) +We now show that (2.3) is equivalent to (2.2). We assume without loss of generality +that s0 = 0, and we choose normal coordinates (θ1, θ2) centered at Z(0). Then for +all α, β, δ = 1, 2, +Gαβ|s=0 = δαβ, +Γµ +βδ|s=0 = 0, +and there exist T1, T2 ∈ R with δαβTαTβ = 1 such that Z(s) = ˆY (sT1, sT2). In +particular, we conclude that +a = γµ +βδTβTδy,µ. +Since F|Z(0) : TZ(0)S → Tz(0)χ(S) is an isomorphism, (2.3) is equivalent to +∀U = UµY ,µ|Z(0) ∈ TZ(0)S, +FU · a +�� +s=0 = 0 +⇐⇒ ∀U = UµY ,µ|Z(0) ∈ TZ(0)S, +gµαUαγµ +βδTβTδ�� +s=0 = 0 +⇐⇒ ∀U = UµY ,µ|Z(0) ∈ TZ(0)S, +L[U ⊗ T ⊗ T] +�� +s=0 = 0. +□ +As an illustrative example, consider +S = {(X1, X2, 0) | X1 ∈ [a, b], X2 ∈ [0, π]} ⊆ B = [a, b] × [0, π] × [0, ∞), +and the deformation +χ(X1, X2, X3) = (eX1 cos X2, eX1 sin X2, X3). + +8 +C. RODRIGUEZ +Then +E = 1 +2(e2X1 − 1) +� +e1 ⊗ e1 + e2 ⊗ e2 +� +, +K = 0, +L = e2X1� +e1 ⊗ e1 ⊗ e1 + e2 ⊗ e1 ⊗ e2 + e2 ⊗ e2 ⊗ e1 − e1 ⊗ e2 ⊗ e2 +� +. +The image of the coordinate curve X2 = d is a straight, radially outward traveling +curve in the x1x2-plane parameterized by X1 ∈ [a, b] with +L[e1 ⊗ e1 ⊗ e1] = e2X1, +L[e2 ⊗ e1 ⊗ e1] = 0. +In particular, the stretch is not constant along the convected curve. The image +of the coordinate curve X1 = c is the upper-half of the circle in the x1x2-plane +centered at (0, 0) of radius ec. The convected curve is parameterized by X2 ∈ [0, π], +has constant stretch but nonzero curvature relative to the convected surface (i.e., +the x1x2-plane), and it satisfies +L[e2 ⊗ e2 ⊗ e2] = 0, +L[e1 ⊗ e2 ⊗ e2] = −e2c. +2.2. Strain energy and material symmetry. In our mathematical model, a +hyperelastic elastic body B with strain-gradient elastic surface S ⊆ ∂B is prescribed +a strain energy of the form +Φ[χ] = +ˆ +B +W(C) dV + +ˆ +S +U(E, K, ∇E) dA, +(2.4) +where we omit listing the possible dependence of W on X ∈ B and of U on Y ∈ S. +We note that the strain energy is frame indifferent, i.e., it is invariant with respect +to super-imposed rigid motions.5 +Although our main motivation for considering a strain-gradient elastic surface +is when it forms part of the boundary of a bulk solid B, we will treat material +symmetry of S independently of that of B. The notion of material symmetry for the +energy density W of the bulk three-dimensional solid B is well-known, see [18,24], +so, we will limit our discussion to that of S. +Consider a material point with reference position Y 0 ∈ S. Our discussion of +material symmetry for the surface energy per unit reference area U at Y 0 is in- +spired by the framework introduced by Murdoch and Cohen [16,17] which was later +advocated for and summarized by Steigmann and Ogden [23] using local coordinate +parameterizations. We will follow Steigmann and Ogden’s style of exposition, and +unless specified otherwise, all quantities in what follows are evaluated at Y 0. +Let λ : E3 → E3 be a rigid motion with deformation gradient R ∈ SO(3) +satisfying λ(Y 0) = Y 0, and RN = N. +We define a second reference surface +S∗ = +� +Y ∗ = λ−1(Y ) | Y ∈ S +� +. It follows that TY 0S∗ = TY 0S, at Y 0 ∈ S∗ the +unit normal N ∗ to S∗ satisfies N ∗ = N = RN, and +R := R|T Y 0S : TY 0S → TY 0S +is a rotation. +A local parameterization on S∗ is given by Y ∗ = ˆY +∗(θ1, θ2) := +λ−1( ˆY (θ1, θ2)), so then +Y ,α = RY ∗ +,α, +Y ,α = RY ∗,α, +α = 1, 2, +and at Y 0, Y ,α = RY ∗ +,α, Y ,α = RY ∗,α, for α = 1, 2. +5In the case of S contained in the plane, Hilgers and Pipkin showed in Section 7 of [7] that +U(E, K, ∇E) is the most general form of a surface energy density that is frame indifferent. + +STRAIN-GRADIENT ELASTIC SURFACES +9 +Let χ : E3 → E3 be a smooth invertible deformation of Euclidean space. Since a +superimposed rigid motion does not affect the value of the surface energy, we will +assume without loss of generality that +χ(Y 0) = Y 0 and at Y 0, F : TY 0S → TY 0S. +Following [17] we will impose the stronger requirement that +χ(Y 0) = Y 0 and at Y 0, F = Fα +βY ,α ⊗ Y ,β + N ⊗ N. +(2.5) +In particular, it follows that at Y 0, F = Fα +βY ,α ⊗ Y ,β and n = N = F N. The +deformation χ∗(·) := χ(λ(·)) when restricted to S∗ has the same image as χ|S, and +in particular, the convected tangent vector fields are the same, +y∗ +,α := +∂ +∂θα χ∗(Y ∗) = +∂ +∂θα χ(Y ) = y,α. +Then the surface deformation gradients of χ and χ∗ at Y 0 are related by +F = yα ⊗ Y ,α = y∗ +α ⊗ RY ∗,α = F∗RT =⇒ C = RC∗RT . +The components of the first and second fundamental forms associated to S and S∗ +satisfy +Gαβ = Y ,α · Y ,β = RY ∗ +,α · RY ∗ +β = Y ∗ +,α · Y ∗ +β = G∗ +αβ =⇒ G = RG∗RT , +Bαβ = Y ,αβ · N = RY ∗ +,αβ · RN ∗ = Y ∗ +,αβ · N ∗ = B∗ +αβ =⇒ B = RB∗RT , +and thus, E = RE∗RT . Since b = bαβy,α ⊗ yβ is the same for both deformations, +we conclude that the relative curvature tensors K and K∗ associated to χ and χ∗ +satisfy at Y 0, +K = RK∗RT . +Finally, since the components of the first fundamental forms associated to χ(S) +and χ∗(S∗) satisfy +gαβ = yα · yβ = y∗ +α · y∗ +β =: g∗ +αβ, +we conclude that ∇E and ∇∗E∗ at Y 0 satisfy +∇E = 1 +2∇δgαβY ,α ⊗ Y ,β ⊗ Y ,δ += 1 +2∇∗ +δg∗ +αβRY ∗,α ⊗ RY ∗,β ⊗ RY ∗,δ += R[(∇∗E∗)T RT ]T RT . +We denote the surface energy per unit reference area relative to S∗ by U ∗. For +the surface energy per unit mass to be independent of the reference surface used, +we must have +U ∗(E∗, K∗, ∇∗E∗) = U(E, H, ∇E) += U +� +RE∗RT , RK∗RT , R +� +(∇∗E∗)T RT �T RT � +. +(2.6) +We now view χ also as a deformation of S∗, but we denote its values by ¯χ, +¯χ(X) = χ(X), +X ∈ E3, +and the associated kinematic variables relative to S∗ are denoted with an over-bar. +We will now derive relationships between ¯E, ¯K, ∇∗¯E and E, K, ∇E. + +10 +C. RODRIGUEZ +We first note that since C = ¯C, we immediately conclude that +E = 1 +2[(C − I)Y ,α · Y ,β]Y ,α ⊗ Y ,β += 1 +2[( ¯C − I)Y ∗ +,α · Y ∗ +,β]Y ∗,α ⊗ Y ∗,β = ¯E. +(2.7) +Let +Grad C = ∂Cij +∂Xa ei ⊗ ej ⊗ ea, +a third order tensor on R3. The identity gαβ = CY ,α · Y ,β, the Gauss equations +Y ,αδ = ΓµαδY ,µ + BαδN on S, and the symmetry of C imply that +gαβ,δ = Grad C[Y ,α ⊗ Y ,β ⊗ Y ,δ] + CY ,αδ · Y ,β + CY ,α · Y ,βδ += Grad C[Y ,α ⊗ Y ,β ⊗ Y ,δ] + Γµ +αδgµβ + BαδCN · Y ,β ++ Γµ +βδgµα + BβδCN · Y ,α. +By (2.5), we have CN = N, and thus, at Y 0, +∇E = 1 +2Grad C[Y ,α ⊗ Y ,β ⊗ Y ,δ]Y ,α ⊗ Y ,β ⊗ Y ,δ. +In particular, since ¯C = C we conclude that +∇∗¯E = ∇E. +(2.8) +Finally, as shown in Section 6 of [23], we have +¯K = K. +(2.9) +Inspired by Murdoch and Cohen’s extension of Noll’s theory of material sym- +metry, we say that Y 0 ∈ S is symmetry related to Y 0 ∈ S∗ if the mechanical +responses to the arbitrary deformation χ are identical, i.e., +U(E, K, ∇E) = U ∗(¯E, ¯K, ∇¯E). +By (2.6), (2.7), (2.8), (2.9), this requirement leads to the following definition: the +rotation R is in the symmetry set of Y 0 relative to S if for every smooth, invertible +deformation χ : E3 → E3 satisfying (2.5), we have +U(E, K, ∇E) = U +� +RERT , RKRT , R +� +∇ET RT �T RT � +. +As in the case of the standard theory of Noll [18], one can verify that the symmetry +set of Y 0 relative to S is a subgroup of the group of proper rotations of TY 0S. +In the case that the symmetry set of Y 0 relative to S equals the group of proper +rotations of TY 0S, we say that the surface energy density U is hemitropic at Y 0. +2.3. Field equations. The field equations for the body B with strain-gradient +elastic surface S ⊆ ∂B are defined to be the Euler-Lagrange equations for the +Lagrangian energy functional +A[χ] = Φ[χ] + V [χ] +(2.10) +where V [χ] is the potential energy associated to the applied forces. In this work, +we assume that the Gˆateaux derivative of the load potential takes the form +˙V = − +ˆ +B +f · u dV − +ˆ +S +t · u dA, + +STRAIN-GRADIENT ELASTIC SURFACES +11 +where f is a prescribed external body force on B and t is a prescribed boundary +traction on S. +Let χ(·; ϵ) be a one-parameter family of deformations of B such that χ(·; ϵ)|∂B\S = +χ0(·), and denote +˙:= d +dϵ +��� +ϵ=0, +u := ˙χ(·; ϵ). +Then u|∂B vanishes to first order on ∂B\S. Using the chain rule and integration +by parts we have the classical identity +ˆ +B +˙W dV = +ˆ +∂B +P N · u dA − +ˆ +B +Div P · u dV +(2.11) +where P = P a +i ei⊗ea is the Piola stress with P a +i += +∂W +∂F ia and Div P = +� +∂XaP a +i +� +ei. +Using the chain rule we have that +ˆ +S +˙U dA = +ˆ +S +� +Tα · u,α + Mαβ · u,αβ +� +dA, +(2.12) +Tα := ∂U +∂ykα +ek, +Mαβ := +∂U +∂yk +,αβ +ek. +We define surface stress vectors Pα by +Pα := Tα − G−1/2(G1/2Mαβ),β, +and use (2.11), (2.12) and integration by parts to obtain the Euler-Lagrange equa- +tions associated to the Lagrangian energy functional (2.10), +Div P + f = 0, +on B, +P N = G−1/2(G1/2Pα),α + t, +on S, +χ(X) = χ0(X), +on ∂B\S. +(2.13) +3. Small strain models +Our principle motivation for modeling an elastic solid with strain-gradient elastic +boundary surface is the study of brittle fracture. In this setting (to be discussed +more in the following section), the surface S possessing strain-gradient surface +elasticity will be the crack front and strains will be linearized, motivating the intro- +duction of a quadratic surface energy density. In this section, we present a model +uniform, hemitropic, quadratic surface energy density that requires the same ma- +terial constants (with the same physical interpretations) as found in the narrower +Steigmann-Ogden theory. In contradistinction, the surface energy incorporates the +surface’s resistance to geodesic distortion and satisfies the strong ellipticity condi- +tion. Moreover, the surface energy density may be viewed as a geometric general- +ization of that introduced and advocated for by Hilgers and Pipkin in [7,8]. +3.1. Hilgers-Pipkin surface energy. For the surface S, we propose the uniform, +quadratic, hemitropic surface energy density +U = λs +2 (Eα +α)2 + µsEαβEαβ + ζ +2 +� +(Kα +α)2 + (g−1)µνL +α +µα L +β +νβ +� ++ η +� +KαβKαβ + (g−1)µνLµαβL αβ +ν +� +. +(3.1) +Here λs, µs, ζ and η are positive numbers that can be interpreted as the surface +Lam´e constants and pure bending moduli. Indices are raised and lowered using the + +12 +C. RODRIGUEZ +reference metric G, but note that y,α are the dual vector fields to y,α and are not +given by y,β(G−1)αβ. In the case that S is contained in a plane with flat coordinates +(θ1, θ2), we have +Eαβ = 1 +2(gαβ − δαβ), +Lµαβ = y,µ · y,αβ, +y,αβ = Lµαβy,µ + Kαβn, +2 +� +α=1 +y,αα = L +α +µα y,µ + Kα +αn, +��� +2 +� +α=1 +y,αα +��� +2 += (Kα +α)2 + (g−1)µνL +α +µα L +β +νβ , +2 +� +α,β=1 +|y,αβ|2 = KαβKαβ + (g−1)µνLµαβL αβ +ν +, +and (3.1) becomes +U = λs +2 (Eα +α)2 + µsEαβEαβ + ζ +2 +��� +2 +� +α=1 +y,αα +��� +2 ++ η +2 +� +α,β=1 +|y,αβ|2. +(3.2) +Up to a choice of constants, the surface energy density (3.2) is precisely that in- +troduced by Hilgers and Pipkin in [7], and therefore, we refer to (3.1) as a quadratic +Hilgers-Pipkin surface energy. In [7,8], Hilgers and Pipkin advocated for the use of +(3.2) over the classical surface energy +U = λs +2 (Eα +α)2 + µsEαβEαβ + ζ +2Kα +α + ηKαβKαβ += λs +2 (Eα +α)2 + µsEαβEαβ + ζ +2 +��� +2 +� +α=1 +(n · y,αα) +��� +2 ++ η +2 +� +α,β=1 +(n · y,αβ)2 +(3.3) +on the basis of (3.2) being analytically simpler than (3.3). Indeed, with little effort +one sees that for (3.2), +Tα = (λsEγ +γδαβ + 2µsEαβ)y,β, +(3.4) +Mαβ = ζ +�� +γ +y,γγ +� +δαβ + 2ηy,αβ. +(3.5) +Moreover, it is simple to see that (3.2) satisfies the strong ellipticity condition, +∀(a1, a2) ∈ R2\{(0, 0)}, b ∈ R3\{0}, +aαaβb · +� +Cαβδγaδaγb +� +> 0, +(3.6) +Cαβγδ := +∂2U +∂yi +,αβ∂yj +,δγ +ei ⊗ ej, +while (3.3) does not (see (3.7) and (3.8) below). Physically, this may be viewed as +a consequence of the surface energy (3.1) incorporating the surface’s resistance to +geodesic distortion via also including dependence on the tensor L. +In general, for (3.1) we have +Tα = ∂U +∂y,α += +∂U +∂Eβγ +∂Eβγ +∂y,α ++ ∂U +∂Kδν +∂Kδν +∂y,α ++ +∂U +∂Lβγδ +∂Lβγδ +∂y,α +. + +STRAIN-GRADIENT ELASTIC SURFACES +13 +Using +∂Eβγ +∂y,α += 1 +2 +� +δα +βy,γ + δα +γy,β +� +, +∂Kµν +∂y,α += −γα +µν n, +∂Kµν +∂y,αβ += 1 +2(δα +µδβ +ν + δα +νδβ +µ)n, +∂Lβγµ +∂y,α += δα +βyγµ − (δα +νy,β + δα +βy,ν)Γν +γµ, +∂Lγµσ +∂y,αβ += 1 +2(δα +µδβ +σ + δα +σδβ +µ)yγ, +we readily compute that +Tα = +� +λsEµ +µ(G−1)αγ + 2µsEαγ +− (g−1)µα(g−1)νγ(ζL +δ +µδ L +σ +νσ + 2ηLµδσL δσ +ν +) +� +y,γ, +− +� +ζKµ +µ(G−1)δνγα +δν + 2ηKδνγα +δν +� +n, ++ +� +ζ(g−1)µαL +ν +µν (G−1)γδ + 2η(g−1)µαL γδ +µ +� +y,γδ +− ζ +� +(g−1)µαL +ν +µν Γβ δ +δ + (g−1)µβL +ν +µν Γα δ +δ +� +y,β +− 2η +� +(g−1)µαL δσ +µ +Γβ +δσ + (g−1)µβL δσ +µ +Γα +δσ +� +y,β. +and +Mαβ = +� +ζKµ +µ(G−1)αβ + 2ηKαβ� +n ++ +� +ζ(g−1)µγL +ν +µν (G−1)αβ + 2η(g−1)µγL αβ +µ +� +y,γ. +To see that the strong ellipticity condition is satisfied for (3.1), we compute +Cαβµν = +� +ζ(G−1)αβ(G−1)µν + 2η[(G−1)αµ(G−1)βν + (G−1)αν(G−1)βµ] +� +I, +and thus, for all (a1, a2) ∈ R2\{(0, 0)} and b ∈ R3\{0}, +aαaβb · +� +Cαβδγaδaγb +� += (ζ + 2η)[(G−1)αβaαaβ]2|b|2 > 0. +(3.7) +For the classical surface energy (3.3) of Steigmann-Ogden type, we have +Cαβµν = +� +ζ(G−1)αβ(G−1)µν + 2η[(G−1)αµ(G−1)βν + (G−1)αν(G−1)βµ] +� +n ⊗ n, +and thus, for all (a1, a2) ∈ R2 and b ∈ R3, +aαaβb · +� +Cαβδγaδaγb +� += (ζ + 2η)[(G−1)αβaαaβ]2(n · b)2. +(3.8) +In particular, (3.8) shows that the surface energy (3.3) satisfies the Legendre- +Hadamard condition but not the strong ellipticity condition since the right side +of (3.8) is 0 for b ̸= 0 and orthogonal to n. + +14 +C. RODRIGUEZ +3.2. Linearized equations. We now compute the linearization of (4.1) about the +reference configuration. For the bulk solid, we adopt a classical quadratic, isotropic +energy density +W = λ +2 (Ei +i)2 + µEijEij. +Here indices are raised using the flat metric on R3, and λ and µ are the Lam´e +constants for the bulk solid. For the surface S, the surface energy density is given +by (3.1). +Let u : B → R3 be a displacement field such that u|∂B\S = 0 and +sup +X∈B +� +|u(X)| + |Grad u(X)| +� ++ +2 +� +α,β=1 +sup +Y ∈S +|u,αβ(Y )| ≤ δ0. +(3.9) +Assume that the body force f, boundary traction t, and Dirichlet condition χ0 +satisfy +|f| = O(δ0), +|t| = O(δ0), +|χ0 − Id| = O(δ0). +If χ(X) = X + u(X), then Eij = εij + O(δ2 +0), Eαβ = ϵαβ + O(δ2 +0), and Kαβ = +kαβ + O(δ2 +0) where +εij = 1 +2 +� +ei · ∂u +∂Xj + ej · ∂u +∂Xi +� += O(δ0), +and on S, +ϵαβ = 1 +2 +� +Y ,α · u,β + Y ,α · u,β +� += O(δ0), +kαβ = N · u;αβ = O(δ0), +see (3.12) in [23]. Now we observe that Lαβδ = lαβδ + O(δ2 +0), with +lαβδ = Y ,α · u,βδ + Y ,βδ · u,α − Γµ +βδ +� +Y ,α · u,µ + Y ,µ · u,α +� += Y ,α · u;βδ + Y ;βδ · u,α += Y ,α · u;βδ + (N · u,α)Bβδ = O(δ0). +Then Tα = tα + O(δ2 +0) and Mαβ = mαβ + O(δ2 +0) where +tα := +� +λsϵµ +µ(G−1)αγ + 2µsϵαγ� +Y ,γ − +� +ζkµ +µΓα ν +ν + 2ηkδνΓα +δν +� +N, ++ +� +ζlα ν +ν Bδ +δ + 2ηlαδσBδσ +� +N − +� +ζlβ ν +ν Γα δ +δ + 2ηlβδσΓα +δσ +� +Y ,β, +mαβ := +� +ζkµ +µ(G−1)αβ + 2ηkαβ� +N + +� +ζlγ ν +ν (G−1)αβ + 2ηlγαβ� +Y ,γ. +The linearization of (2.13) about the reference configuration is obtained by omit- +ting the O(δ2 +0) terms from P , Tα and Mαβ, yielding +Div σ + f = 0, +on B, +σN = G−1/2(G1/2pα),α + t, +on S, +u = 0, +on ∂B\S, +(3.10) + +STRAIN-GRADIENT ELASTIC SURFACES +15 +where σ = λ(trε)I + 2µε and pα = tα − G−1/2(G1/2mαβ),β. We observe that solu- +tions to the linearized equations (3.10) are critical points of the energy functional +AL[u] = +ˆ +B +�λ +2 (εi +i)2 + µεijεij� +dV − +ˆ +B +f · u dV + +ˆ +S +�λs +2 (ϵα +α)2 + µsϵαβϵαβ� +dS ++ +ˆ +S +�ζ +2 +� +(kα +α)2 + l +α +µα lµ β +β +� ++ η +� +kαβkαβ + lµαβlµαβ�� +dA − +ˆ +S +t · u dA +over the set of u satisfying u|∂B\S = 0. +In the case of the classical Hilgers-Pipkin surface energy (3.2), we see from (3.4) +and (3.5) that +tα = (λsϵγ +γδαβ + 2µsϵαβ)Y ,β, +mαβ = ζδαβu +γ +,γ + 2ηu,αβ. +Writing u = u + u3N = uγY γ + u3N, it follows that (3.10) becomes +Div σ + f = 0, +on B, +σN = µsu +α +,α + (λs + µs)uγ +,αγY α − (ζ + 2η)∂α∂βu,αβ + t, +on S, +u = 0, +on ∂B\S. +(3.11) +4. Mode-III Fracture Problem +In this section, we apply the linearized theory (3.11) to the problem of a brittle +infinite plate, with a straight crack C of length 2ℓ, under far-field anti-plane shear +loading σ. As discussed in the Introduction and in contrast to ascribing either a +quadratic Gurtin-Murdoch or Steigmann-Ogden surface energy (3.3) to the crack +fronts, the use of (3.1) yields a model that predicts bounded strains and stresses +up to the crack tips (see Theorem 4.4). +4.1. Formulation and governing equations. We consider a brittle, infinite +plate under anti-plane shear loading limx2→±∞ σ23 = ±σ, with a straight crack +C = {(x1, 0, x3) | x1 ∈ [−ℓ, ℓ]} of length 2ℓ (see Figure 2). For anti-plane shear, the +displacement field takes the form +u(x1, x2, x3) = u(x1, x2)e3, +Then the only nonzero components of the stress are +σ13 = µu,1, +σ23 = µu,2 +By the symmetry of the problem, u will be odd in x2, so we will focus only on +the strain and stress fields for x2 ≥ 0. The governing field equations are (3.11) on +B = {(x1, x2, x3) | x2 ≥ 0} with S = {(x1, 0, x3) | x1 ∈ [−ℓ, ℓ]}, t = 0 and f = 0. +We define dimensionless variables +x = x1 +ℓ , +y = x2 +ℓ , +z = x3 +ℓ , +w(x, y, z) = 1 +ℓ +� +u(x1, x2, x3) − σ +µx2� +. +Then the field equations take the dimensionless form +∆w(x, y) = 0, +y > 0, +− wy(x, 0) = αwxx(x, 0) − βwxxxx(x, 0) + γ, +x ∈ (−1, 1), +w(x, 0) = 0, +|x| ≥ 1, +wx(±1, 0) = 0, +(4.1) + +16 +C. RODRIGUEZ +with the decay condition limy→∞ wy(x, y) = 0. We note that the boundary con- +ditions wx(±1, 0) = 0 imply that the crack opening is cusp shaped rather then +blunted. The dimensionless parameters α, β and γ are given by +α = µs +µℓ > 0, +β = ζ + 2η +µℓ3 +> 0, +γ = σ +µ, +(4.2) +and in particular, we see from (4.2) that the behavior of the displacement w depends +on the length of the crack, ℓ. For macro cracks satisfying β ≪ α ≪ 1, we expect +w(x, 0) to be well-approximated by the singular, rounded opening profile from the +classical linear elastic fracture mechanics except in small regions near the crack tips +(boundary layers). See Figure 4. +We remark that in using the Steigmann-Ogden surface energy (3.3) rather than +(3.1), the boundary conditions at y = 0 are replaced by +− wy(x, 0) = αwxx(x, 0) + γ, +x ∈ (−1, 1), +w(x, 0) = 0, +|x| ≥ 1. +(4.3) +One may view this loss of higher order derivatives in the boundary conditions as a +consequence of the fact that the Steigmann-Ogden surface energy does not satisfy +the strong-ellipticity condition (3.6): for anti-plane shear, b = u,11(x1, 0, x3) is +orthogonal to the surface’s normal n = −e2 (see (3.8)). As discussed in [9,25], the +boundary conditions (4.3) do not lead to a model predicting bounded strains up to +the crack tips x = ±1 (see [9,25]), i.e., the displacement field satisfies +sup +y>0 +|∇w(±1, y)| = ∞. +We see that (4.1) is the system of Euler-Lagrange equations for the energy func- +tional +AL[w] = 1 +2 +ˆ ∞ +0 +ˆ ∞ +−∞ +|∇w(x, y)|2dxdy + +ˆ ∞ +−∞ +�α +2 |wx(x, 0)|2 + β +2 |wxx(x, 0)|2� +dx +− γ +ˆ ∞ +−∞ +w(x, 0)dx +defined for w with ∇w ∈ L2({y > 0}), w(·, 0) ∈ H2(R) and w(x, 0) = 0 for all +|x| ≥ 1. Motivated by this observation, we define the Hilbert space H to be the +x3 +x1 +x2 +σe3 +−σe3 +(−ℓ, 0, 0) +(ℓ, 0, 0) +Figure 2. Schematic of the mode-III problem with the crack C +appearing in blue. + +STRAIN-GRADIENT ELASTIC SURFACES +17 +completion of C∞ +c ((−1, 1)) under the norm +∥f∥2 +H := +ˆ ∞ +−∞ +� +α|f ′(x)|2 + β|f ′′(x)|2� +dx. +It is straightforward to verify the following facts using the fundamental theorem of +calculus and Cauchy-Schwarz inequality: +• (Sobolev embedding) If f ∈ H then f ∈ C1,1/2− +c +(R) and f(x) = 0 for all +|x| ≥ 1, and for all δ ∈ [0, 1/2), there exists a constant A > 0 depending on +α, β and δ, such that for all f ∈ H, +∥f∥C1,γ(R) ≤ A∥f∥H. +• f ∈ H if and only if f ∈ H2(R) and f(x) = 0 for all |x| ≥ 1. Moreover, +there exist b, B > 0 depending on α and β such that for all f ∈ H, +b∥f∥H ≤ ∥f∥H2(R) ≤ B∥f∥H. +(4.4) +The problem (4.1) can be reduced completely to a problem on the boundary by +using the Dirichlet-to-Neumann map −wy(x, 0) = Hwx(x, 0) where H is the Hilbert +transform +Hf(x) = 1 +π p.v. +ˆ ∞ +−∞ +f(s) +x − sds, +f ∈ H. +Then finding w with ∇w ∈ L2({y > 0}) and w(·, 0) ∈ H satisfying (4.1) is equiva- +lent to determining w(·, 0) =: f ∈ H satisfying6 +βf ′′′′(x) − αf ′′(x) + Hf ′(x) = γ, +x ∈ (−1, 1). +(4.5) +By using the Plancherel theorem, the Fourier representation of the Hilbert trans- +form (see (4.13)), and (4.4), we have for all f ∈ H, +∥Hf ′∥H1(R) ≤ ∥f ′∥H1(R) ≤ ∥f∥H2(R) ≤ B∥f∥H. +(4.6) +Definition 4.1. A function f ∈ H is a weak solution to the integro-differential +equation (4.5) if for all g ∈ H +ˆ ∞ +−∞ +[βf ′′(x)g′′(x) + αf ′(x)g′(x) + Hf ′(x)g(x) +� +dx = +ˆ ∞ +−∞ +γg(x)dx. +(4.7) +We remark that since f, g ∈ H, the integrals appearing in (4.7) are in fact over the +interval (−1, 1). +A function f ∈ H is a classical solution to (4.5) if f ∈ C4((−1, 1)) ∩ H and f +satisfies (4.5) pointwise. +We note that by (4.6) and Cauchy-Schwarz, (4.7) is well-defined for each f, g ∈ +H. +6Once f is found, w is determined on the upper half plane using the standard Poisson kernel +for the upper half plane. + +18 +C. RODRIGUEZ +Figure 3. Numerical solutions for the equivalent formulation of +(4.5) as a Fredholm problem (4.10). The parameters (β, α, γ) range +over (1, 1, 1), (5, 1, 5) and (10, 1, 10). For γ = β and β ≫ 1 ≃ α, +the opening profile is well approximated by the limiting opening +profile f∞(x) = +1 +24(1 − x2)2 on [−1, 1]. +4.2. Solution of the integro-differential equation. We now establish that +there exists a unique classical solution to (4.5), and the solution’s behavior is con- +sistent with the linearization assumption (3.9). +We denote the following Green +function +G(x, τ) = +� +1 +24(x − 1)2(τ + 1)2(1 + 2x − 2τ − xτ) +τ ∈ [−1, x], +1 +24(τ − 1)2(x + 1)2(1 + 2τ − 2x − xτ) +τ ∈ [x, 1], +satisfying Gxxxx(x, τ) = δ(x − τ), G(±1, τ) = 0, Gx(±1, τ) = 0. We note that +G(x, τ) = G(τ, x) for all τ, x ∈ [−1, 1], G ∈ C2([−1, 1]×[−1, 1]) and +´ 1 +−1 G(x, τ)dτ = +1 +24(1 − x2)2. In particular, we have for all f ∈ H, +ˆ 1 +−1 +Gττ(x, τ)f ′′(τ)dτ = f(x). +(4.8) +Lemma 4.2. A function f ∈ H is a weak solution to (4.5) if and only if f satisfies +βf(x) + +ˆ 1 +−1 +G(x, τ)(−αf ′′(τ) + Hf ′(τ))dτ = γ +24(1 − x2)2, +x ∈ [−1, 1]. (4.9) +Proof. Let h ∈ L2(R) with h(x) = 0 for all |x| > 1, and set +g(x) = +�´ 1 +−1 G(x, τ)h(τ)dτ +if |x| ≤ 1, +0 +if |x| > 1. +Since G ∈ C2([−1, 1] × [−, 1, 1]), G(±1, τ) = 0, and Gx(±1, τ) = 0, g is twice +continuously differentiable on R\{±1} and continuously differentiable on R with +g′(x) = χ{|x|≤1}(x) +ˆ 1 +−1 +Gx(x, τ)h(τ)dτ, +g′′(x) = χ{|x|≤1}(x) +ˆ 1 +−1 +Gxx(x, τ)h(τ)dτ, + +0.05 +β=1 +..β=5 +-β= 10 +0.04 +f +0.03 +J +0.02 +0.01 +0- +-1 +-0.8 +-0.6 +-0.4 +-0.2 +0 +0.2 +0.4 +0.6 +0.8 +xSTRAIN-GRADIENT ELASTIC SURFACES +19 +where χE is the indicator function of a subset E ⊆ R. In particular, we conclude +that g ∈ H2(R) and thus, g ∈ H. Inserting g into (4.7), integrating by parts in the +second term and using that g(±1) = 0 yield +β +ˆ 1 +−1 +ˆ 1 +−1 +Gxx(x, τ)f ′′(x)h(τ)dτdx ++ +ˆ 1 +−1 +ˆ 1 +−1 +G(x, τ)(−αf ′′(x) + Hf ′(x))h(τ)dτdx += γ +ˆ 1 +−1 +ˆ 1 +−1 +G(x, τ)h(τ)dτdx. +Interchanging the order of integration, using the symmetry of G and relabeling the +integration variables lead to +β +ˆ 1 +−1 +ˆ 1 +−1 +Gττ(x, τ)f ′′(τ)dτ h(x)dx ++ +ˆ 1 +−1 +ˆ 1 +−1 +G(x, τ)(−αf ′′(τ) + Hf ′(τ))dτ h(x)dx += +ˆ 1 +−1 +γ +24(1 − x2)2h(x)dx. +Finally, by (4.8) we conclude that +ˆ 1 +−1 +� +βf(x) + +ˆ 1 +−1 +G(x, τ)(−αf ′′(τ) + Hf ′(τ))dτ +� +h(x)dx += +ˆ 1 +−1 +γ +24(1 − x2)2h(x)dx, +for all h(x) ∈ L2(R) with h(x) = 0 for all |x| > 1, proving (4.9). +Conversely, if f ∈ H and (4.9) holds, then for all g ∈ H, we have +β +ˆ ∞ +−∞ +f ′′(x)g′′(x)dx = +ˆ 1 +−1 +ˆ 1 +−1 +Gxx(x, τ)(αf ′′(τ) − Hf ′(τ))g′′(x)dx ++ +ˆ 1 +−1 +γ +6 (3x2 − 1)g′′(x)dx. +We again interchange the order of integration and use integration by parts and +´ 1 +−1 Gxx(x, τ)g′′(x)dx = g(τ) to conclude that +ˆ 1 +−1 +ˆ 1 +−1 +Gxx(x, τ)(αf ′′(τ) − Hf ′(τ))g′′(x)dx + +ˆ 1 +−1 +γ +6 (3x2 − 1)g′′(x)dx += +ˆ 1 +−1 +(αf ′′(τ) − Hf ′(τ))g(τ)dτ + +ˆ 1 +−1 +γg(x)dx += − +ˆ 1 +−1 +(αf ′(x)g′(x) + Hf ′(x)g(x))dx + +ˆ 1 +−1 +γg(x)dx +This proves f ∈ H satisfies (4.7) and concludes the proof. +□ +Via integration by parts and straightforward computations, we conclude from +Lemma 4.2 that the crack opening profile f must satisfy the following Fredholm +integral equation of the second kind (4.10). We will show in Theorem 4.4 that this + +20 +C. RODRIGUEZ +Figure 4. Numerical solutions for the macro-crack regime β ≪ +α ≪ 1. +The parameters (β, α, γ) range over (10−1, 10−1, 1), +(10−2, 10−1, 1), (10−5, 10−2, 1) and (10−6, 10−3, 1). For β ≪ α ≪ +1, we expect the crack opening f(x) to be well-approximated by the +singular, rounded opening profile predicted by classical linear elas- +tic fracture mechanics away from the crack tips where f ′(±1) = 0 +(and the profile is cusped). +equation is uniquely solvable for arbitrary α, β > 0 and γ. We remark that since +the kernel extends to a continuous function on [−1, 1] × [−1, 1] (the singularities +are removable), the numerical computation of solutions is relatively straightforward +via the N¨ystrom method with the trapezoidal rule to approximate the integral (see +Figure 3 and Figure 4). +Corollary 4.3. A function f ∈ H is a weak solution to (4.5) if and only if f +satisfies the Fredholm equation +βf(x) + +ˆ 1 +−1 +K(x, s)f(s)ds = γ +24(1 − x2)2, +x ∈ [−1, 1], +(4.10) +where K(x, s) = −αGss(x, s) + 1 +πp.v. +´ 1 +−1 +Gτ (x,τ) +s−τ +dτ, +Gss(x, s) = +� +− 1 +4(x − 1)2(2s + xs + 1) +s ∈ [−1, x], +− 1 +4(x + 1)2(−2s + xs + 1) +s ∈ [x, 1], + +1 +-β=0.1 +-β = 0.01 +--β = 0.00001 +0.8 +-β = 0.000001 +0.6 +f +0.4 +0.2 +0 +-0.8 +-0.6 +-0.4 +-0.2 +0 +0.2 +0.4 +0.6 +0.80.5 +- β= 0.1 +-β = 0.01 +0.4 +--β= 0.00001 +β = 0.000001 +0.3 +0.1 +0 +-0.1 +-0.99 +-0.98 +-0.97 +-0.96 +-0.95 +-0.94 +-0.93 +-0.92 +-0.91 +-0.9 +-1STRAIN-GRADIENT ELASTIC SURFACES +21 +and +1 +π p.v. +ˆ 1 +−1 +Gτ(x, τ) +s − τ +dτ = 1 +4π (sx − 1)(x2 − 1) + 1 +2π (s − x)2 log |x − s| +− 1 +8π (x − 1)2(−x + 2s + sx)(1 + s) log(1 + s) +− 1 +8π (x + 1)2(x − 2s + sx)(1 − s) log(1 − s). +Theorem 4.4. There exists C > 0 depending on α and β such that the following +hold. There exists a unique classical solution f to (4.5), and f satisfies +∥f∥C4([−1,1]) ≤ C|γ|. +(4.11) +Moreover, the displacement field w(x, y) = +´ ∞ +−∞ Py(x − s)f(s)ds, where Py(·) is the +Poisson kernel for the upper half plane, has bounded strains up to the crack tips: +∥w∥C1({y≥0}) ≤ C|γ|. +(4.12) +Proof. In what follows, C will denote a positive constant depending only on α and +β that may change from line to line, and we denote the Fourier transform and +inverse Fourier transform by +ˆf(ξ) = +ˆ ∞ +−∞ +f(x)e−2πixξdx, +ˇf(x) = +ˆ ∞ +−∞ +f(ξ)e2πixξ dξ. +We recall that +�f ′(ξ) = 2πiξ ˆf(ξ), +� +Hf(ξ) = −isgn(ξ) ˆf(ξ), +(4.13) +the latter relation following from the Fourier representation of w on the upper half +plane (see (4.15)). +We define a bilinear form B(·, ·) : H × H → R by +B(f, g) = +ˆ ∞ +−∞ +[βf ′′(x)g′′(x) + αf ′(x)g′(x) + Hf ′(x)g(x) +� +dx, +f, g ∈ H. +By Cauchy-Schwarz, (4.6), and (4.4) we conclude that for all f, g ∈ H, +|B(f, g)| ≤ (1 + B2)∥f∥H∥g∥H, +so that B(·, ·) is a bounded bilinear form on H. +Moreover, by the Plancherel +theorem and (4.13), we have for all f ∈ H, +ˆ ∞ +−∞ +Hf ′(x)f(x)dx = +ˆ ∞ +−∞ +2π|ξ|| ˆf(ξ)|2dξ ≥ 0. +Thus, the bilinear form is coercive. Since for all g ∈ H, +��� +ˆ ∞ +−∞ +γg(x)dx +��� ≤ |γ|∥g∥C([−1,1]) ≤ |γ|A∥g∥H, +the classical Lax-Milgram theorem implies that there exists a unique weak solution +f ∈ H to (4.5), and moreover, +∥f∥H ≤ A|γ|. +(4.14) +To prove (4.12), we express w via the Fourier transform, +w(x, y) = +ˆ ∞ +−∞ +e−2πy|ξ|e2πixξ ˆf(ξ)dξ. +(4.15) + +22 +C. RODRIGUEZ +Since f ∈ H ⊂ H2(R), we have +ˆ +(1 + |ξ|)4| ˆf(ξ)|2dξ ≤ C0∥f∥2 +H2(R) ≤ C∥f∥2 +H. +Thus, by Cauchy-Schwarz +|w(x, y)| + |∇w(x, y)| +≤ +ˆ ∞ +−∞ +(1 + 2π|ξ| +√ +2)| ˆf(ξ)|2 dξ +≤ 2π +√ +2 +�ˆ ∞ +−∞ +|ξ|2(1 + |ξ|)−4dξ +�1/2�ˆ ∞ +−∞ +(1 + |ξ|)4| ˆf(ξ)|2dξ +�1/2 +≤ C∥f∥H ≤ C|γ|. +We now show that the weak solution f is a classical solution, and (4.11) holds. By +a density argument in H, we have that +´ 1 +−1 G(·, τ)(αf ′′(τ)−Hf ′(τ))dτ ∈ H3([−1, 1]) +with +dk +dxk +ˆ 1 +−1 +G(x, τ)(αf ′′(τ) − Hf ′(τ))dτ += +ˆ 1 +−1 +∂k +xG(x, τ)(αf ′′(τ) − Hf ′(τ))dτ, +k = 1, 2, +d3 +dx3 +ˆ 1 +−1 +G(x, τ)(αf ′′(τ) − Hf ′(τ))dτ += +ˆ x +−1 +1 +4(2 − τ)(τ + 1)2(αf ′′(τ) − Hf ′(τ))dτ +− +ˆ 1 +x +1 +4(2 + τ)(τ − 1)2(αf ′′(τ) − Hf ′(τ))dτ. +(4.16) +By Lemma 4.2, we conclude that f ∈ H3([−1, 1]), and by (4.9) (4.16), Cauchy- +Schwarz, and (4.14) we have +∥f ′′′∥L2([−1,1]) ≤ C(∥f ′′∥L2(R) + ∥Hf ′∥L2(R) + |γ|) +≤ C(∥f∥H + |γ|) ≤ C|γ|. +(4.17) +Moreover, by (4.14), (4.17) and the fundamental theorem of calculus, f ∈ C2([−1, 1]) +with +∥f∥C2([−1,1]) ≤ C|γ|. +(4.18) +By (4.7) and integration by parts, it follows that for all g ∈ C∞ +c ((−1, 1)) ⊂ H, +ˆ 1 +−1 +f ′′′(x)g′(x)dx = 1 +β +ˆ 1 +−1 +[−αf ′′(x) + Hf ′(x) − γ]g(x)dx, +and, thus, f ∈ H4([−1, 1]) and +βf ′′′′(x) − αf ′′(x) + Hf ′(x) = γ, +for a.e. x ∈ (−1, 1). +(4.19) + +STRAIN-GRADIENT ELASTIC SURFACES +23 +Then by (4.19), (4.18), and the fact that Hf ′ ∈ H1(R) �→ C(R), we conclude that +f ′′′′ ∈ C([−1, 1]) and +∥f ′′′′∥C([−1,1]) ≤ C +� +∥f∥C2([−1,1]) + ∥Hf ′∥C([−1,1]) + |γ| +� +≤ C +� +∥Hf ′∥H1(R) + |γ| +� +≤ C +� +∥f∥H + |γ| +� +≤ C|γ|. +(4.20) +By (4.17), (4.18) and (4.20), it follows that f ∈ C4([−1, 1]) is a classical solution +to (4.5) and (4.11) holds. +□ +References +[1] K. 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Axisymmetric frictionless indentation of a rigid stamp +into a semi-space with a surface energetic boundary. Math. Mech. Solids, 27(2):334–347, 2022. +C. Rodriguez +Department of Mathematics, University of North Carolina +Chapel Hill, NC 27599, USA +crodrig@email.unc.edu + diff --git a/odFST4oBgHgl3EQfMjgS/content/tmp_files/load_file.txt b/odFST4oBgHgl3EQfMjgS/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5de89feb56952906726a50df45b4098de0c3619a --- /dev/null +++ b/odFST4oBgHgl3EQfMjgS/content/tmp_files/load_file.txt @@ -0,0 +1,915 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf,len=914 +page_content='ELASTIC SOLIDS WITH STRAIN-GRADIENT ELASTIC BOUNDARY SURFACES C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' RODRIGUEZ Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Recent works have shown that in contrast to classical linear elastic fracture mechanics, endowing crack fronts in a brittle solid with Steigmann- Ogden surface elasticity yields a model that predicts bounded strains at the crack tips for plane-strain problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' However, a logarithmic singularity is still present in general for anti-plane shear (mode-III fracture) even when Steigmann-Ogden surface elasticity is incorporated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Motivated by obtaining a model of brittle fracture capable of predicting bounded strains for all modes of loading, we formulate an exact general the- ory of a bulk solid containing a boundary surface with strain-gradient surface elasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' For planar reference surfaces, the form of surface elasticity reduces to that introduced by Hilgers and Pipkin, and when the surface energy is independent of the surface gradient of the stretching, the theory reduces to that of Steigmann and Ogden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' We give a full discussion of material symmetry using Murdoch and Cohen’s extension of Noll’s theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' We present a model quadratic surface energy that incorporates resistance to geodesic distortion, satisfies strong ellipticity, and requires the same material constants found in the Steigmann-Ogden theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Finally, we derive and apply the linearized theory to mode-III fracture in an infinite plate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' We prove that there always exists a unique classical solu- tion to the governing integro-differential equation, and in contrast to using Steigmann-Ogden surface elasticity, our model is consistent with the lineariza- tion assumption in predicting finite strains at the crack tips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Introduction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Surface stressed solid bodies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' The study of surface tension for solids was initiated by Gibbs in 1857, and it is now widely accepted that surface tension and more general surfaces stresses must be accounted for when modeling mechanical structures at small length scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' In particular, interfaces1 between a bulk solid and its environment can form due to various mechanisms including coating or atomic rearrangement during fracture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' A way to mathematically model such an interface is by endowing part of the three-dimensional body’s two-dimensional boundary with it’s own thermodynamic properties that are distinct from the bulk (such as energy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Motivated by modeling the apparent compressive surface stresses in certain cleaved crystals, Gurtin and Murdoch [4,5] developed a rigorous general theory of material surface stresses that accounts for the surface’s resistance to stretching (but not flexure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Their celebrated theory has been used to model a wide arrange of phe- nomena over the past 45 years, especially recently due to advances in nanoscience and nanotechnology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' In the special case of a hyperelastic, three-dimensional bulk 1By an interface, we mean a thin region separating either two distinct materials or two distinct phases of a material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='13744v1 [math-ph] 31 Jan 2023 2 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' RODRIGUEZ solid with reference configuration B and material surface S ⊆ ∂B, the field equa- tions for the current configuration χ : B → χ(B) are the Euler-Lagrange equations associated to the total strain energy Φ[χ] = ˆ B W(C) dA + ˆ S U(C) dA, where we omit the possible explicit dependence of the functions W and U on points in B and S (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Here C is the left Cauchy-Green stretch tensor and C is the left Cauchy-Green surface stretch tensor, the pullbacks by χ of the metric tensors on χ(B) and χ(S), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' In particular, the material surface stress tensor is derived from the surface energy density U(C) in analogy with the bulk’s Piola stress being derived from the bulk energy density W(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' However, Steigmann and Ogden showed in their seminal works [22, 23] that equilibrium states under compressive surface stresses obtained from the Gurtin- Murdoch theory do not satisfy an associated Legendre-Hadamard condition, and thus, these equilibrium states cannot be local energy minimizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='2 Steigmann and Ogden [22, 23] rectified this inadequacy and incorporated the material surface’s resistance to flexure by including curvature dependence in the surface energy: Φ[χ] = ˆ B W(C) dA + ˆ S U(C, κ) dA, where κ is the pullback of the second fundamental form on χ(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Since κ depends on the second derivatives of χ, the surface energy is of strain-gradient type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' In recent years, the Steigmann-Ogden theory has attracted considerable interest from various perspectives including the study of contact problems [10–13, 31, 32, 37], inclusion problems [2, 6, 14, 15, 27, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' The theory has also been used to model fracture in brittle materials [30,33,34], the main phenomenon motivating this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' One of the most successful and practical theories for modeling fracture in brittle materials is classical linear elastic fracture mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' The governing linear par- tial differential equations are derived from finite elasticity under the assumption of infinitesimal strains (linearized elasticity), but the theory predicts unbounded singular strains at the crack tips, a striking inconsistency and physically impossi- ble prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' There have been a vast number of suggestions for supplementing 2The fact that a pure membrane, a special type of Gurtin-Murdoch material surface, in equi- lbrium and under compressive surface stresses cannot by locally energy minimizing is due to Pipkin [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' S B Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' The reference configuration of a bulk solid B with strain-gradient elastic surface S ⊆ ∂B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' STRAIN-GRADIENT ELASTIC SURFACES 3 classical linear elastic fracture mechanics to correct this defect in the theory (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=', [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' A more recent approach aimed at eliminating these crack tip singularities is to modify the boundary conditions of classical linear elastic fracture mechanics by endowing the crack fronts with material surface stresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Beginning with [19] and developed further by Sendova and Walton [21], one line of thought has been to pre- scribe the crack front’s surface stresses as a curvature dependent surface tension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Although able to remove the singularities completely in a diverse range of settings (see [3,26,28,29,36]), it is unclear if the surface stresses from the Sendova-Walton theory can be derived from a surface energy density,3 a reasonable definition of “elastic-like” behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Deriving the material stresses for the crack fronts from a Gurtin-Murdoch surface energy does not completely remove the crack tip sin- gularities (see [9, 25]), but however, endowing the crack fronts with Steigmann- Ogden surface energy does completely remove the singularities for plane-strain problems [30, 33] and axisymmetric penny shaped cracks [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Unfortunately, for anti-plane shear (mode-III loading), Steigmann-Ogden surface energy reduces to Gurtin-Murdoch surface energy and crack tip singularities persist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' The fact that the Steigmann-Ogden theory reduces to the Gurtin-Murdoch theory for mode-III loading is due to the fact that the linearized curvature vanishes for anti-plane shear, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=', for displacement fields tangent to the material surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Main results and outline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Inspired by [3] and motivated by obtaining a model of brittle fracture capable of predicting bounded crack tip strains for all modes of loading, this work proposes an augmentation of the Steigmann-Ogden theory that is of strain-gradient type and includes the derivative of stretching in the surface energy: Φ[χ] = ˆ B W(C) dA + ˆ S U(C, κ, ∇C) dA, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='1) where ∇ is the Levi-Cevita connection on S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' For S contained in a plane, the surface energy is equivalent to that introduced by Hilgers and Pipkin [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' In Section 2, the relevant kinematics of the boundary surface S convected by a deformation of the bulk solid B are first summarized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' In particular, we introduce a third order tensor L that is obtained from certain transposes of ∇C and has components in terms of the difference of Christoffel symbols on S and χ(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' This tensor is used in a model surface energy proposed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Physically, the tensor L furnishes the rate of stretching of convected geodesics and locally characterizes geodesic distortion, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=', how convected geodesics fail to be geodesics on χ(S) (see Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' The general form of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='1) that we consider is then introduced (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='4)), and inspired by Murdoch and Cohen’s extension to surfaces of Noll’s classical theory of material symmetry, we introduce material symmetry for the surface energy density U (see 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' In the final subsection of Section 2, we derive the field equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='13) governing equilibrium states of the solid B with strain-gradient elastic surface S ⊆ ∂B from a Lagrangian energy functional (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='10) including the boundary relations between the boundary tractions and the surface stresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' 3Equivalently, it is unclear if the governing field equations from the Sendova-Walton model can be derived from a Lagrangian energy functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' 4 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' RODRIGUEZ In Section 3, we present a model hemitropic, quadratic surface energy that re- quires the same number of material constants (with the same physical interpreta- tions) as found in the Steigmann-Ogden theory (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' In contradistinction, however, the surface energy we propose satisfies the strong ellipticity condition (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='6), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='7)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' We then derive the linearized field equations governing infinites- imal displacements of B and S, including the boundary relations connecting the linearized boundary tractions and the linearized surface stresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Finally, in Section 4, we apply the linearized theory (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='11) to the problem of a brittle infinite solid with a straight, non-interfacial crack of finite length, under mode-III loading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Using the explicit Dirichlet-to-Neumann map, the problem is reduced to solving a fourth order integro-differential equation for the crack profile along the boundary crack front (see (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='5)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Analytically, it is the surface energy satisfying the strong ellipticity condition that implies that this integro-differential equation is fourth order in the surface derivative of the displacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='4 The dimen- sionless form of the equation implies that the behavior of the displacement depends on the size of the crack, and for macro cracks, we expect the displacement to be well-approximated by the solution given by classical linear elastic fracture mechan- ics except in small regions near the crack tips (boundary layers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Finally, using the Lax-Milgram theorem and regularity afforded by the presence of the fourth order derivative, we prove that there always exist a unique classical solution to the gov- erning integro-differential equation, and in contrast to using the Steigmann-Ogden theory, our model predicts bounded strains up to the crack tips (see Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' The author would like to thank Jay R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Walton for several fruitful discussions related to fracture, surface elasticity and interfacial physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' He would also like to thank Jeremy Marzuola for his help generating numerical solutions to the mode-III fracture problem discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Kinematics and field equations In this section formulate a general mathematical model for a hyperelastic bulk solid containing a boundary surface with strain-gradient surface elasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' For planar reference surfaces, the form of surface elasticity reduces to that introduced by Hilgers and Pipkin [7], and when the surface energy is independent of the surface gradient of the stretching, the theory reduces to that of Steigmann and Ogden [22,23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Kinematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Let E3 be three-dimensional Euclidean space with Cartesian coordinates (X1, X2, X3) = X ∈ E3 and coordinate vector fields given by a fixed orthonormal basis {ei}3 i=1 of R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' We define the following operations for elementary 4Physically, it is the model incorporating resistance to geodesic distortion that implies that this integro-differential equation is fourth order in the surface derivative of the displacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' STRAIN-GRADIENT ELASTIC SURFACES 5 tensor products of vectors in R3, (a1 ⊗ a2)b = (a2 · b)a1, (a1 ⊗ a2 ⊗ a3)b = (a3 · b)a1 ⊗ a2, (a1 ⊗ a2 ⊗ a3)(b1 ⊗ b2) = (a3 · b1)a1 ⊗ a2 ⊗ b2, (b1 ⊗ b2)(a1 ⊗ a2 ⊗ a3) = (b2 · a1)b1 ⊗ a2 ⊗ a3, (a1 ⊗ a2 ⊗ a3)[b1 ⊗ b2] = (a2 · b1)(a3 · b2)a1, (a1 ⊗ a2)[b1 ⊗ b2] = (a1 · b1)(a2 · b2), (a1 ⊗ a2 ⊗ a3)[b1 ⊗ b2 ⊗ b3] = (a1 · b1)(a2 · b2)(a3 · b3), (a1 ⊗ a2)T = a2 ⊗ a1, (a1 ⊗ a2 ⊗ a3)T = a1 ⊗ a3 ⊗ a2, (a1 ⊗ a2 ⊗ a3)∼ = a2 ⊗ a1 ⊗ a3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' These operations are extended to general second and third order tensors on R3 by linearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Let B ⊆ E3 be a domain with smooth boundary ∂B, the reference configuration of a three-dimensional body, and let S ⊆ ∂B be a closed surface with smooth boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Let χ : B → χ(B) ⊆ E3 be a smooth, invertible deformation of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' We denote the current position of the reference particle X ∈ B by x = χ(X) = (χ1(X), χ2(X), χ3(X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' The deformation gradient F : R3 → R3 is the second order tensor field F = ∂χi ∂Xa (X)ei ⊗ ea where ea := ea, a = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' We denote the left Cauchy-Green stretch tensor by C = F T F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Let Y = ˆY (θ1, θ2) be a local parameterization of the reference surface S ⊆ ∂B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Then y = ˆy(θ1, θ2) = χ( ˆY (θ1, θ2)) is a local parameterization of the current surface χ(S) ⊆ χ(∂B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' The (local) tangent vector fields on the reference and current surfaces are then given by Y ,α ∈ TY S, y,α = F Y ,α ∈ Tyχ(S), where ·,α := ∂ ∂θα .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' The dual tangent vector fields on the reference and current surfaces are denoted by Y ,β and y,β respectively and satisfy Y ,β · Y ,α = δβ α, y,β · y,α = δβ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' We remark that we may also write y,α = FY ,α where F = F k β ek ⊗ Y ,β := yk ,βek ⊗ Y ,β = y,β ⊗ Y ,β is the surface deformation gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' The first fundamental forms for the reference and current surfaces are then given by G = GαβY ,α ⊗ Y ,β, Gαβ = Y ,α · Y ,β, g = gαβy,α ⊗ y,β, gαβ = y,α · y,β, and we note that Y ,α = (G−1)αβY ,β, y,α = (g−1)αβy,β 6 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' RODRIGUEZ The Christoffel symbols of the second kind for the reference and current surfaces are denoted by Γα βδ = Y ,α · Y ,βδ, γα βδ = y,α · y,βδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' The left Cauchy-Green surface stretch tensor on S is the second order tensor C := FT F = gαβY ,α ⊗ Y ,β, and the surface Green-St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Venant tensor is the second order tensor E = 1 2(C − G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' We assume that the reference and current surfaces are orientable with unit normals to the reference and current surfaces (locally) given by N = |Y ,1 × Y ,2|−1Y ,1 × Y ,2, n = |y,1 × y,2|−1y,1 × y,2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' The second fundamental forms on the reference and current surfaces are given by B = BαβY ,α ⊗ Y ,β, Bαβ := Y ,αβ · N, b = bαβy,α ⊗ y,β, bαβ := y,αβ · n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' The relative curvature tensor on S is the second order tensor K = KαβY ,α ⊗ Y ,β given by K = FT bF − B = [bαβ − Bαβ]Y ,α ⊗ Y ,β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' As discussed by Steigmann and Ogden [23], E and K furnish local differences in length and scaled extrinsic normal curvature between a given curve on S and the convected curve on χ(S), where ˙ := d ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Indeed, if Z(s) is a a curve on S with unit tangent vector field T and z(s) = χ(Z(s)) is the convected curve, then the tangent to the convected curve is given by t = ˙z = F T = FT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' The stretch ν of the convected curve is ν = |t|2 − 1 = FT · FT − 1 = CT · T − 1 = 2ET · T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' The extrinsic normal curvature of the convected curve is κ = bt · t = |t|−2bFT · FT = (ν + 1)−1(FT bF)T, and thus, the difference in scaled extrinsic normal curvature of the convected curve and the original curve is given by |t|2κ − |T|2BT · T = KT · T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' The Levi-Cevita connection on S is denoted by ∇, so ∇E = 1 2∇C = 1 2∇δgαβY ,α ⊗ Y ,β ⊗ Y ,δ, where ∇δgαβ := gαβ,δ − Γµ αδgµβ − Γµ βδgµα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' For later use, we define the following third order tensor on S, L = ∇E + (∇E)T − (∇E∼)T , with components Lαβδ = 1 2(∇δgαβ + ∇βgαδ − ∇αgβδ) = (γµ βδ − Γµ βδ)gµα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' STRAIN-GRADIENT ELASTIC SURFACES 7 Physically, the tensor L (and thus ∇E) furnishes the rate of stretching of convected geodesics, and more generally, it quantifies geodesic distortion, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' how convected geodesics fail to be geodesics on χ(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' More precisely, we have the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Let Z(s) : I → S be a geodesic on S with unit tangent vector field T = ˙Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Then L yields the rate of stretching of the convected curve z(s) = χ(Z(s)), 2L[T ⊗ T ⊗ T] = d ds| ˙z|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='1) Moreover, the convected curve z(s) is a geodesic on χ(S) if and only if for each s0 ∈ I, ∀U ∈ TZ(s0)S, L[U ⊗ T ⊗ T] �� s=s0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='2) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' To prove (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='1), we simply note that since Z is a geodesic, ∇TT = 0 so d ds| ˙z|2 = d ds[CT · T] = ∇C[T ⊗ T ⊗ T] + 2C∇TT · T = ∇C[T ⊗ T ⊗ T] = 2L[T ⊗ T ⊗ T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' We now prove (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' The convected curve z on χ(S) has tangent vector field t = FT = tµy,µ and acceleration vector field a = �˙tµ + γµ βδtβtδ� y,µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' The acceleration is zero and z is a geodesic if and only if for each s0 ∈ I, ∀u ∈ Tz(s0)χ(S), u · a|s=s0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='3) We now show that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='3) is equivalent to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' We assume without loss of generality that s0 = 0, and we choose normal coordinates (θ1, θ2) centered at Z(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Then for all α, β, δ = 1, 2, Gαβ|s=0 = δαβ, Γµ βδ|s=0 = 0, and there exist T1, T2 ∈ R with δαβTαTβ = 1 such that Z(s) = ˆY (sT1, sT2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' In particular, we conclude that a = γµ βδTβTδy,µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Since F|Z(0) : TZ(0)S → Tz(0)χ(S) is an isomorphism, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='3) is equivalent to ∀U = UµY ,µ|Z(0) ∈ TZ(0)S, FU · a �� s=0 = 0 ⇐⇒ ∀U = UµY ,µ|Z(0) ∈ TZ(0)S, gµαUαγµ βδTβTδ�� s=0 = 0 ⇐⇒ ∀U = UµY ,µ|Z(0) ∈ TZ(0)S, L[U ⊗ T ⊗ T] �� s=0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' □ As an illustrative example, consider S = {(X1, X2, 0) | X1 ∈ [a, b], X2 ∈ [0, π]} ⊆ B = [a, b] × [0, π] × [0, ∞), and the deformation χ(X1, X2, X3) = (eX1 cos X2, eX1 sin X2, X3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' 8 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' RODRIGUEZ Then E = 1 2(e2X1 − 1) � e1 ⊗ e1 + e2 ⊗ e2 � , K = 0, L = e2X1� e1 ⊗ e1 ⊗ e1 + e2 ⊗ e1 ⊗ e2 + e2 ⊗ e2 ⊗ e1 − e1 ⊗ e2 ⊗ e2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' The image of the coordinate curve X2 = d is a straight, radially outward traveling curve in the x1x2-plane parameterized by X1 ∈ [a, b] with L[e1 ⊗ e1 ⊗ e1] = e2X1, L[e2 ⊗ e1 ⊗ e1] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' In particular, the stretch is not constant along the convected curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' The image of the coordinate curve X1 = c is the upper-half of the circle in the x1x2-plane centered at (0, 0) of radius ec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' The convected curve is parameterized by X2 ∈ [0, π], has constant stretch but nonzero curvature relative to the convected surface (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=', the x1x2-plane), and it satisfies L[e2 ⊗ e2 ⊗ e2] = 0, L[e1 ⊗ e2 ⊗ e2] = −e2c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Strain energy and material symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' In our mathematical model, a hyperelastic elastic body B with strain-gradient elastic surface S ⊆ ∂B is prescribed a strain energy of the form Φ[χ] = ˆ B W(C) dV + ˆ S U(E, K, ∇E) dA, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='4) where we omit listing the possible dependence of W on X ∈ B and of U on Y ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' We note that the strain energy is frame indifferent, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=', it is invariant with respect to super-imposed rigid motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='5 Although our main motivation for considering a strain-gradient elastic surface is when it forms part of the boundary of a bulk solid B, we will treat material symmetry of S independently of that of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' The notion of material symmetry for the energy density W of the bulk three-dimensional solid B is well-known, see [18,24], so, we will limit our discussion to that of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Consider a material point with reference position Y 0 ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Our discussion of material symmetry for the surface energy per unit reference area U at Y 0 is in- spired by the framework introduced by Murdoch and Cohen [16,17] which was later advocated for and summarized by Steigmann and Ogden [23] using local coordinate parameterizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' We will follow Steigmann and Ogden’s style of exposition, and unless specified otherwise, all quantities in what follows are evaluated at Y 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Let λ : E3 → E3 be a rigid motion with deformation gradient R ∈ SO(3) satisfying λ(Y 0) = Y 0, and RN = N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' We define a second reference surface S∗ = � Y ∗ = λ−1(Y ) | Y ∈ S � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' It follows that TY 0S∗ = TY 0S, at Y 0 ∈ S∗ the unit normal N ∗ to S∗ satisfies N ∗ = N = RN, and R := R|T Y 0S : TY 0S → TY 0S is a rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' A local parameterization on S∗ is given by Y ∗ = ˆY ∗(θ1, θ2) := λ−1( ˆY (θ1, θ2)), so then Y ,α = RY ∗ ,α, Y ,α = RY ∗,α, α = 1, 2, and at Y 0, Y ,α = RY ∗ ,α, Y ,α = RY ∗,α, for α = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' 5In the case of S contained in the plane, Hilgers and Pipkin showed in Section 7 of [7] that U(E, K, ∇E) is the most general form of a surface energy density that is frame indifferent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' STRAIN-GRADIENT ELASTIC SURFACES 9 Let χ : E3 → E3 be a smooth invertible deformation of Euclidean space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Since a superimposed rigid motion does not affect the value of the surface energy, we will assume without loss of generality that χ(Y 0) = Y 0 and at Y 0, F : TY 0S → TY 0S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Following [17] we will impose the stronger requirement that χ(Y 0) = Y 0 and at Y 0, F = Fα βY ,α ⊗ Y ,β + N ⊗ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='5) In particular, it follows that at Y 0, F = Fα βY ,α ⊗ Y ,β and n = N = F N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' The deformation χ∗(·) := χ(λ(·)) when restricted to S∗ has the same image as χ|S, and in particular, the convected tangent vector fields are the same, y∗ ,α := ∂ ∂θα χ∗(Y ∗) = ∂ ∂θα χ(Y ) = y,α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Then the surface deformation gradients of χ and χ∗ at Y 0 are related by F = yα ⊗ Y ,α = y∗ α ⊗ RY ∗,α = F∗RT =⇒ C = RC∗RT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' The components of the first and second fundamental forms associated to S and S∗ satisfy Gαβ = Y ,α · Y ,β = RY ∗ ,α · RY ∗ β = Y ∗ ,α · Y ∗ β = G∗ αβ =⇒ G = RG∗RT , Bαβ = Y ,αβ · N = RY ∗ ,αβ · RN ∗ = Y ∗ ,αβ · N ∗ = B∗ αβ =⇒ B = RB∗RT , and thus, E = RE∗RT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Since b = bαβy,α ⊗ yβ is the same for both deformations, we conclude that the relative curvature tensors K and K∗ associated to χ and χ∗ satisfy at Y 0, K = RK∗RT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Finally, since the components of the first fundamental forms associated to χ(S) and χ∗(S∗) satisfy gαβ = yα · yβ = y∗ α · y∗ β =: g∗ αβ, we conclude that ∇E and ∇∗E∗ at Y 0 satisfy ∇E = 1 2∇δgαβY ,α ⊗ Y ,β ⊗ Y ,δ = 1 2∇∗ δg∗ αβRY ∗,α ⊗ RY ∗,β ⊗ RY ∗,δ = R[(∇∗E∗)T RT ]T RT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' We denote the surface energy per unit reference area relative to S∗ by U ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' For the surface energy per unit mass to be independent of the reference surface used, we must have U ∗(E∗, K∗, ∇∗E∗) = U(E, H, ∇E) = U � RE∗RT , RK∗RT , R � (∇∗E∗)T RT �T RT � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='6) We now view χ also as a deformation of S∗, but we denote its values by ¯χ, ¯χ(X) = χ(X), X ∈ E3, and the associated kinematic variables relative to S∗ are denoted with an over-bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' We will now derive relationships between ¯E, ¯K, ∇∗¯E and E, K, ∇E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' 10 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' RODRIGUEZ We first note that since C = ¯C, we immediately conclude that E = 1 2[(C − I)Y ,α · Y ,β]Y ,α ⊗ Y ,β = 1 2[( ¯C − I)Y ∗ ,α · Y ∗ ,β]Y ∗,α ⊗ Y ∗,β = ¯E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='7) Let Grad C = ∂Cij ∂Xa ei ⊗ ej ⊗ ea, a third order tensor on R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' The identity gαβ = CY ,α · Y ,β, the Gauss equations Y ,αδ = ΓµαδY ,µ + BαδN on S, and the symmetry of C imply that gαβ,δ = Grad C[Y ,α ⊗ Y ,β ⊗ Y ,δ] + CY ,αδ · Y ,β + CY ,α · Y ,βδ = Grad C[Y ,α ⊗ Y ,β ⊗ Y ,δ] + Γµ αδgµβ + BαδCN · Y ,β + Γµ βδgµα + BβδCN · Y ,α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='5), we have CN = N, and thus, at Y 0, ∇E = 1 2Grad C[Y ,α ⊗ Y ,β ⊗ Y ,δ]Y ,α ⊗ Y ,β ⊗ Y ,δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' In particular, since ¯C = C we conclude that ∇∗¯E = ∇E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='8) Finally, as shown in Section 6 of [23], we have ¯K = K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='9) Inspired by Murdoch and Cohen’s extension of Noll’s theory of material sym- metry, we say that Y 0 ∈ S is symmetry related to Y 0 ∈ S∗ if the mechanical responses to the arbitrary deformation χ are identical, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=', U(E, K, ∇E) = U ∗(¯E, ¯K, ∇¯E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='6), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='7), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='8), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='9), this requirement leads to the following definition: the rotation R is in the symmetry set of Y 0 relative to S if for every smooth, invertible deformation χ : E3 → E3 satisfying (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='5), we have U(E, K, ∇E) = U � RERT , RKRT , R � ∇ET RT �T RT � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' As in the case of the standard theory of Noll [18], one can verify that the symmetry set of Y 0 relative to S is a subgroup of the group of proper rotations of TY 0S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' In the case that the symmetry set of Y 0 relative to S equals the group of proper rotations of TY 0S, we say that the surface energy density U is hemitropic at Y 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Field equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' The field equations for the body B with strain-gradient elastic surface S ⊆ ∂B are defined to be the Euler-Lagrange equations for the Lagrangian energy functional A[χ] = Φ[χ] + V [χ] (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='10) where V [χ] is the potential energy associated to the applied forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' In this work, we assume that the Gˆateaux derivative of the load potential takes the form ˙V = − ˆ B f · u dV − ˆ S t · u dA, STRAIN-GRADIENT ELASTIC SURFACES 11 where f is a prescribed external body force on B and t is a prescribed boundary traction on S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Let χ(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' ϵ) be a one-parameter family of deformations of B such that χ(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' ϵ)|∂B\\S = χ0(·), and denote ˙:= d dϵ ��� ϵ=0, u := ˙χ(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Then u|∂B vanishes to first order on ∂B\\S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Using the chain rule and integration by parts we have the classical identity ˆ B ˙W dV = ˆ ∂B P N · u dA − ˆ B Div P · u dV (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='11) where P = P a i ei⊗ea is the Piola stress with P a i = ∂W ∂F ia and Div P = � ∂XaP a i � ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Using the chain rule we have that ˆ S ˙U dA = ˆ S � Tα · u,α + Mαβ · u,αβ � dA, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='12) Tα := ∂U ∂ykα ek, Mαβ := ∂U ∂yk ,αβ ek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' We define surface stress vectors Pα by Pα := Tα − G−1/2(G1/2Mαβ),β, and use (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='11), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='12) and integration by parts to obtain the Euler-Lagrange equa- tions associated to the Lagrangian energy functional (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='10), Div P + f = 0, on B, P N = G−1/2(G1/2Pα),α + t, on S, χ(X) = χ0(X), on ∂B\\S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='13) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Small strain models Our principle motivation for modeling an elastic solid with strain-gradient elastic boundary surface is the study of brittle fracture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' In this setting (to be discussed more in the following section), the surface S possessing strain-gradient surface elasticity will be the crack front and strains will be linearized, motivating the intro- duction of a quadratic surface energy density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' In this section, we present a model uniform, hemitropic, quadratic surface energy density that requires the same ma- terial constants (with the same physical interpretations) as found in the narrower Steigmann-Ogden theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' In contradistinction, the surface energy incorporates the surface’s resistance to geodesic distortion and satisfies the strong ellipticity condi- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Moreover, the surface energy density may be viewed as a geometric general- ization of that introduced and advocated for by Hilgers and Pipkin in [7,8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Hilgers-Pipkin surface energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' For the surface S, we propose the uniform, quadratic, hemitropic surface energy density U = λs 2 (Eα α)2 + µsEαβEαβ + ζ 2 � (Kα α)2 + (g−1)µνL α µα L β νβ � + η � KαβKαβ + (g−1)µνLµαβL αβ ν � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='1) Here λs, µs, ζ and η are positive numbers that can be interpreted as the surface Lam´e constants and pure bending moduli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Indices are raised and lowered using the 12 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' RODRIGUEZ reference metric G, but note that y,α are the dual vector fields to y,α and are not given by y,β(G−1)αβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' In the case that S is contained in a plane with flat coordinates (θ1, θ2), we have Eαβ = 1 2(gαβ − δαβ), Lµαβ = y,µ · y,αβ, y,αβ = Lµαβy,µ + Kαβn, 2 � α=1 y,αα = L α µα y,µ + Kα αn, ��� 2 � α=1 y,αα ��� 2 = (Kα α)2 + (g−1)µνL α µα L β νβ , 2 � α,β=1 |y,αβ|2 = KαβKαβ + (g−1)µνLµαβL αβ ν , and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='1) becomes U = λs 2 (Eα α)2 + µsEαβEαβ + ζ 2 ��� 2 � α=1 y,αα ��� 2 + η 2 � α,β=1 |y,αβ|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='2) Up to a choice of constants, the surface energy density (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='2) is precisely that in- troduced by Hilgers and Pipkin in [7], and therefore, we refer to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='1) as a quadratic Hilgers-Pipkin surface energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' In [7,8], Hilgers and Pipkin advocated for the use of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='2) over the classical surface energy U = λs 2 (Eα α)2 + µsEαβEαβ + ζ 2Kα α + ηKαβKαβ = λs 2 (Eα α)2 + µsEαβEαβ + ζ 2 ��� 2 � α=1 (n · y,αα) ��� 2 + η 2 � α,β=1 (n · y,αβ)2 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='3) on the basis of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='2) being analytically simpler than (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Indeed, with little effort one sees that for (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='2), Tα = (λsEγ γδαβ + 2µsEαβ)y,β, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='4) Mαβ = ζ �� γ y,γγ � δαβ + 2ηy,αβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='5) Moreover, it is simple to see that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='2) satisfies the strong ellipticity condition, ∀(a1, a2) ∈ R2\\{(0, 0)}, b ∈ R3\\{0}, aαaβb · � Cαβδγaδaγb � > 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='6) Cαβγδ := ∂2U ∂yi ,αβ∂yj ,δγ ei ⊗ ej, while (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='3) does not (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='7) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='8) below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Physically, this may be viewed as a consequence of the surface energy (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='1) incorporating the surface’s resistance to geodesic distortion via also including dependence on the tensor L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' In general, for (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='1) we have Tα = ∂U ∂y,α = ∂U ∂Eβγ ∂Eβγ ∂y,α + ∂U ∂Kδν ∂Kδν ∂y,α + ∂U ∂Lβγδ ∂Lβγδ ∂y,α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' STRAIN-GRADIENT ELASTIC SURFACES 13 Using ∂Eβγ ∂y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='α = 1 2 � δα βy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='γ + δα γy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='β � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' ∂Kµν ∂y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='α = −γα µν n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' ∂Kµν ∂y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='αβ = 1 2(δα µδβ ν + δα νδβ µ)n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' ∂Lβγµ ∂y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='α = δα βyγµ − (δα νy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='β + δα βy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='ν)Γν γµ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' ∂Lγµσ ∂y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='αβ = 1 2(δα µδβ σ + δα σδβ µ)yγ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' we readily compute that Tα = � λsEµ µ(G−1)αγ + 2µsEαγ − (g−1)µα(g−1)νγ(ζL δ µδ L σ νσ + 2ηLµδσL δσ ν ) � y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' − � ζKµ µ(G−1)δνγα δν + 2ηKδνγα δν � n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' + � ζ(g−1)µαL ν µν (G−1)γδ + 2η(g−1)µαL γδ µ � y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='γδ − ζ � (g−1)µαL ν µν Γβ δ δ + (g−1)µβL ν µν Γα δ δ � y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='β − 2η � (g−1)µαL δσ µ Γβ δσ + (g−1)µβL δσ µ Γα δσ � y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' and Mαβ = � ζKµ µ(G−1)αβ + 2ηKαβ� n + � ζ(g−1)µγL ν µν (G−1)αβ + 2η(g−1)µγL αβ µ � y,γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' To see that the strong ellipticity condition is satisfied for (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='1), we compute Cαβµν = � ζ(G−1)αβ(G−1)µν + 2η[(G−1)αµ(G−1)βν + (G−1)αν(G−1)βµ] � I, and thus, for all (a1, a2) ∈ R2\\{(0, 0)} and b ∈ R3\\{0}, aαaβb · � Cαβδγaδaγb � = (ζ + 2η)[(G−1)αβaαaβ]2|b|2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='7) For the classical surface energy (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='3) of Steigmann-Ogden type, we have Cαβµν = � ζ(G−1)αβ(G−1)µν + 2η[(G−1)αµ(G−1)βν + (G−1)αν(G−1)βµ] � n ⊗ n, and thus, for all (a1, a2) ∈ R2 and b ∈ R3, aαaβb · � Cαβδγaδaγb � = (ζ + 2η)[(G−1)αβaαaβ]2(n · b)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='8) In particular, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='8) shows that the surface energy (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='3) satisfies the Legendre- Hadamard condition but not the strong ellipticity condition since the right side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='8) is 0 for b ̸= 0 and orthogonal to n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' 14 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' RODRIGUEZ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Linearized equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' We now compute the linearization of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='1) about the reference configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' For the bulk solid, we adopt a classical quadratic, isotropic energy density W = λ 2 (Ei i)2 + µEijEij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Here indices are raised using the flat metric on R3, and λ and µ are the Lam´e constants for the bulk solid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' For the surface S, the surface energy density is given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Let u : B → R3 be a displacement field such that u|∂B\\S = 0 and sup X∈B � |u(X)| + |Grad u(X)| � + 2 � α,β=1 sup Y ∈S |u,αβ(Y )| ≤ δ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='9) Assume that the body force f, boundary traction t, and Dirichlet condition χ0 satisfy |f| = O(δ0), |t| = O(δ0), |χ0 − Id| = O(δ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' If χ(X) = X + u(X), then Eij = εij + O(δ2 0), Eαβ = ϵαβ + O(δ2 0), and Kαβ = kαβ + O(δ2 0) where εij = 1 2 � ei · ∂u ∂Xj + ej · ∂u ∂Xi � = O(δ0), and on S, ϵαβ = 1 2 � Y ,α · u,β + Y ,α · u,β � = O(δ0), kαβ = N · u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='αβ = O(δ0), see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='12) in [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Now we observe that Lαβδ = lαβδ + O(δ2 0), with lαβδ = Y ,α · u,βδ + Y ,βδ · u,α − Γµ βδ � Y ,α · u,µ + Y ,µ · u,α � = Y ,α · u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='βδ + Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='βδ · u,α = Y ,α · u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='βδ + (N · u,α)Bβδ = O(δ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Then Tα = tα + O(δ2 0) and Mαβ = mαβ + O(δ2 0) where tα := � λsϵµ µ(G−1)αγ + 2µsϵαγ� Y ,γ − � ζkµ µΓα ν ν + 2ηkδνΓα δν � N, + � ζlα ν ν Bδ δ + 2ηlαδσBδσ � N − � ζlβ ν ν Γα δ δ + 2ηlβδσΓα δσ � Y ,β, mαβ := � ζkµ µ(G−1)αβ + 2ηkαβ� N + � ζlγ ν ν (G−1)αβ + 2ηlγαβ� Y ,γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' The linearization of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='13) about the reference configuration is obtained by omit- ting the O(δ2 0) terms from P , Tα and Mαβ, yielding Div σ + f = 0, on B, σN = G−1/2(G1/2pα),α + t, on S, u = 0, on ∂B\\S, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='10) STRAIN-GRADIENT ELASTIC SURFACES 15 where σ = λ(trε)I + 2µε and pα = tα − G−1/2(G1/2mαβ),β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' We observe that solu- tions to the linearized equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='10) are critical points of the energy functional AL[u] = ˆ B �λ 2 (εi i)2 + µεijεij� dV − ˆ B f · u dV + ˆ S �λs 2 (ϵα α)2 + µsϵαβϵαβ� dS + ˆ S �ζ 2 � (kα α)2 + l α µα lµ β β � + η � kαβkαβ + lµαβlµαβ�� dA − ˆ S t · u dA over the set of u satisfying u|∂B\\S = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' In the case of the classical Hilgers-Pipkin surface energy (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='2), we see from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='4) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='5) that tα = (λsϵγ γδαβ + 2µsϵαβ)Y ,β, mαβ = ζδαβu γ ,γ + 2ηu,αβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Writing u = u + u3N = uγY γ + u3N, it follows that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='10) becomes Div σ + f = 0, on B, σN = µsu α ,α + (λs + µs)uγ ,αγY α − (ζ + 2η)∂α∂βu,αβ + t, on S, u = 0, on ∂B\\S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='11) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Mode-III Fracture Problem In this section, we apply the linearized theory (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='11) to the problem of a brittle infinite plate, with a straight crack C of length 2ℓ, under far-field anti-plane shear loading σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' As discussed in the Introduction and in contrast to ascribing either a quadratic Gurtin-Murdoch or Steigmann-Ogden surface energy (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='3) to the crack fronts, the use of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='1) yields a model that predicts bounded strains and stresses up to the crack tips (see Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Formulation and governing equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' We consider a brittle, infinite plate under anti-plane shear loading limx2→±∞ σ23 = ±σ, with a straight crack C = {(x1, 0, x3) | x1 ∈ [−ℓ, ℓ]} of length 2ℓ (see Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' For anti-plane shear, the displacement field takes the form u(x1, x2, x3) = u(x1, x2)e3, Then the only nonzero components of the stress are σ13 = µu,1, σ23 = µu,2 By the symmetry of the problem, u will be odd in x2, so we will focus only on the strain and stress fields for x2 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' The governing field equations are (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='11) on B = {(x1, x2, x3) | x2 ≥ 0} with S = {(x1, 0, x3) | x1 ∈ [−ℓ, ℓ]}, t = 0 and f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' We define dimensionless variables x = x1 ℓ , y = x2 ℓ , z = x3 ℓ , w(x, y, z) = 1 ℓ � u(x1, x2, x3) − σ µx2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Then the field equations take the dimensionless form ∆w(x, y) = 0, y > 0, − wy(x, 0) = αwxx(x, 0) − βwxxxx(x, 0) + γ, x ∈ (−1, 1), w(x, 0) = 0, |x| ≥ 1, wx(±1, 0) = 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='1) 16 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' RODRIGUEZ with the decay condition limy→∞ wy(x, y) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' We note that the boundary con- ditions wx(±1, 0) = 0 imply that the crack opening is cusp shaped rather then blunted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' The dimensionless parameters α, β and γ are given by α = µs µℓ > 0, β = ζ + 2η µℓ3 > 0, γ = σ µ, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='2) and in particular, we see from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='2) that the behavior of the displacement w depends on the length of the crack, ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' For macro cracks satisfying β ≪ α ≪ 1, we expect w(x, 0) to be well-approximated by the singular, rounded opening profile from the classical linear elastic fracture mechanics except in small regions near the crack tips (boundary layers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' See Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' We remark that in using the Steigmann-Ogden surface energy (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='3) rather than (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='1), the boundary conditions at y = 0 are replaced by − wy(x, 0) = αwxx(x, 0) + γ, x ∈ (−1, 1), w(x, 0) = 0, |x| ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='3) One may view this loss of higher order derivatives in the boundary conditions as a consequence of the fact that the Steigmann-Ogden surface energy does not satisfy the strong-ellipticity condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='6): for anti-plane shear, b = u,11(x1, 0, x3) is orthogonal to the surface’s normal n = −e2 (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='8)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' As discussed in [9,25], the boundary conditions (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='3) do not lead to a model predicting bounded strains up to the crack tips x = ±1 (see [9,25]), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=', the displacement field satisfies sup y>0 |∇w(±1, y)| = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' We see that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='1) is the system of Euler-Lagrange equations for the energy func- tional AL[w] = 1 2 ˆ ∞ 0 ˆ ∞ −∞ |∇w(x, y)|2dxdy + ˆ ∞ −∞ �α 2 |wx(x, 0)|2 + β 2 |wxx(x, 0)|2� dx − γ ˆ ∞ −∞ w(x, 0)dx defined for w with ∇w ∈ L2({y > 0}), w(·, 0) ∈ H2(R) and w(x, 0) = 0 for all |x| ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Motivated by this observation, we define the Hilbert space H to be the x3 x1 x2 σe3 −σe3 (−ℓ, 0, 0) (ℓ, 0, 0) Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Schematic of the mode-III problem with the crack C appearing in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' STRAIN-GRADIENT ELASTIC SURFACES 17 completion of C∞ c ((−1, 1)) under the norm ∥f∥2 H := ˆ ∞ −∞ � α|f ′(x)|2 + β|f ′′(x)|2� dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' It is straightforward to verify the following facts using the fundamental theorem of calculus and Cauchy-Schwarz inequality: (Sobolev embedding) If f ∈ H then f ∈ C1,1/2− c (R) and f(x) = 0 for all |x| ≥ 1, and for all δ ∈ [0, 1/2), there exists a constant A > 0 depending on α, β and δ, such that for all f ∈ H, ∥f∥C1,γ(R) ≤ A∥f∥H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' f ∈ H if and only if f ∈ H2(R) and f(x) = 0 for all |x| ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Moreover, there exist b, B > 0 depending on α and β such that for all f ∈ H, b∥f∥H ≤ ∥f∥H2(R) ≤ B∥f∥H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='4) The problem (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='1) can be reduced completely to a problem on the boundary by using the Dirichlet-to-Neumann map −wy(x, 0) = Hwx(x, 0) where H is the Hilbert transform Hf(x) = 1 π p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' ˆ ∞ −∞ f(s) x − sds, f ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Then finding w with ∇w ∈ L2({y > 0}) and w(·, 0) ∈ H satisfying (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='1) is equiva- lent to determining w(·, 0) =: f ∈ H satisfying6 βf ′′′′(x) − αf ′′(x) + Hf ′(x) = γ, x ∈ (−1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='5) By using the Plancherel theorem, the Fourier representation of the Hilbert trans- form (see (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='13)), and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='4), we have for all f ∈ H, ∥Hf ′∥H1(R) ≤ ∥f ′∥H1(R) ≤ ∥f∥H2(R) ≤ B∥f∥H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='6) Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' A function f ∈ H is a weak solution to the integro-differential equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='5) if for all g ∈ H ˆ ∞ −∞ [βf ′′(x)g′′(x) + αf ′(x)g′(x) + Hf ′(x)g(x) � dx = ˆ ∞ −∞ γg(x)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='7) We remark that since f, g ∈ H, the integrals appearing in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='7) are in fact over the interval (−1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' A function f ∈ H is a classical solution to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='5) if f ∈ C4((−1, 1)) ∩ H and f satisfies (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='5) pointwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' We note that by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='6) and Cauchy-Schwarz, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='7) is well-defined for each f, g ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' 6Once f is found, w is determined on the upper half plane using the standard Poisson kernel for the upper half plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' 18 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' RODRIGUEZ Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Numerical solutions for the equivalent formulation of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='5) as a Fredholm problem (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' The parameters (β, α, γ) range over (1, 1, 1), (5, 1, 5) and (10, 1, 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' For γ = β and β ≫ 1 ≃ α, the opening profile is well approximated by the limiting opening profile f∞(x) = 1 24(1 − x2)2 on [−1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Solution of the integro-differential equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' We now establish that there exists a unique classical solution to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='5), and the solution’s behavior is con- sistent with the linearization assumption (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' We denote the following Green function G(x, τ) = � 1 24(x − 1)2(τ + 1)2(1 + 2x − 2τ − xτ) τ ∈ [−1, x], 1 24(τ − 1)2(x + 1)2(1 + 2τ − 2x − xτ) τ ∈ [x, 1], satisfying Gxxxx(x, τ) = δ(x − τ), G(±1, τ) = 0, Gx(±1, τ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' We note that G(x, τ) = G(τ, x) for all τ, x ∈ [−1, 1], G ∈ C2([−1, 1]×[−1, 1]) and ´ 1 −1 G(x, τ)dτ = 1 24(1 − x2)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' In particular, we have for all f ∈ H, ˆ 1 −1 Gττ(x, τ)f ′′(τ)dτ = f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='8) Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' A function f ∈ H is a weak solution to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='5) if and only if f satisfies βf(x) + ˆ 1 −1 G(x, τ)(−αf ′′(τ) + Hf ′(τ))dτ = γ 24(1 − x2)2, x ∈ [−1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='9) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Let h ∈ L2(R) with h(x) = 0 for all |x| > 1, and set g(x) = �´ 1 −1 G(x, τ)h(τ)dτ if |x| ≤ 1, 0 if |x| > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Since G ∈ C2([−1, 1] × [−, 1, 1]), G(±1, τ) = 0, and Gx(±1, τ) = 0, g is twice continuously differentiable on R\\{±1} and continuously differentiable on R with g′(x) = χ{|x|≤1}(x) ˆ 1 −1 Gx(x, τ)h(τ)dτ, g′′(x) = χ{|x|≤1}(x) ˆ 1 −1 Gxx(x, τ)h(τ)dτ, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='05 β=1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='.β=5 β= 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='04 f 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='03 J 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='01 0- 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='8 xSTRAIN-GRADIENT ELASTIC SURFACES 19 where χE is the indicator function of a subset E ⊆ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' In particular, we conclude that g ∈ H2(R) and thus, g ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Inserting g into (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='7), integrating by parts in the second term and using that g(±1) = 0 yield β ˆ 1 −1 ˆ 1 −1 Gxx(x, τ)f ′′(x)h(τ)dτdx + ˆ 1 −1 ˆ 1 −1 G(x, τ)(−αf ′′(x) + Hf ′(x))h(τ)dτdx = γ ˆ 1 −1 ˆ 1 −1 G(x, τ)h(τ)dτdx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Interchanging the order of integration, using the symmetry of G and relabeling the integration variables lead to β ˆ 1 −1 ˆ 1 −1 Gττ(x, τ)f ′′(τ)dτ h(x)dx + ˆ 1 −1 ˆ 1 −1 G(x, τ)(−αf ′′(τ) + Hf ′(τ))dτ h(x)dx = ˆ 1 −1 γ 24(1 − x2)2h(x)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Finally, by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='8) we conclude that ˆ 1 −1 � βf(x) + ˆ 1 −1 G(x, τ)(−αf ′′(τ) + Hf ′(τ))dτ � h(x)dx = ˆ 1 −1 γ 24(1 − x2)2h(x)dx, for all h(x) ∈ L2(R) with h(x) = 0 for all |x| > 1, proving (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Conversely, if f ∈ H and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='9) holds, then for all g ∈ H, we have β ˆ ∞ −∞ f ′′(x)g′′(x)dx = ˆ 1 −1 ˆ 1 −1 Gxx(x, τ)(αf ′′(τ) − Hf ′(τ))g′′(x)dx + ˆ 1 −1 γ 6 (3x2 − 1)g′′(x)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' We again interchange the order of integration and use integration by parts and ´ 1 −1 Gxx(x, τ)g′′(x)dx = g(τ) to conclude that ˆ 1 −1 ˆ 1 −1 Gxx(x, τ)(αf ′′(τ) − Hf ′(τ))g′′(x)dx + ˆ 1 −1 γ 6 (3x2 − 1)g′′(x)dx = ˆ 1 −1 (αf ′′(τ) − Hf ′(τ))g(τ)dτ + ˆ 1 −1 γg(x)dx = − ˆ 1 −1 (αf ′(x)g′(x) + Hf ′(x)g(x))dx + ˆ 1 −1 γg(x)dx This proves f ∈ H satisfies (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='7) and concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' □ Via integration by parts and straightforward computations, we conclude from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='2 that the crack opening profile f must satisfy the following Fredholm integral equation of the second kind (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' We will show in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='4 that this 20 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' RODRIGUEZ Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Numerical solutions for the macro-crack regime β ≪ α ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' The parameters (β, α, γ) range over (10−1, 10−1, 1), (10−2, 10−1, 1), (10−5, 10−2, 1) and (10−6, 10−3, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' For β ≪ α ≪ 1, we expect the crack opening f(x) to be well-approximated by the singular, rounded opening profile predicted by classical linear elas- tic fracture mechanics away from the crack tips where f ′(±1) = 0 (and the profile is cusped).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' equation is uniquely solvable for arbitrary α, β > 0 and γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' We remark that since the kernel extends to a continuous function on [−1, 1] × [−1, 1] (the singularities are removable), the numerical computation of solutions is relatively straightforward via the N¨ystrom method with the trapezoidal rule to approximate the integral (see Figure 3 and Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' A function f ∈ H is a weak solution to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='5) if and only if f satisfies the Fredholm equation βf(x) + ˆ 1 −1 K(x, s)f(s)ds = γ 24(1 − x2)2, x ∈ [−1, 1], (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='10) where K(x, s) = −αGss(x, s) + 1 πp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' ´ 1 −1 Gτ (x,τ) s−τ dτ, Gss(x, s) = � − 1 4(x − 1)2(2s + xs + 1) s ∈ [−1, x], − 1 4(x + 1)2(−2s + xs + 1) s ∈ [x, 1], 1 β=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='1 β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='01 --β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='00001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='8 β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='000001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='6 f 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='5 β= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='1 β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='4 --β= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='00001 β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='000001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='93 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='92 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='91 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='9 1STRAIN-GRADIENT ELASTIC SURFACES 21 and 1 π p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' ˆ 1 −1 Gτ(x, τ) s − τ dτ = 1 4π (sx − 1)(x2 − 1) + 1 2π (s − x)2 log |x − s| − 1 8π (x − 1)2(−x + 2s + sx)(1 + s) log(1 + s) − 1 8π (x + 1)2(x − 2s + sx)(1 − s) log(1 − s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' There exists C > 0 depending on α and β such that the following hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' There exists a unique classical solution f to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='5), and f satisfies ∥f∥C4([−1,1]) ≤ C|γ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='11) Moreover, the displacement field w(x, y) = ´ ∞ −∞ Py(x − s)f(s)ds, where Py(·) is the Poisson kernel for the upper half plane, has bounded strains up to the crack tips: ∥w∥C1({y≥0}) ≤ C|γ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='12) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' In what follows, C will denote a positive constant depending only on α and β that may change from line to line, and we denote the Fourier transform and inverse Fourier transform by ˆf(ξ) = ˆ ∞ −∞ f(x)e−2πixξdx, ˇf(x) = ˆ ∞ −∞ f(ξ)e2πixξ dξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' We recall that �f ′(ξ) = 2πiξ ˆf(ξ), � Hf(ξ) = −isgn(ξ) ˆf(ξ), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='13) the latter relation following from the Fourier representation of w on the upper half plane (see (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='15)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' We define a bilinear form B(·, ·) : H × H → R by B(f, g) = ˆ ∞ −∞ [βf ′′(x)g′′(x) + αf ′(x)g′(x) + Hf ′(x)g(x) � dx, f, g ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' By Cauchy-Schwarz, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='6), and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='4) we conclude that for all f, g ∈ H, |B(f, g)| ≤ (1 + B2)∥f∥H∥g∥H, so that B(·, ·) is a bounded bilinear form on H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Moreover, by the Plancherel theorem and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='13), we have for all f ∈ H, ˆ ∞ −∞ Hf ′(x)f(x)dx = ˆ ∞ −∞ 2π|ξ|| ˆf(ξ)|2dξ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Thus, the bilinear form is coercive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Since for all g ∈ H, ��� ˆ ∞ −∞ γg(x)dx ��� ≤ |γ|∥g∥C([−1,1]) ≤ |γ|A∥g∥H, the classical Lax-Milgram theorem implies that there exists a unique weak solution f ∈ H to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='5), and moreover, ∥f∥H ≤ A|γ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='14) To prove (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='12), we express w via the Fourier transform, w(x, y) = ˆ ∞ −∞ e−2πy|ξ|e2πixξ ˆf(ξ)dξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='15) 22 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' RODRIGUEZ Since f ∈ H ⊂ H2(R), we have ˆ (1 + |ξ|)4| ˆf(ξ)|2dξ ≤ C0∥f∥2 H2(R) ≤ C∥f∥2 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Thus, by Cauchy-Schwarz |w(x, y)| + |∇w(x, y)| ≤ ˆ ∞ −∞ (1 + 2π|ξ| √ 2)| ˆf(ξ)|2 dξ ≤ 2π √ 2 �ˆ ∞ −∞ |ξ|2(1 + |ξ|)−4dξ �1/2�ˆ ∞ −∞ (1 + |ξ|)4| ˆf(ξ)|2dξ �1/2 ≤ C∥f∥H ≤ C|γ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' We now show that the weak solution f is a classical solution, and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='11) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' By a density argument in H, we have that ´ 1 −1 G(·, τ)(αf ′′(τ)−Hf ′(τ))dτ ∈ H3([−1, 1]) with dk dxk ˆ 1 −1 G(x, τ)(αf ′′(τ) − Hf ′(τ))dτ = ˆ 1 −1 ∂k xG(x, τ)(αf ′′(τ) − Hf ′(τ))dτ, k = 1, 2, d3 dx3 ˆ 1 −1 G(x, τ)(αf ′′(τ) − Hf ′(τ))dτ = ˆ x −1 1 4(2 − τ)(τ + 1)2(αf ′′(τ) − Hf ′(τ))dτ − ˆ 1 x 1 4(2 + τ)(τ − 1)2(αf ′′(τ) − Hf ′(τ))dτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='16) By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='2, we conclude that f ∈ H3([−1, 1]), and by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='9) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='16), Cauchy- Schwarz, and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='14) we have ∥f ′′′∥L2([−1,1]) ≤ C(∥f ′′∥L2(R) + ∥Hf ′∥L2(R) + |γ|) ≤ C(∥f∥H + |γ|) ≤ C|γ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='17) Moreover, by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='14), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='17) and the fundamental theorem of calculus, f ∈ C2([−1, 1]) with ∥f∥C2([−1,1]) ≤ C|γ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='18) By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='7) and integration by parts, it follows that for all g ∈ C∞ c ((−1, 1)) ⊂ H, ˆ 1 −1 f ′′′(x)g′(x)dx = 1 β ˆ 1 −1 [−αf ′′(x) + Hf ′(x) − γ]g(x)dx, and, thus, f ∈ H4([−1, 1]) and βf ′′′′(x) − αf ′′(x) + Hf ′(x) = γ, for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' x ∈ (−1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='19) STRAIN-GRADIENT ELASTIC SURFACES 23 Then by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='19), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='18), and the fact that Hf ′ ∈ H1(R) �→ C(R), we conclude that f ′′′′ ∈ C([−1, 1]) and ∥f ′′′′∥C([−1,1]) ≤ C � ∥f∥C2([−1,1]) + ∥Hf ′∥C([−1,1]) + |γ| � ≤ C � ∥Hf ′∥H1(R) + |γ| � ≤ C � ∥f∥H + |γ| � ≤ C|γ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='20) By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='17), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='18) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='20), it follows that f ∈ C4([−1, 1]) is a classical solution to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='5) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='11) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' □ References [1] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Broberg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Cracks and Fracture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Academic Press, San Diego, 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' [2] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Dai, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Gharahi, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Schiavone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Analytic solution for a circular nano-inhomogeneity with interface stretching and bending resistance in plane strain deformations.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content=' Rodriguez Department of Mathematics, University of North Carolina Chapel Hill, NC 27599, USA crodrig@email.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='unc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} +page_content='edu' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFST4oBgHgl3EQfMjgS/content/2301.13744v1.pdf'} diff --git a/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf b/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..e5458a29eb40cbbc8fd02eb5b83d7a78d90fcfd5 --- /dev/null +++ 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100644 index 0000000000000000000000000000000000000000..c8cdc4597ace7d6473abb27d4bf68941b53626c9 --- /dev/null +++ b/s9E3T4oBgHgl3EQfNAmL/content/tmp_files/2301.04379v1.pdf.txt @@ -0,0 +1,948 @@ +Mass testing of the JUNO experiment +20-inch PMTs readout electronics +Alberto Coppia, Beatrice Jelminia,b,∗, Marco Bellatob, Antonio Bergnolib, +Matteo Bolognesia,b, Riccardo Brugneraa,b, Vanessa Cerronea, Chao Chenc, +Barbara Clerbauxd, Daniele Cortib, Flavio dal Corsob, Jianmeng Donge, Wei Doue, +Lei Fanc, Alberto Garfagninia,b, Arsenii Gavrikova,b, Guanghua Gonge, +Marco Grassia,b, Rosa Maria Guizzettia, Shuang Hangd,f, Cong Hec, Jun Huc, +Roberto Isocrateb, Xiaolu Jic, Xiaoshan Jiangc,g, Fei Lic, Zehong Liangc, Ivano Lippib, +Hongbang Liuh, Hongbin Liuc, Shenghui Liuc, Xuewei Liue, Daibin Luoc, +Ronghua Luoh, Filippo Marinia,b, Daniele Mazzarob, Luciano Modeneseb, +Marta Colomer Mollad, Zhe Ningc, Yu Pengc, Pierre-Alexandre Petitjeand, +Alberto Pitaccob, Mengyao Qic, Loris Raminab, Mirco Rampazzob, +Massimo Rebeschinib, Mariia Redchukb, Andrea Serafinia,b, Yunhua Sunc, +Andrea Triossia,b, Riccardo Triozzia, Fabio Veroneseb, Katharina von Sturma,b, +Peiliang Wangc, Peng Wangd,f, Yangfu Wangc, Yusheng Wangc, Yuyi Wange, +Zheng Wangc, Ping Weih, Jun Wenge, Shishen Xiani,j, Xiaochuan Xiec, Benda Xue, +Chuang Xue, Donglian Xui,j, Hai Xuh, Xiongbo Yanc,g, Ziyue Yanc, Fengfan Yangc, +Yan Yangh, Yifan Yangd, Mei Yec, Tingxuan Zengc, Shuihan Zhangc, Wei Zhangc, +Aiqiang Zhange, Bin Zhange, Siyao Zhaoh, Changge Zic, Sebastiano Aiellok, +Giuseppe Andronicok, Vito Antonellil, Andrea Barresim, Davide Basilicol, +Marco Berettal, Augusto Brigattil, Riccardo Brunok, Antonio Budanon, +Barbara Caccianigal, Antonio Cammio, Stefano Campesea,b, Davide Chiesam, +Catia Clementip, Marco Cordelliq, Stefano Dusinib, Andrea Fabbrin, Giulietto Feliciq, +Federico Ferrarol, Marco Giulio Giammarchil, Cecilia Landinil, Paolo Lombardil, +Claudio Lombardor,k, Andrea Mainos,t, Fabio Mantovanis,t, Stefano Maria Marin, +Agnese Martiniq, Emanuela Meronil, Lino Miramontil, Michele Montuschis,t, +Massimiliano Nastasim, Domizia Orestanon, Fausto Orticap, Alessandro Paoloniq, +Sergio Parmeggianol, Fabrizio Petruccin, Ezio Previtalim, Gioacchino Ranuccil, +Alessandra Carlotta Rel, Barbara Riccis,t, Aldo Romanip, Paolo Saggesel, +Simone Sanfilippon,∗∗, Chiara Sirignanoa,b, Monica Sistim, Luca Stancob, +Virginia Stratis,t, Francesco Tortoricir,k, Cristina Tuv´er,k, Carlo Venettaccin, +Giuseppe Verdek, Lucia Votanoq +aUniversit`a di Padova, Dipartimento di Fisica e Astronomia, Padova, Italy +bINFN Sezione di Padova, Padova, Italy +cInstitute of High Energy Physics, Beijing, China +dUniversit´e Libre de Bruxelles, Brussels, Belgium +eTsinghua University, Beijing, China +fNanjing University of Aeronautics and Astronautics, Nanjing, China +gUniversity of Chinese Academy of Sciences, Beijing, China +hGuangxi University, Nanning, China +iSchool of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai, China +jTsung-Dao Lee Institute, Shanghai Jiao Tong University, Shanghai, China +kINFN Sezione di Catania, Catania, Italy +lINFN Sezione di Milano e Universit`a di Milano, Dipartimento di Fisica, Milano, Italy +mINFN Sezione di Milano Bicocca, e Universit`a di Milano Bicocca, Dipartimento di Fisica, Milano, +Italy +1 +arXiv:2301.04379v1 [physics.ins-det] 11 Jan 2023 + +nINFN Sezione di Roma Tre e Universit`a di Roma Tre, Dipartimento di Matematica e Fisica, Roma, +Italy +oINFN, Sezione di Milano Bicocca e Politecnico di Milano, Dipartimento di Energetica, Milano, Italy +pINFN Sezione di Perugia e Universit`a di Perugia, Dipartimento di Chimica, Biologia e +Biotecnologie, Perugia, Italy +qLaboratori Nazionali dell’INFN di Frascati, Italy +rUniversit`a di Catania, Dipartimento di Fisica e Astronomia, Catania, Italy +sINFN Sezione di Ferrara, Ferrara, Italy +tUniversit`a degli Studi di Ferrara, Dipartimento di Fisica e Scienze della Terra, Italy +Abstract +The Jiangmen Underground Neutrino Observatory (JUNO) is a multi-purpose, large +size, liquid scintillator experiment under construction in China. +JUNO will perform +leading measurements detecting neutrinos from different sources (reactor, terrestrial and +astrophysical neutrinos) covering a wide energy range (from 200 keV to several GeV). +This paper focuses on the design and development of a test protocol for the 20-inch +PMT underwater readout electronics, performed in parallel to the mass production line. +In a time period of about ten months, a total number of 6950 electronic boards were +tested with an acceptance yield of 99.1 %. +Keywords: +Read-Out electronics, photomultiplier, liquid scintillator, large scale +neutrino experiment +1. Introduction +The Jiangmen Underground Neutrino Observatory [1] (JUNO) is a 20 kton neutrino +medium baseline experiment under construction in southern China. +The JUNO ex- +periment has been proposed [2] with the main goals of determining the neutrino mass +ordering with a significance of 3 σ within the first six years of data taking and measuring +the oscillation parameters, ∆m2 +21, ∆m2 +31, and sin2 θ12, with sub-percent precision [3]. To +achieve these goals, JUNO is located about 53 km away from two nuclear power plants +and will detect electron antineutrinos produced by the beta decays of fission products +∗Corresponding author +∗∗Now at INFN Laboratori Nazionali del Sud, Italy +Preprint submitted to Elsevier +January 12, 2023 + +inside the nuclear cores. JUNO will also be able to address many other topics in particle +and astroparticle physics, by detecting neutrinos from natural sources: solar neutrinos, +atmospheric neutrinos, geo-neutrinos, neutrinos from core-collapse supernovae, and from +the diffuse supernovae neutrino background. An updated overview of the JUNO physics +reach can be found here [1]. +The JUNO Central Detector (CD) consists of 20 kton of liquid scintillator contained +in a spherical acrylic vessel with a 35.4 m diameter, supported by a stainless Steel Truss. +A double system of 17 612 20-inch large-PMTs (LPMTs) [4, 5] and 25 600 3-inch small- +PMTs (SPMTs) [6] is employed to detect the scintillation and Cherenkov light produced +by neutrino interactions with the liquid scintillator. +The liquid scintillator target is +surrounded by a 35-kt pure Water Pool, which is instrumented with 2400 20-inch LPMTs; +the Water Pool shields the inner part of the detector from environmental radioactivity, +and is part of the muon Veto system, together with the Top Tracker on top of the whole +structure. +The JUNO LPMT underwater readout electronic system is responsible to sample and +to process the LPMT output current [7, 8]. Spotting hardware failures and evaluating +the performance of the underwater readout electronics before the actual installation is of +paramount importance, because it will be impossible to repair or to change an electronics +module after its deployment. Furthermore, the required loss rate of the electronics chan- +nels is less than 0.5% in 6 years [1]. To this end, we designed and developed a dedicated +test protocol [9] to be carried out during the mass production, held in a dedicated facility +in Kunshan, China. +The rest of the paper is organized as follows: in Section 2 we describe the JUNO +LPMT readout electronics; in Section 3 we discuss the mass production and the mass +testing setup at the dedicated facility in Kunshan; in Section 4 a detailed description of +the developed test protocol is presented; conclusions are drawn in Section 5. +2. JUNO LPMT readout electronics +A scheme of the JUNO LPMT electronics is given in Figure 1 [8]; the design is an +optimization of previous developments [7]. The full electronics chain is composed of two +parts: the front-end (FE), or wet, electronics [10] located very close to the LPMT output, +3 + +High +Voltage +Unit +High +Voltage +Unit +High +Voltage +Unit +Custom HV +(0 - 3 kV) / 300µA +Global Control Unit (GCU) +FPGA +Under Water Box (UWBox) +Custom ADC +14 bit, 1 Gsps +1.5 - 3.5 m cables +(signal and HV) +2 GB RAM for SN bursts +Gbit +Enterpise +Switch +DAQ +LV +Up to 100 m +CAT6 + low Z +power cables +Dry electronics +Wet electronics +Back End +Card +Trigger +Electronics +CLK +DDR3 +Sync +link +Async +link +Up to 100 m +CAT5 cables +ADC +ADC +Front +End +Chip +HG +LG +ADC +ADC +Front +End +Chip +HG +LG +ADC +ADC +Front +End +Chip +High gain (HG) +Low gain (LG) +Figure 1: JUNO LPMT readout electronics scheme. A description of the different parts is given in the +text. +inside the JUNO Water Pool; and the dry electronics, installed in the electronics rooms +of the JUNO underground laboratories, which consists of the back-end (BE), or trigger, +electronics and the data acquisition (DAQ) system. +The FE electronics will be installed underwater on the JUNO Steel Truss structure, +inside a stainless steel, water-tight box, the so-called Under Water Box (UWBox). In +total, the JUNO detector is instrumented with 6681 UWBoxes, 5878 for the CD and 803 +for the Water Pool as part of the JUNO Veto system. Three LPMT output signals are +fed to one UWBox which contains +• three High Voltage Units (HVU): programmable modules which provide the bias +voltage to the LPMT voltage divider. Each HVU independently powers one LPMT. +The HVUs are mounted on a custom Printed Circuit Board (PCB), the splitter +board, that provides mechanical stability to the modules, and decouples the PMT +signal current from the high voltage. +• a Global Control Unit (GCU): a motherboard incorporating the front-end and +readout electronics components. The three LPMT signals reaching the GCU are +processed though independent readout chains. +The LPMT analog signal reaching the GCU is processed by a custom Front-End +4 + +Chip (FEC), which duplicates the input signal and inject it in two parallel streams with +different gains, referred to as high-gain stream and low-gain stream (see Figure 1). The +signal from each stream is further converted to a digital waveform by a 14-bit, 1 GS/s, +custom Flash Analog-to-Digital Converter (FADC). +The usage of two FADCs per readout channel has been driven by the design require- +ment of providing a wide dynamic range in terms of reconstructed photo-electrons (PE): +from 1 PE to 100 PE (high-gain stream) with a resolution of 0.1 PE or better, and from +100 PE to 1000 PE (low-gain stream) with a resolution of 1 PE or better [2, 11], with a +nominal LPMT gain of 107. +A Xilinx Kintex-7 FPGA (XC7K325T) is the core of the GCU and allows to further +process the digital signal (local trigger generation, charge reconstruction and timestamp +tagging) and temporarily store it in a local memory buffer before sending it to the data +acquisition (DAQ). Besides the local memory available in the readout-board FPGA, a +2 GBytes DDR3 memory is available and used to provide a larger memory buffer in the +exceptional case of a sudden increase of the input rate, which overruns the current data +transfer bandwidth between the FE electronics and the DAQ [12]. +The BE electronics is composed of the following active elements: +• the Back End Card (BEC) with the Trigger and Time Interface Mezzanine (TTIM) +• the Reorganize and Multiplex Units (RMU) and the Central Trigger Unit (CTU), +which are part of the Trigger Electronics (see Figure 1). +The LPMTs are connected to the UWBox electronics with a 50 Ω, coaxial cable, with +a length ranging between 1.5 m and 3.5 m. The electronics inside the UWBox has two +independent connections to the dry electronics: a so-called synchronous link (S-link) for +the connection to BE electronics, which provides the clock and synchronization to the +boards and handles the trigger primitives; and an asynchronous link (A-link) which is +fully dedicated to the DAQ and slow-control, or Detector Control System (DCS). These +connections are realized using commercially available CAT-5 and CAT-6 Ethernet cables +for the A-link and S-link, respectively; the length of the cables ranges between 30 m and +100 m. An additional, low-resistance, power cable will be used to bring power to the +electronics inside the UWBox. +5 + +The LPMT electronics can run with a centralized global trigger mode, where the +information from the single fired PMTs is collected and processed in the CTU. The +latter validates the trigger based on a simple PMT multiplicity condition or a more +refined topological distribution of the fired LPMTs in JUNO [13]. Upon a trigger request, +validated waveforms are sent to the DAQ event builder through the A-link. The IPBus +Core protocol [14] is used for data transfer [15], slow control monitoring, and electronics +configurations. +An alternative scheme is possible where all readout boards send their locally trig- +gered waveforms to the DAQ, independently of each other, without passing through the +BE trigger electronics and the S-link. With this approach, all the digitized waveforms, +including those generated by dark noise photo-electrons, would be sent to the DAQ. +The BE electronics was only used in the tests presented in this paper to provide the +UWBox with the clock needed to operate properly, while it was not used to handle trigger +information; hence, the boards were operated in the locally triggered setup explained +above. We chose this configuration because we used the internal test pulse generator, +described in the next section, to generate the input signals, but the internal generator +could only be activated through the A-link, thus generating non-synchronized waveforms. +So, for the purpose of the electronics mass testing, it was easier to just read every +waveform without relying on any triggering logic. +2.1. Internal test pulse generator +Each GCU is equipped with three test pulse generator circuits, one per channel; a +scheme of the circuit is presented in Figure 2. +The main components of the circuit +are a 16-bit digital-to-analog converter (DAC), a switch, and a RC circuit acting as a +differentiator, or high-pass filter, with C1 = 390 pF and R2 = 24.9 Ω, with a 5 % and +1 % tolerances, respectively; the values of C1 and R2 were chosen to produce a signal +mimicking a PMT signal. +The amplitude of the generated pulse can be adjusted via IPbus protocol [14] by +changing the input digital amplitude of the DAC (ADAC), which uses a reference voltage +of 5 V to convert the digital value to a voltage value. The pulse is generated by closing +the switch and connecting the node between the DAC and the differentiator to ground, +generating a step voltage, as shown in Figure 2. +The step function goes through a +6 + +R1 +4.7 kΩ +R2 +24.9 Ω +C1 +390 pF +SWITCH +TS5A3166-Q1 + +FEC +2.5 V +R3 +49.9 Ω +DAC +16 +bit +Vref +5 V +RC +differentiator + +C2 +0.1 μF +PMT +Figure 2: Scheme of the internal test pulse generator. +Each channel is equipped with one internal +generator circuit, which is connected directly to the Front-End Chip (FEC). The main components of +the circuit are a 16-bit digital-to-analog converter (DAC), a switch, and a RC circuit with C1 = 390 pF +and R2 = 24.9 Ω. The connection from the PMT to the FEC is also shown; arrows are used to indicate +the direction of the signals. +differentiator, or high-pass filter, generating a PMT-like pulse which is injected directly +into the FEC of the channel. The switch is also controlled via the IPbus protocol: to +generate one pulse, we need to close and then open again the switch, hence two IPbus +commands are needed; in this way it is possible to control the frequency at which the +switch is closed/open and the test pulses are generated. +The injected input charge, which is the area of each generated pulse, corresponds to +the charge accumulated by the capacitor C1 under a potential difference equal to the +DAC output, evaluated as follows: +Qin = ADAC · 5 V +216 · C1, +(1) +where Qin is in unit of pC if C1 and ADAC are in units of pF and DAC counts, respectively. +The value 5 V/216 ≃ 76 µV/DAC counts is the conversion factor from DAC counts to a +tension in volts. +3. Mass production and testing at the Kunshan site +A facility in Kunshan, China, was devoted to the mass production and testing of the +20-inch PMTs readout electronics. +7 + +3.1. Production process +During mass production, the first step was the welding of the stainless steel bellow +to the UWBox, followed by a leakage test. After that, the threading of the cables for the +S-link, A-link and the power line through the bellow was done. Then, the GCU board +and the three HV units were assembled inside the threaded UWBox and soldered. Each +box was then tested for at least five days, and if it passed the tests, it was finally laser +welded, and, after a leakage test, was stored before being sent to the JUNO experimental +site. A picture of an assembled UWBox before laser welding is shown in Figure 3a. +Before the beginning of the mass production, tests were performed on a small number +of boxes to assess the possible damage and risks from the laser welding procedure; it was +found that no damage is expected from this procedure. Nonetheless, a quicker version of +the tests was performed on each board after the laser welding. +3.2. Testing of the GCUs +During the test, the assembled UWBoxes and the bellows were located on shelves in +a dedicated testing room, as shown in Figure 3b; in the front of the picture, a rack with +power supplies, switches for the network connection, and the trigger electronics is also +visible. The room had places to locate a maximum number of 344 GCUs on nine shelves. +All the tests described in Section 4 were performed in parallel on all the GCUs available +in the testing room. +The test procedure was automatized in order to minimize human errors during the +shifts. Shifts were organized exploiting time zone differences between China, where the +boxes were located, and Europe, so that the European part of the collaboration could +take part in the mass testing remotely, since it was not possible to travel to China due +to COVID-19 restrictions. During day time in China, local shifters were in charge of +assembling between 40 and 60 new UWBoxes per day and replacing them in the testing +room; at the end of the Chinese working day, an European shifter took over to perform +the tests, in this way it was possible to have shifts covering all 24 hours each day. Data +analysis on the acquired data from the tests was performed on the following day, in order +to provide a fast feedback on the tested boards. The mass testing of all 6950 GCUs lasted +for about 10 months from October 2021 to July 2022. Figure 4 shows the cumulative +number of tested boards as a function of time. +8 + +(a) Assembled UWBox before laser welding. +(b) Shelf with UWBoxes in the testing room in Kunshan. +Figure 3: (a) Picture of an assembled Underwater Box before laser welding. The three HVUs are clearly +visible, one near each of the connectors at which the LPMTs will be connected. The GCU board is +located on the bottom. (b) A shelf full of UWBoxes in the testing room at the Kunshan facility. In the +front of the picture, a rack with power supplies, switches, and back-end and trigger electronics is also +visible. +9 + +Oct−2021 +Nov−2021 +Dec−2021 +Jan−2022 +Feb−2022 +Mar−2022 +Apr−2022 +May−2022 +Jun−2022 +Jul−2022 +Aug−2022 +time +0 +1000 +2000 +3000 +4000 +5000 +6000 +7000 +number of tested GCUs [#] +Figure 4: Cumulative number of tested GCUs as a function of time. The production and testing campaign +started in October 2021 and ended in July 2022. A total of 6900 boards were tested. Two breaks in the +production, the first due to Chinese New Year Holidays and the second due to a COVID-19 outbreak, +are clearly visible. +3.3. Network and connection details at the Kunshan site +In the testing room, GCUs were connected to the BECs in batches of 40 in order +to provide the clock to the tested boards through the synchronous link. For the asyn- +chronous link, 40 GCUs were connected to a level 1 (L1) switch through a 1 Gb link, +for a total of nine L1 switches; L1 switches were then connected to a level 2 (L2) switch +through a 4x10 Gb link; the L2 switch was finally connected to the DAQ server via a +4x100 Gb link. The DAQ server consisted of a Dell PowerEdge C6400, with a total of 24 +cores and 48 threads, 2.7 GHz processor and 192 GB RAM. A dedicated local network +was used for the communication between the GCUs and the server. +4. Test Protocol for the LPTM readout electronics +We designed and implemented the test protocol [9] according to the following criteria: +(1) it had to be controlled remotely and to be run in parallel to the production line; (2) +it had to be easy to operate, in order to have non-expert shifters being able to join the +10 + +testing campaign; (3) it had to provide the shifter with a fast and visual feedback of the +performance of the tested components. +The test protocol was performed on each electronics card after all the components got +assembled together, as described in Section 3.1, and before the UWBox was finally sealed +by means of laser welding and then sent to a storage warehouse near the JUNO site. It +is made of several steps: (1) a ping test (sec. 4.3), to check the connection of the board +to the local network; (2) a linearity (sec. 4.4) and a stability (sec. 4.5) tests investigating +the properties of the digitized waveforms to validate the performance and the reliability +of the whole readout chain; (3) a DCS test (sec. 4.6) to monitor the temperature and the +status of the board. Each test is presented in more details in the following subsections. +The tests of step (2) were performed separately on the high-gain and on the low-gain +streams. Input signals were generated in both cases by the internal test pulse generator, +but either the high-gain stream, or the low-gain stream was selected for the readout of +the digitized waveform. +4.1. Properties of the digitized waveform +Figure 5 shows an example of a digitized waveform generated by the internal test pulse +generator described in Section 2.1, where the high-gain stream was selected. During the +tests, the length of the readout window, and hence the length of the waveform, is fixed to +304 samples which correspond to 304 ns, given the FADC sampling frequency of 1 GS/s. +For each digitized waveform, baseline and noise are evaluated. +The baseline, B, +is defined as the average of the first 85 samples; the noise, σbaseline, is defined as the +standard deviation computed on the same samples. +Another property which is monitored during the test is the waveform integrated +charge. The waveform integrated charge, Qout, corresponds to the shadowed region in +Figure 5 and it is evaluated offline as in the following equation: +Qout = +Ns +� +i +75 µV · (B − Ni) · ∆ts +R +, +(2) +where Ns is the number of bins in the integration window, Ni is the amplitude in ADC +counts of the i-th bin, B the baseline value as defined above, 75 µV is the voltage cor- +responding to 1 ADC count, R = 50 Ω is the input impedance, and ∆ts is the width of +11 + +0 +50 +100 +150 +200 +250 +300 +time [ns] +0 +2000 +4000 +6000 +8000 +10000 +12000 +14000 +16000 +ADC counts +Integration window +Baseline and noise +Figure 5: Example of a digitized waveform from GCU 3133 channel 0, generated with the internal test +pulse generator, as described in Section 2.1, and obtained by selecting the high-gain stream. The first 85 +samples are used to evaluate the values of the baseline and the noise. The limits of the charge integration +window are shown as dashed black lines. +12 + +a single bin; in our case ∆ts = 1 ns. The integration window, shown in Figure 5, starts +5 ns before the minimum, or peak, of the waveform, up to 50 ns after the minimum. +In eq. (2), the conversion factor between ADC counts and voltage, 75 µV/ADC count, +is a characteristic of the FADCs which depends on the dynamic range and the number of +bits, and it is the same for the high-gain and low-gain streams. In this way, eq. (2) does +not take into account the gain of the amplification step in the FEC, which in turn has to +be determined through the linearity test of the test protocol, as explained in Section 4.4. +4.2. Configuration of the GCUs +The following GCU parameters needed to be set through the slow control before each +test: (1) the length of the readout window; (2) the value of the pre-trigger; (3) the value +of the trigger threshold; (4) and the trigger mode. For the mass production tests, we fixed +the length of the readout window to 304 ns to optimize the total size of the acquired data. +The pre-trigger is the time interval between the moment at which the signal exceeds the +threshold and the beginning of the readout window, i.e., the region that precedes the +pulse. +There are two possibilities for the trigger threshold: the threshold is either fixed to a +given value in ADC counts, and is the same for all channels; or it is evaluated for each +channel in terms of σbaseline from the baseline. During the tests, the trigger threshold +was fixed to a common value for all channels. The trigger modes have already been +described in Section 2; during the tests, the trigger mode was set to locally triggered +approach, therefore each channel triggers independently from each other and the BE +trigger electronics is not employed. +4.3. Ping test +The first step of the test protocol is the ping test, meant to check that all the GCUs +are properly connected to the local network and responding. A non-responding board +would imply either that the cables are not properly plugged in, which is an easy issue +to solve, or that the assembling procedure was not successful, thus requiring further +investigation on the production side. +For this test, we used the default Linux ping command and sent 100 56-byte packets +in 1 s from the DAQ server to each GCU, so that it was possible to test in a few sec- +onds the connection to the local network of hundreds of boards; the IP addresses were +13 + +automatically recovered by the input GCU ID number. Note that the ping command +directly calculates the mean response time and its standard deviation, which were both +stored, together with the fraction of lost packets. +As a quick visual feedback for the shifter, the mean response time and its standard +deviation were recovered and plotted versus the GCU ID number; an example with a +batch of 160 GCUs is shown in Figure 6. The mean response time depends on the length +of the asynchronous link cables and on the network configuration. +0 +25 +50 +75 +100 +125 +150 +GCU number +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 +0.07 +0.08 +0.09 +0.10 +0.11 +response time [ms] +0 +25 +counts/0.004 ms +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 +0.07 +0.08 +0.09 +0.10 +0.11 +Figure 6: Ping test results for a batch of 160 GCUs. The plot on the left shows the mean response time +and its standard deviation for each GCU; the plot on the right show the distribution of the response +time of the batch of GCUs. A step in the response time is visible around GCU 80, pointing at differences +in cable lengths and network configuration between the first 80 GCUs and the other 80 boards of the +batch. +4.4. Linearity test +The linearity test was meant to test the linear response of the two FADCs serving +each channel and evaluate the gain factors of the two data streams in the FEC. The +test was performed by generating PMT-like signals with the internal test pulse circuit +described in Section 2.1. Before this test, the channel linearity was studied with external +physics sources and by reading PMT signals on a small set of boxes [16, 17]. +For the test, values of the test pulse amplitude were chosen to cover a wide range. For +the high-gain stream, the range starts at 1 PE up to about 160 PE, before the beginning +of the saturation regime. For the low-gain stream, the range starts at about 90 PE up +14 + +to the maximum possible value of the DAC, corresponding to about 1200 PE. The two +ranges overlap, allowing us to check the cross range between the two streams. +The +frequency of pulses generation and the acquisition time were set to provide more than +2000 waveforms for each linearity point. Parameter settings, test pulse generation, and +data acquisition were completely automatized. +Raw data were then processed and saved in ROOT [18] files as TTree objects. For each +channel and each input DAC amplitude, the integrated output charge was evaluated +according to eq. (2); the evaluated values were then collected into an histogram and +the mean value was taken as the output charge corresponding to the given input DAC +amplitude. Finally, for each channel, a quadratic fit was done for both data streams to +extract the gain factor of the two FEC streams, with the fit function defined as: +Qout = c2 · Q2 +in + G · Qin + c0, +(3) +where Qout and Qin are the output and input charge defined by eq. (2) and (1), +respectively, G is the dimensionless gain of the FEC stream, c0 is the intercept, and c2 is +the coefficient of the quadratic term. A quadratic function was used because the response +is not perfectly linear, due to the differential non-linearity (DNL) which is characteristic +of ADCs and DACs; we expect the quadratic term to be subdominant with respect to +the linear term. The gain G is expected to be < 1; the reason for this design choice is +that the FEC input signal is expected to reach amplitudes exceeding the typical FADC +dynamic range, hence the necessity to attenuate and not amplify the signal. +Figure 7 shows the results of the linearity test for one channel of a typical GCU for +the high-gain stream (circles, light red) and the low-gain stream (squares, dark red), +respectively. The corrected output charge shown in the plot was first evaluated through +eq. (2) and then corrected with the gain obtained from the quadratic fit; as it can be seen, +after the gain correction the two data streams lie on the same line. In the figure, input +and output charges are expressed in picocoulomb on the primary axes and in terms of +number of photo-electrons (PE) on the secondary axes, with 1 PE = qe · GPMT = 1.6 pC, +where qe is the electron charge, and GPMT = 107 is the assumed nominal PMT gain of +the 20-inch PMTs in JUNO [1, 5]. +During the analysis, we also checked for the saturation amplitude of the high-gain +15 + +stream, while for the low-gain stream we could not reach saturation with the internal test +pulse generator. In the high-gain configuration, channels saturate for an input signal of +about 16500 DAC counts, corresponding to an input charge of about 450 pC ≃ 280 PE. +Data points above the saturation threshold are not used in the linear fit and are not +shown in Figure 7. +100 +101 +102 +103 +Qin [PE] +100 +101 +102 +103 +corrected Qout [PE] +101 +102 +103 +corrected Qout [pC] +High-gain stream +Low-gain stream +100 +101 +102 +103 +Qin [pC] +−10 +−8 +−6 +−4 +−2 +0 +2 +(data-fit)/fit [%] +Figure 7: Results from the linearity test for one channel of a typical GCU for the high-gain (circles, +light red) and low-gain (squares, dark red) streams. Results from the quadratic fit for the high-stream +are: c0 = (−0.01 ± 0.02) pC, G = 0.5856 ± 0.0006, and c2 = (−4.5 × 10−5 ± 0.3 × 10−5) pC−1; while +for the low-gain stream are: c0 = (0.90 ± 0.08) pC, G = 0.0850 ± 0.0002, and c2 = (−3.3 × 10−6 ± +0.9 × 10−6) pC−1. The fit ranges are [1.6, 257] pC and [149, 1934] pC for the high-gain and the low-gain +streams, respectively. The input charge is evaluated by using eq. (1), while the output charge is first +evaluated through eq. (2) and then corrected for the gain obtained from the quadratic fit. Charges are +also expressed in number of PEs on the secondary axes, where 1 PE = 1.6 pC. +4.5. Stability test +The stability test consists in firing the internal test pulse generator with a fixed +amplitude over a time period lasting several hours, and to check that the waveform +16 + +properties listed below do not change. The input amplitude was set to 12000 DAC counts +for the high-gain stream and to 45000 DAC counts for the low-gain stream. The frequency +of the test pulses was set to 1 Hz, while the data acquisition time was determined by the +available time during the shift. +The waveform monitored parameters are: baseline, noise, minimum value of the wave- +form, and minimum position in the readout window. The baseline and noise are obtained +as described in Section 4.1. These quantities were obtained by processing raw data and +saved in ROOT files as TTree objects. +As an example, Figure 8 shows the results of the stability test for the noise of a typical +GCU. The value of the noise as a function of time is shown for the three channels in three +different panels; distributions of the values are shown as well. The accepted noise level is +between 2 and 4.5 ADC counts, corresponding to about 0.03 PE and 0.08 PE respectively, +and, as it can be seen in Figure 8, the evaluated values lie within these limits. +4.6. Slow control monitoring +The slow control monitoring is meant to read several internal parameters and sensors +installed on the GCU and to monitor the overall status of the board. All sensors were +read through the IPbus protocol [14] in parallel to the DAQ and over the same transport +layer. +For each GCU, the following parameters were read during the slow control monitoring: +the temperature of the FPGA, the temperature and the high voltage value of each HVU, +and several FPGA internal reference voltages [19]. +As an example, Figure 9 shows a plot of the evolution of the FPGA temperature for +five GCUs. For all GCUs, the FPGA temperature is stable over time. The difference +in the absolute values is due to the different positions of the GCUs on the racks in the +testing room (see Section 3). The testing room was equipped with an air conditioning +system with a constant temperature of about 26 °C. +4.7. Storing of test results into a database +The information on the configuration and parameters used for the tests, together +with the results of the tests, are saved in a MySQL database which is available on +the local server at the Kunshan site. Storing these kinds of information is important +17 + +0 +1 +2 +3 +4 +5 +time [h] +2.0 +2.5 +3.0 +3.5 +4.0 +channel 2 +0 +1 +2 +3 +4 +5 +2.0 +2.5 +3.0 +3.5 +4.0 +channel 0 +0 +1 +2 +3 +4 +5 +2.0 +2.5 +3.0 +3.5 +4.0 +σbaseline [ADC counts] +channel 1 +0 +100 +entries/0.2 ADC counts +2.0 +2.5 +3.0 +3.5 +4.0 +0 +100 +2.0 +2.5 +3.0 +3.5 +4.0 +0 +100 +2.0 +2.5 +3.0 +3.5 +4.0 +Figure 8: Evolution of the noise over a 5-hour stability run for the three channels of a typical GCU. The +left plots show the noise evaluated on single waveforms as a function of time; the right plots show the +distribution of the noise values. For all three channels, noise is within the acceptance interval. +18 + +20:00 +21:00 +22:00 +23:00 +Feb−26 +time +2022−Feb−26 +54 +55 +56 +57 +58 +temperature [°C] +GCU ID +3952 +4330 +4946 +5328 +6629 +Figure 9: The figure shows the evolution of the FPGA temperature for 5 GCUs, recorded from the slow +control monitoring. The different temperature values are due to the different positions of the GCUs in +the testing room in the dedicated facility at Kunshan. +to have an history of the performances of each GCU, and to compare the results during +mass production with the tests foreseen for the upcoming installation and commissioning +phases. +Figure 10 is obtained by accessing the local database and shows the value of the noise +from the stability test for several days and runs for GCU 3333; each panel shows results +for one of the three GCU channels. The runs shown in the figure span a time period of +more than 25 days, during which the noise is stable and within the acceptance range. +Figures 11a and 11b show the distributions of the high-gain and the low-gain values, +respectively, obtained in the linearity test by using eq. (3). The distribution for the high- +gain stream has a mean of 0.599 and a standard deviation of 0.007, while the distribution +for the low-gain stream has a mean of 0.0883 and a standard deviation of 0.0013. +5. Conclusion +A test protocol was developed to evaluate the performance of the 20-inch PMT read- +out electronics for the JUNO experiment during mass production. A total of 6950 devices +19 + +2.0 +2.5 +3.0 +3.5 +4.0 +4.5 +noise [ADC counts] ch. 0 +GCU 3333 +2.0 +2.5 +3.0 +3.5 +4.0 +4.5 +noise [ADC counts] ch. 1 +20220330 - run2 +20220331 - run0 +20220402 - run0 +20220402 - run1 +20220403 - run0 +20220404 - run0 +20220405 - run0 +20220406 - run0 +20220407 - run0 +20220408 - run0 +20220409 - run0 +20220410 - run1 +20220410 - run0 +20220411 - run1 +20220412 - run0 +20220412 - run1 +20220413 - run0 +20220420 - run0 +20220420 - run1 +20220420 - run2 +20220420 - run3 +20220421 - run1 +20220425 - run0 +date [YYYYMMDD] +2.0 +2.5 +3.0 +3.5 +4.0 +4.5 +noise [ADC counts] ch. 2 +Figure 10: Results of the stability test from several runs for GCU 3333 are shown. The three panels +show the values of the noise for channel 0 (top), channel 1 (middle) and channel 2 (bottom). The noise +for all channels is stable and within the acceptance range over a period of more than 25 days. +20 + +0.57 +0.58 +0.59 +0.60 +0.61 +0.62 +0.63 +high gain +0 +200 +400 +600 +800 +1000 +1200 +1400 +counts/0.0012 +mean: 0.599 +sigma: 0.007 +(a) Gain distribution for the high-gain stream. +0.084 +0.086 +0.088 +0.090 +0.092 +low gain +0 +200 +400 +600 +800 +1000 +1200 +counts/0.0002 +mean: 0.0883 +sigma: 0.0013 +(b) Gain distribution for the low-gain stream. +Figure 11: Distributions of the gain obtained from the linearity test for (a) the high-gain stream and +(b) the low-gain stream. The distribution of the high gain has mean and standard deviation equal to +0.599 and 0.007, respectively; the distribution of the low gain has mean and standard deviation equal to +0.0883 and 0.0013, respectively. +21 + +Table 1: Acceptance range for the baseline, noise, high gain, and low gain, used as acceptance criteria +for the evaluation of the performance of each GCU. +Parameter +Acceptance range +baseline +11000 - 12000 ADC counts +noise +2 - 4.5 ADC counts +high gain +0.5 - 0.65 +low gain +0.05 - 0.095 +were tested in about ten months. Only eight GCUs were discarded on the basis of the +tests presented in this work and the criteria shown in Table 1. Other 56 GCUs were +discarded due to issues arisen during the assembling procedure. In total, 6886 GCUs +were accepted, while only 64 were rejected, providing a final acceptance yield of 99.1 %. +Out of the 6886 accepted cards, 6681 will be used in the CD and Water Pool veto system, +25 will be used by OSIRIS [20], while the remaining 180 will be kept as backup. The +test protocol described in this paper will be used as a reference for the upcoming tests +during the installation and commissioning phases, where a few adjustments are needed +given the different environmental conditions and setup. +Acknowledgements +Part of this work has been supported by the Italian-Chinese collaborative research +program jointly funded by the Italian Ministry of Foreign Affairs and International Co- +operation (MAECI) and the National Natural Science Foundation of China (NSFC). +References +[1] A. Abusleme, et al., JUNO physics and detector, Prog. Part. Nucl. Phys. 123 (2022) 103927. +arXiv:2104.02565, doi:10.1016/j.ppnp.2021.103927. +[2] F. An, et al., Neutrino Physics with JUNO, J. 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' Roma,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' Italy oINFN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' Sezione di Milano Bicocca e Politecnico di Milano,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' Dipartimento di Energetica,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' Milano,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' Italy pINFN Sezione di Perugia e Universit`a di Perugia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' Dipartimento di Chimica,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' Biologia e Biotecnologie,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' Perugia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' Italy qLaboratori Nazionali dell’INFN di Frascati,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' Italy rUniversit`a di Catania,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' Dipartimento di Fisica e Astronomia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' Catania,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' Italy sINFN Sezione di Ferrara,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' Ferrara,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' Italy tUniversit`a degli Studi di Ferrara,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' Dipartimento di Fisica e Scienze della Terra,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' Italy Abstract The Jiangmen Underground Neutrino Observatory (JUNO) is a multi-purpose,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' large size,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' liquid scintillator experiment under construction in China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' JUNO will perform leading measurements detecting neutrinos from different sources (reactor, terrestrial and astrophysical neutrinos) covering a wide energy range (from 200 keV to several GeV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' This paper focuses on the design and development of a test protocol for the 20-inch PMT underwater readout electronics, performed in parallel to the mass production line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' In a time period of about ten months, a total number of 6950 electronic boards were tested with an acceptance yield of 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='1 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' Keywords: Read-Out electronics, photomultiplier, liquid scintillator, large scale neutrino experiment 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' Introduction The Jiangmen Underground Neutrino Observatory [1] (JUNO) is a 20 kton neutrino medium baseline experiment under construction in southern China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' The JUNO ex- periment has been proposed [2] with the main goals of determining the neutrino mass ordering with a significance of 3 σ within the first six years of data taking and measuring the oscillation parameters, ∆m2 21, ∆m2 31, and sin2 θ12, with sub-percent precision [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' To achieve these goals, JUNO is located about 53 km away from two nuclear power plants and will detect electron antineutrinos produced by the beta decays of fission products ∗Corresponding author ∗∗Now at INFN Laboratori Nazionali del Sud, Italy Preprint submitted to Elsevier January 12, 2023 inside the nuclear cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' JUNO will also be able to address many other topics in particle and astroparticle physics, by detecting neutrinos from natural sources: solar neutrinos, atmospheric neutrinos, geo-neutrinos, neutrinos from core-collapse supernovae, and from the diffuse supernovae neutrino background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' An updated overview of the JUNO physics reach can be found here [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' The JUNO Central Detector (CD) consists of 20 kton of liquid scintillator contained in a spherical acrylic vessel with a 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='4 m diameter, supported by a stainless Steel Truss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' A double system of 17 612 20-inch large-PMTs (LPMTs) [4, 5] and 25 600 3-inch small- PMTs (SPMTs) [6] is employed to detect the scintillation and Cherenkov light produced by neutrino interactions with the liquid scintillator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' The liquid scintillator target is surrounded by a 35-kt pure Water Pool, which is instrumented with 2400 20-inch LPMTs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' the Water Pool shields the inner part of the detector from environmental radioactivity, and is part of the muon Veto system, together with the Top Tracker on top of the whole structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' The JUNO LPMT underwater readout electronic system is responsible to sample and to process the LPMT output current [7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' Spotting hardware failures and evaluating the performance of the underwater readout electronics before the actual installation is of paramount importance, because it will be impossible to repair or to change an electronics module after its deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' Furthermore, the required loss rate of the electronics chan- nels is less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='5% in 6 years [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' To this end, we designed and developed a dedicated test protocol [9] to be carried out during the mass production, held in a dedicated facility in Kunshan, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' The rest of the paper is organized as follows: in Section 2 we describe the JUNO LPMT readout electronics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' in Section 3 we discuss the mass production and the mass testing setup at the dedicated facility in Kunshan;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' in Section 4 a detailed description of the developed test protocol is presented;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' conclusions are drawn in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' JUNO LPMT readout electronics A scheme of the JUNO LPMT electronics is given in Figure 1 [8];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' the design is an optimization of previous developments [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' The full electronics chain is composed of two parts: the front-end (FE), or wet, electronics [10] located very close to the LPMT output, 3 High Voltage Unit High Voltage Unit High Voltage Unit Custom HV (0 - 3 kV) / 300µA Global Control Unit (GCU) FPGA Under Water Box (UWBox) Custom ADC 14 bit, 1 Gsps 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='5 - 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='5 m cables (signal and HV) 2 GB RAM for SN bursts Gbit Enterpise Switch DAQ LV Up to 100 m CAT6 + low Z power cables Dry electronics Wet electronics Back End Card Trigger Electronics CLK DDR3 Sync link Async link Up to 100 m CAT5 cables ADC ADC Front End Chip HG LG ADC ADC Front End Chip HG LG ADC ADC Front End Chip High gain (HG) Low gain (LG) Figure 1: JUNO LPMT readout electronics scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' A description of the different parts is given in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' inside the JUNO Water Pool;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' and the dry electronics, installed in the electronics rooms of the JUNO underground laboratories, which consists of the back-end (BE), or trigger, electronics and the data acquisition (DAQ) system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' The FE electronics will be installed underwater on the JUNO Steel Truss structure, inside a stainless steel, water-tight box, the so-called Under Water Box (UWBox).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' In total, the JUNO detector is instrumented with 6681 UWBoxes, 5878 for the CD and 803 for the Water Pool as part of the JUNO Veto system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' Three LPMT output signals are fed to one UWBox which contains three High Voltage Units (HVU): programmable modules which provide the bias voltage to the LPMT voltage divider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' Each HVU independently powers one LPMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' The HVUs are mounted on a custom Printed Circuit Board (PCB), the splitter board, that provides mechanical stability to the modules, and decouples the PMT signal current from the high voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' a Global Control Unit (GCU): a motherboard incorporating the front-end and readout electronics components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' The three LPMT signals reaching the GCU are processed though independent readout chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' The LPMT analog signal reaching the GCU is processed by a custom Front-End 4 Chip (FEC), which duplicates the input signal and inject it in two parallel streams with different gains, referred to as high-gain stream and low-gain stream (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' The signal from each stream is further converted to a digital waveform by a 14-bit, 1 GS/s, custom Flash Analog-to-Digital Converter (FADC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' The usage of two FADCs per readout channel has been driven by the design require- ment of providing a wide dynamic range in terms of reconstructed photo-electrons (PE): from 1 PE to 100 PE (high-gain stream) with a resolution of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='1 PE or better, and from 100 PE to 1000 PE (low-gain stream) with a resolution of 1 PE or better [2, 11], with a nominal LPMT gain of 107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' A Xilinx Kintex-7 FPGA (XC7K325T) is the core of the GCU and allows to further process the digital signal (local trigger generation, charge reconstruction and timestamp tagging) and temporarily store it in a local memory buffer before sending it to the data acquisition (DAQ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' Besides the local memory available in the readout-board FPGA, a 2 GBytes DDR3 memory is available and used to provide a larger memory buffer in the exceptional case of a sudden increase of the input rate, which overruns the current data transfer bandwidth between the FE electronics and the DAQ [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' The BE electronics is composed of the following active elements: the Back End Card (BEC) with the Trigger and Time Interface Mezzanine (TTIM) the Reorganize and Multiplex Units (RMU) and the Central Trigger Unit (CTU), which are part of the Trigger Electronics (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' The LPMTs are connected to the UWBox electronics with a 50 Ω, coaxial cable, with a length ranging between 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='5 m and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='5 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' The electronics inside the UWBox has two independent connections to the dry electronics: a so-called synchronous link (S-link) for the connection to BE electronics, which provides the clock and synchronization to the boards and handles the trigger primitives;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' and an asynchronous link (A-link) which is fully dedicated to the DAQ and slow-control, or Detector Control System (DCS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' These connections are realized using commercially available CAT-5 and CAT-6 Ethernet cables for the A-link and S-link, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' the length of the cables ranges between 30 m and 100 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' An additional, low-resistance, power cable will be used to bring power to the electronics inside the UWBox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' 5 The LPMT electronics can run with a centralized global trigger mode, where the information from the single fired PMTs is collected and processed in the CTU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' The latter validates the trigger based on a simple PMT multiplicity condition or a more refined topological distribution of the fired LPMTs in JUNO [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' Upon a trigger request, validated waveforms are sent to the DAQ event builder through the A-link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' The IPBus Core protocol [14] is used for data transfer [15], slow control monitoring, and electronics configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' An alternative scheme is possible where all readout boards send their locally trig- gered waveforms to the DAQ, independently of each other, without passing through the BE trigger electronics and the S-link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' With this approach, all the digitized waveforms, including those generated by dark noise photo-electrons, would be sent to the DAQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' The BE electronics was only used in the tests presented in this paper to provide the UWBox with the clock needed to operate properly, while it was not used to handle trigger information;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' hence, the boards were operated in the locally triggered setup explained above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' We chose this configuration because we used the internal test pulse generator, described in the next section, to generate the input signals, but the internal generator could only be activated through the A-link, thus generating non-synchronized waveforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' So, for the purpose of the electronics mass testing, it was easier to just read every waveform without relying on any triggering logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' Internal test pulse generator Each GCU is equipped with three test pulse generator circuits, one per channel;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' a scheme of the circuit is presented in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' The main components of the circuit are a 16-bit digital-to-analog converter (DAC), a switch, and a RC circuit acting as a differentiator, or high-pass filter, with C1 = 390 pF and R2 = 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='9 Ω, with a 5 % and 1 % tolerances, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' the values of C1 and R2 were chosen to produce a signal mimicking a PMT signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' The amplitude of the generated pulse can be adjusted via IPbus protocol [14] by changing the input digital amplitude of the DAC (ADAC), which uses a reference voltage of 5 V to convert the digital value to a voltage value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' The pulse is generated by closing the switch and connecting the node between the DAC and the differentiator to ground, generating a step voltage, as shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' The step function goes through a 6 R1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='7 kΩ R2 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='9 Ω C1 390 pF SWITCH TS5A3166-Q1 FEC 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='5 V R3 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='9 Ω DAC 16 bit Vref 5 V RC differentiator C2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='1 μF PMT Figure 2: Scheme of the internal test pulse generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' Each channel is equipped with one internal generator circuit, which is connected directly to the Front-End Chip (FEC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' The main components of the circuit are a 16-bit digital-to-analog converter (DAC), a switch, and a RC circuit with C1 = 390 pF and R2 = 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='9 Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' The connection from the PMT to the FEC is also shown;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' arrows are used to indicate the direction of the signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' differentiator, or high-pass filter, generating a PMT-like pulse which is injected directly into the FEC of the channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' The switch is also controlled via the IPbus protocol: to generate one pulse, we need to close and then open again the switch, hence two IPbus commands are needed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' in this way it is possible to control the frequency at which the switch is closed/open and the test pulses are generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' The injected input charge, which is the area of each generated pulse, corresponds to the charge accumulated by the capacitor C1 under a potential difference equal to the DAC output, evaluated as follows: Qin = ADAC · 5 V 216 · C1, (1) where Qin is in unit of pC if C1 and ADAC are in units of pF and DAC counts, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' The value 5 V/216 ≃ 76 µV/DAC counts is the conversion factor from DAC counts to a tension in volts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' Mass production and testing at the Kunshan site A facility in Kunshan, China, was devoted to the mass production and testing of the 20-inch PMTs readout electronics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' Production process During mass production, the first step was the welding of the stainless steel bellow to the UWBox, followed by a leakage test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' After that, the threading of the cables for the S-link, A-link and the power line through the bellow was done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' Then, the GCU board and the three HV units were assembled inside the threaded UWBox and soldered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' Each box was then tested for at least five days, and if it passed the tests, it was finally laser welded, and, after a leakage test, was stored before being sent to the JUNO experimental site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' A picture of an assembled UWBox before laser welding is shown in Figure 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' Before the beginning of the mass production, tests were performed on a small number of boxes to assess the possible damage and risks from the laser welding procedure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' it was found that no damage is expected from this procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' Nonetheless, a quicker version of the tests was performed on each board after the laser welding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' Testing of the GCUs During the test, the assembled UWBoxes and the bellows were located on shelves in a dedicated testing room, as shown in Figure 3b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' in the front of the picture, a rack with power supplies, switches for the network connection, and the trigger electronics is also visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' The room had places to locate a maximum number of 344 GCUs on nine shelves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' All the tests described in Section 4 were performed in parallel on all the GCUs available in the testing room.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' The test procedure was automatized in order to minimize human errors during the shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' Shifts were organized exploiting time zone differences between China, where the boxes were located, and Europe, so that the European part of the collaboration could take part in the mass testing remotely, since it was not possible to travel to China due to COVID-19 restrictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' During day time in China, local shifters were in charge of assembling between 40 and 60 new UWBoxes per day and replacing them in the testing room;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' at the end of the Chinese working day, an European shifter took over to perform the tests, in this way it was possible to have shifts covering all 24 hours each day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' Data analysis on the acquired data from the tests was performed on the following day, in order to provide a fast feedback on the tested boards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' The mass testing of all 6950 GCUs lasted for about 10 months from October 2021 to July 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' Figure 4 shows the cumulative number of tested boards as a function of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' 8 (a) Assembled UWBox before laser welding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' (b) Shelf with UWBoxes in the testing room in Kunshan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' Figure 3: (a) Picture of an assembled Underwater Box before laser welding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' The three HVUs are clearly visible, one near each of the connectors at which the LPMTs will be connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' The GCU board is located on the bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' (b) A shelf full of UWBoxes in the testing room at the Kunshan facility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' In the front of the picture, a rack with power supplies, switches, and back-end and trigger electronics is also visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' 9 Oct−2021 Nov−2021 Dec−2021 Jan−2022 Feb−2022 Mar−2022 Apr−2022 May−2022 Jun−2022 Jul−2022 Aug−2022 time 0 1000 2000 3000 4000 5000 6000 7000 number of tested GCUs [#] Figure 4: Cumulative number of tested GCUs as a function of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' The production and testing campaign started in October 2021 and ended in July 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' A total of 6900 boards were tested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' Two breaks in the production, the first due to Chinese New Year Holidays and the second due to a COVID-19 outbreak, are clearly visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' Network and connection details at the Kunshan site In the testing room, GCUs were connected to the BECs in batches of 40 in order to provide the clock to the tested boards through the synchronous link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' For the asyn- chronous link, 40 GCUs were connected to a level 1 (L1) switch through a 1 Gb link, for a total of nine L1 switches;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' L1 switches were then connected to a level 2 (L2) switch through a 4x10 Gb link;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' the L2 switch was finally connected to the DAQ server via a 4x100 Gb link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' The DAQ server consisted of a Dell PowerEdge C6400, with a total of 24 cores and 48 threads, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='7 GHz processor and 192 GB RAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' A dedicated local network was used for the communication between the GCUs and the server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' Test Protocol for the LPTM readout electronics We designed and implemented the test protocol [9] according to the following criteria: (1) it had to be controlled remotely and to be run in parallel to the production line;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' (2) it had to be easy to operate, in order to have non-expert shifters being able to join the 10 testing campaign;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' (3) it had to provide the shifter with a fast and visual feedback of the performance of the tested components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' The test protocol was performed on each electronics card after all the components got assembled together, as described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='1, and before the UWBox was finally sealed by means of laser welding and then sent to a storage warehouse near the JUNO site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' It is made of several steps: (1) a ping test (sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='3), to check the connection of the board to the local network;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' (2) a linearity (sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='4) and a stability (sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='5) tests investigating the properties of the digitized waveforms to validate the performance and the reliability of the whole readout chain;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' (3) a DCS test (sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='6) to monitor the temperature and the status of the board.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' Each test is presented in more details in the following subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' The tests of step (2) were performed separately on the high-gain and on the low-gain streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' Input signals were generated in both cases by the internal test pulse generator, but either the high-gain stream, or the low-gain stream was selected for the readout of the digitized waveform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' Properties of the digitized waveform Figure 5 shows an example of a digitized waveform generated by the internal test pulse generator described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='1, where the high-gain stream was selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' During the tests, the length of the readout window, and hence the length of the waveform, is fixed to 304 samples which correspond to 304 ns, given the FADC sampling frequency of 1 GS/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' For each digitized waveform, baseline and noise are evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' The baseline, B, is defined as the average of the first 85 samples;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' the noise, σbaseline, is defined as the standard deviation computed on the same samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' Another property which is monitored during the test is the waveform integrated charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' The waveform integrated charge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' Qout,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' corresponds to the shadowed region in Figure 5 and it is evaluated offline as in the following equation: Qout = Ns � i 75 µV · (B − Ni) · ∆ts R ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' (2) where Ns is the number of bins in the integration window,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' Ni is the amplitude in ADC counts of the i-th bin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' B the baseline value as defined above,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' 75 µV is the voltage cor- responding to 1 ADC count,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' R = 50 Ω is the input impedance,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' and ∆ts is the width of 11 0 50 100 150 200 250 300 time [ns] 0 2000 4000 6000 8000 10000 12000 14000 16000 ADC counts Integration window Baseline and noise Figure 5: Example of a digitized waveform from GCU 3133 channel 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' generated with the internal test pulse generator,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' as described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='1, and obtained by selecting the high-gain stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' The first 85 samples are used to evaluate the values of the baseline and the noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' The limits of the charge integration window are shown as dashed black lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' 12 a single bin;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' in our case ∆ts = 1 ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' The integration window, shown in Figure 5, starts 5 ns before the minimum, or peak, of the waveform, up to 50 ns after the minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' In eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' (2), the conversion factor between ADC counts and voltage, 75 µV/ADC count, is a characteristic of the FADCs which depends on the dynamic range and the number of bits, and it is the same for the high-gain and low-gain streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' In this way, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' (2) does not take into account the gain of the amplification step in the FEC, which in turn has to be determined through the linearity test of the test protocol, as explained in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' Configuration of the GCUs The following GCU parameters needed to be set through the slow control before each test: (1) the length of the readout window;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' (2) the value of the pre-trigger;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' (3) the value of the trigger threshold;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' (4) and the trigger mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' For the mass production tests, we fixed the length of the readout window to 304 ns to optimize the total size of the acquired data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' The pre-trigger is the time interval between the moment at which the signal exceeds the threshold and the beginning of the readout window, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=', the region that precedes the pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' There are two possibilities for the trigger threshold: the threshold is either fixed to a given value in ADC counts, and is the same for all channels;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' or it is evaluated for each channel in terms of σbaseline from the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' During the tests, the trigger threshold was fixed to a common value for all channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' The trigger modes have already been described in Section 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' during the tests, the trigger mode was set to locally triggered approach, therefore each channel triggers independently from each other and the BE trigger electronics is not employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' Ping test The first step of the test protocol is the ping test, meant to check that all the GCUs are properly connected to the local network and responding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' A non-responding board would imply either that the cables are not properly plugged in, which is an easy issue to solve, or that the assembling procedure was not successful, thus requiring further investigation on the production side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' For this test, we used the default Linux ping command and sent 100 56-byte packets in 1 s from the DAQ server to each GCU, so that it was possible to test in a few sec- onds the connection to the local network of hundreds of boards;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' the IP addresses were 13 automatically recovered by the input GCU ID number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' Note that the ping command directly calculates the mean response time and its standard deviation, which were both stored, together with the fraction of lost packets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' As a quick visual feedback for the shifter, the mean response time and its standard deviation were recovered and plotted versus the GCU ID number;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' an example with a batch of 160 GCUs is shown in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' The mean response time depends on the length of the asynchronous link cables and on the network configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' 0 25 50 75 100 125 150 GCU number 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='11 response time [ms] 0 25 counts/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='004 ms 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='11 Figure 6: Ping test results for a batch of 160 GCUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' The plot on the left shows the mean response time and its standard deviation for each GCU;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' the plot on the right show the distribution of the response time of the batch of GCUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' A step in the response time is visible around GCU 80, pointing at differences in cable lengths and network configuration between the first 80 GCUs and the other 80 boards of the batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' Linearity test The linearity test was meant to test the linear response of the two FADCs serving each channel and evaluate the gain factors of the two data streams in the FEC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' The test was performed by generating PMT-like signals with the internal test pulse circuit described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' Before this test, the channel linearity was studied with external physics sources and by reading PMT signals on a small set of boxes [16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' For the test, values of the test pulse amplitude were chosen to cover a wide range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' For the high-gain stream, the range starts at 1 PE up to about 160 PE, before the beginning of the saturation regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' For the low-gain stream, the range starts at about 90 PE up 14 to the maximum possible value of the DAC, corresponding to about 1200 PE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' The two ranges overlap, allowing us to check the cross range between the two streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' The frequency of pulses generation and the acquisition time were set to provide more than 2000 waveforms for each linearity point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' Parameter settings, test pulse generation, and data acquisition were completely automatized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' Raw data were then processed and saved in ROOT [18] files as TTree objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' For each channel and each input DAC amplitude, the integrated output charge was evaluated according to eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' (2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' the evaluated values were then collected into an histogram and the mean value was taken as the output charge corresponding to the given input DAC amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' Finally, for each channel, a quadratic fit was done for both data streams to extract the gain factor of the two FEC streams, with the fit function defined as: Qout = c2 · Q2 in + G · Qin + c0, (3) where Qout and Qin are the output and input charge defined by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' (2) and (1), respectively, G is the dimensionless gain of the FEC stream, c0 is the intercept, and c2 is the coefficient of the quadratic term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' A quadratic function was used because the response is not perfectly linear, due to the differential non-linearity (DNL) which is characteristic of ADCs and DACs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' we expect the quadratic term to be subdominant with respect to the linear term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' The gain G is expected to be < 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' the reason for this design choice is that the FEC input signal is expected to reach amplitudes exceeding the typical FADC dynamic range, hence the necessity to attenuate and not amplify the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' Figure 7 shows the results of the linearity test for one channel of a typical GCU for the high-gain stream (circles, light red) and the low-gain stream (squares, dark red), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' The corrected output charge shown in the plot was first evaluated through eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' (2) and then corrected with the gain obtained from the quadratic fit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' as it can be seen, after the gain correction the two data streams lie on the same line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' In the figure, input and output charges are expressed in picocoulomb on the primary axes and in terms of number of photo-electrons (PE) on the secondary axes, with 1 PE = qe · GPMT = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='6 pC, where qe is the electron charge, and GPMT = 107 is the assumed nominal PMT gain of the 20-inch PMTs in JUNO [1, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' During the analysis, we also checked for the saturation amplitude of the high-gain 15 stream, while for the low-gain stream we could not reach saturation with the internal test pulse generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' In the high-gain configuration, channels saturate for an input signal of about 16500 DAC counts, corresponding to an input charge of about 450 pC ≃ 280 PE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' Data points above the saturation threshold are not used in the linear fit and are not shown in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' 100 101 102 103 Qin [PE] 100 101 102 103 corrected Qout [PE] 101 102 103 corrected Qout [pC] High-gain stream Low-gain stream 100 101 102 103 Qin [pC] −10 −8 −6 −4 −2 0 2 (data-fit)/fit [%] Figure 7: Results from the linearity test for one channel of a typical GCU for the high-gain (circles, light red) and low-gain (squares, dark red) streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' Results from the quadratic fit for the high-stream are: c0 = (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='01 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='02) pC, G = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='5856 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='0006, and c2 = (−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='5 × 10−5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='3 × 10−5) pC−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' while for the low-gain stream are: c0 = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='90 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='08) pC, G = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='0850 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='0002, and c2 = (−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='3 × 10−6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='9 × 10−6) pC−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' The fit ranges are [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='6, 257] pC and [149, 1934] pC for the high-gain and the low-gain streams, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' The input charge is evaluated by using eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' (1), while the output charge is first evaluated through eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' (2) and then corrected for the gain obtained from the quadratic fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' Charges are also expressed in number of PEs on the secondary axes, where 1 PE = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='6 pC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' Stability test The stability test consists in firing the internal test pulse generator with a fixed amplitude over a time period lasting several hours, and to check that the waveform 16 properties listed below do not change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' The input amplitude was set to 12000 DAC counts for the high-gain stream and to 45000 DAC counts for the low-gain stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' The frequency of the test pulses was set to 1 Hz, while the data acquisition time was determined by the available time during the shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' The waveform monitored parameters are: baseline, noise, minimum value of the wave- form, and minimum position in the readout window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' The baseline and noise are obtained as described in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' These quantities were obtained by processing raw data and saved in ROOT files as TTree objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' As an example, Figure 8 shows the results of the stability test for the noise of a typical GCU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' The value of the noise as a function of time is shown for the three channels in three different panels;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' distributions of the values are shown as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' The accepted noise level is between 2 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='5 ADC counts, corresponding to about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='03 PE and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='08 PE respectively, and, as it can be seen in Figure 8, the evaluated values lie within these limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' Slow control monitoring The slow control monitoring is meant to read several internal parameters and sensors installed on the GCU and to monitor the overall status of the board.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' All sensors were read through the IPbus protocol [14] in parallel to the DAQ and over the same transport layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' For each GCU, the following parameters were read during the slow control monitoring: the temperature of the FPGA, the temperature and the high voltage value of each HVU, and several FPGA internal reference voltages [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' As an example, Figure 9 shows a plot of the evolution of the FPGA temperature for five GCUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' For all GCUs, the FPGA temperature is stable over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' The difference in the absolute values is due to the different positions of the GCUs on the racks in the testing room (see Section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' The testing room was equipped with an air conditioning system with a constant temperature of about 26 °C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' Storing of test results into a database The information on the configuration and parameters used for the tests, together with the results of the tests, are saved in a MySQL database which is available on the local server at the Kunshan site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' Storing these kinds of information is important 17 0 1 2 3 4 5 time [h] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='0 channel 2 0 1 2 3 4 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='0 channel 0 0 1 2 3 4 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='0 σbaseline [ADC counts] channel 1 0 100 entries/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='2 ADC counts 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='0 0 100 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='0 0 100 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='0 Figure 8: Evolution of the noise over a 5-hour stability run for the three channels of a typical GCU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' The left plots show the noise evaluated on single waveforms as a function of time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' the right plots show the distribution of the noise values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' For all three channels, noise is within the acceptance interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' 18 20:00 21:00 22:00 23:00 Feb−26 time 2022−Feb−26 54 55 56 57 58 temperature [°C] GCU ID 3952 4330 4946 5328 6629 Figure 9: The figure shows the evolution of the FPGA temperature for 5 GCUs, recorded from the slow control monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' The different temperature values are due to the different positions of the GCUs in the testing room in the dedicated facility at Kunshan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' to have an history of the performances of each GCU, and to compare the results during mass production with the tests foreseen for the upcoming installation and commissioning phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' Figure 10 is obtained by accessing the local database and shows the value of the noise from the stability test for several days and runs for GCU 3333;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' each panel shows results for one of the three GCU channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' The runs shown in the figure span a time period of more than 25 days, during which the noise is stable and within the acceptance range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' Figures 11a and 11b show the distributions of the high-gain and the low-gain values, respectively, obtained in the linearity test by using eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' The distribution for the high- gain stream has a mean of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='599 and a standard deviation of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='007, while the distribution for the low-gain stream has a mean of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='0883 and a standard deviation of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='0013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' Conclusion A test protocol was developed to evaluate the performance of the 20-inch PMT read- out electronics for the JUNO experiment during mass production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' A total of 6950 devices 19 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='5 noise [ADC counts] ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' 0 GCU 3333 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='5 noise [ADC counts] ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' 1 20220330 - run2 20220331 - run0 20220402 - run0 20220402 - run1 20220403 - run0 20220404 - run0 20220405 - run0 20220406 - run0 20220407 - run0 20220408 - run0 20220409 - run0 20220410 - run1 20220410 - run0 20220411 - run1 20220412 - run0 20220412 - run1 20220413 - run0 20220420 - run0 20220420 - run1 20220420 - run2 20220420 - run3 20220421 - run1 20220425 - run0 date [YYYYMMDD] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='5 noise [ADC counts] ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' 2 Figure 10: Results of the stability test from several runs for GCU 3333 are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' The three panels show the values of the noise for channel 0 (top), channel 1 (middle) and channel 2 (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' The noise for all channels is stable and within the acceptance range over a period of more than 25 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='57 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='58 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='59 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='61 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='63 high gain 0 200 400 600 800 1000 1200 1400 counts/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='0012 mean: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='599 sigma: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='007 (a) Gain distribution for the high-gain stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='084 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='086 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='088 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='090 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='092 low gain 0 200 400 600 800 1000 1200 counts/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='0002 mean: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='0883 sigma: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='0013 (b) Gain distribution for the low-gain stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' Figure 11: Distributions of the gain obtained from the linearity test for (a) the high-gain stream and (b) the low-gain stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' The distribution of the high gain has mean and standard deviation equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='599 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='007, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' the distribution of the low gain has mean and standard deviation equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='0883 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='0013, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' 21 Table 1: Acceptance range for the baseline, noise, high gain, and low gain, used as acceptance criteria for the evaluation of the performance of each GCU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' Parameter Acceptance range baseline 11000 - 12000 ADC counts noise 2 - 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='5 ADC counts high gain 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='5 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='65 low gain 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='05 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='095 were tested in about ten months.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' Only eight GCUs were discarded on the basis of the tests presented in this work and the criteria shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' Other 56 GCUs were discarded due to issues arisen during the assembling procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' In total, 6886 GCUs were accepted, while only 64 were rejected, providing a final acceptance yield of 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='1 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' Out of the 6886 accepted cards, 6681 will be used in the CD and Water Pool veto system, 25 will be used by OSIRIS [20], while the remaining 180 will be kept as backup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' The test protocol described in this paper will be used as a reference for the upcoming tests during the installation and commissioning phases, where a few adjustments are needed given the different environmental conditions and setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' Acknowledgements Part of this work has been supported by the Italian-Chinese collaborative research program jointly funded by the Italian Ministry of Foreign Affairs and International Co- operation 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content='1140/epjc/s10052-021-09544-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} +page_content=' 23' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfNAmL/content/2301.04379v1.pdf'} diff --git a/stE0T4oBgHgl3EQfbABY/content/tmp_files/2301.02342v1.pdf.txt b/stE0T4oBgHgl3EQfbABY/content/tmp_files/2301.02342v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..612170261731e7c84c60887f4a18e4fac95f87ef --- /dev/null +++ b/stE0T4oBgHgl3EQfbABY/content/tmp_files/2301.02342v1.pdf.txt @@ -0,0 +1,2952 @@ +Instantons on Young diagrams with matters +Yanyan Chen,a Jiaqun Jiang,a Satoshi Nawataa and Yilu Shaob +aDepartment of Physics and Center for Field Theory & Particle Physics, Fudan University, +20005, Songhu Road, 200438 Shanghai, China +bInstitut de Mathématiques de Bourgogne, Université de Bourgogne Franche-Comté, +9 avenue Alain Savary, Dijon, France +E-mail: yanyanchen235@gmail.com, jiangjiaqun@gmail.com, +snawata@gmail.com, shaoyilu1999@gmail.com +Abstract: We present the unrefined instanton partition functions of various 5d +gauge theories with matter beyond the fundamental representation as sums over +Young diagrams. By using these partition functions, we verify a range of dualities +predicted by the fivebrane web and representation theory. +arXiv:2301.02342v1 [hep-th] 6 Jan 2023 + +Contents +1 +Introduction +1 +2 +SU(N) gauge group with (anti-)symmetric hypermultiplet +2 +3 +SO(n) gauge group with spinor hypermultiplet +6 +3.1 +SO(2N + 1) with one spinor +9 +3.2 +SO(2N) with one (conjugate) spinor +11 +3.3 +Isomorphisms of representations +12 +4 +SO(n) N = 1∗ theory +14 +5 +Sp(N) gauge group with antisymmetric hypermultiplet +15 +A Notations and definitions +18 +B Integral expressions of instanton partition functions +20 +B.1 SO(2N + 1) with spinor hypermultiplet +20 +B.2 SO(2N) with (conjugate) spinor hypermultiplet +21 +B.3 Sp(N) with antisymmetric hypermultiplet +23 +C Multiplicity coefficients +24 +D Gopakumar-Vafa invariants for D-type singularities +26 +D.1 SO(4) +27 +D.2 SO(6) +28 +D.3 SO(8) +29 +1 +Introduction +One of the ultimate goals of quantum field theory is to obtain exact results, including +non-perturbative effects. In recent decades, significant progress has been made in +understanding the non-perturbative nature of quantum field theory. Starting with +the seminal work by Seiberg and Witten [1, 2], deep insights have been gained into +supersymmetric theories with eight supercharges. In particular, Nekrasov performed +the first instance of supersymmetric localization for instanton partition functions [3], +providing exact results for certain observables in the Seiberg-Witten theory, such as +the low-energy effective prepotential. +– 1 – + +More remarkably, the fixed points of equivariant actions on the instanton mod- +uli spaces are classified by a set of Young diagrams, and the partition function is +expressed as a sum over Young diagrams. Nekrasov’s exact result has far-reaching +consequences in both physics and mathematics. For example, the expression as a +sum over Young diagrams is important because it provides a connection to the topo- +logical vertex [4–6]. It is also particularly useful in the context of the AGT relation +[7], where the instanton partition function is identified with conformal blocks of a 2d +CFT. +These developments were exclusively made in the gauge groups of A type until +recent years. However, closed-form expressions of the unrefined instanton partition +functions for pure Yang-Mills theory with gauge groups of type BCD were recently +obtained as sums over Young diagrams [8]. +Additionally, the connection to the +topological vertex [9] of an O5-plane was uncovered. +In this paper, we push forward the research direction in [8], aiming to obtain un- +refined instanton partition functions with a hypermultiplet beyond the fundamental +representation as sums over 2d and 4d Young diagrams. We combine two approaches +to studying 5d supersymmetric gauge theories with matters beyond the fundamen- +tal representation. The first approach is the ADHM description, which has been +studied in previous works such as [10–14]. The second approach is the fivebrane +construction, which has been extensively studied in [15–26]. Our main results are +as follows: first, we demonstrate that the poles of the JK residue integral arising +from the ADHM descriptions can be classified by 2d and 4d Young diagrams in the +unrefined limit. Second, we use the instanton partition functions to verify dualities +predicted by fivebrane web diagrams and representation theory. +The paper is structured as follows. In §2, we investigate the SU(N) gauge the- +ory with (anti-)symmetric hypermultiplet, and we represent the unrefined instanton +partition functions as sums over 2d Young diagrams. In §3, we study the SO(N) +gauge theory with (conjugate) spinor hypermultiplet and similarly express the un- +refined instanton partition functions as sums over 2d Young diagrams. In §4, we +consider the N = 1∗ SO(N) gauge theory, and in §5, we investigate the Sp(N) gauge +theory with antisymmetric hypermultiplet. In these latter cases, the unrefined in- +stanton partition functions are represented as sums over 4d Young diagrams. In all +of these cases, we verify the identities of the instanton partition functions predicted +by fivebrane web dualities and representation theory. +2 +SU(N) gauge group with (anti-)symmetric hypermultiplet +The instanton partition function can be obtained from the equivariant Chern charac- +ters of the universal bundle E over the instanton moduli space [3]. Let p be a generic +element of the equivariant torus action on the universal bundle E. The equivari- +ant Chern characters for (rank-two) symmetric and antisymmetric representations +– 2 – + +of SU(N) can be obtained by +Chsym +p +(E) = Chp +� +Sym2E +� += 1 +2 +� +(Chp(E))2 + Chp2(E) +� +, +Chant +p +(E) = Chp +� +∧2E +� += 1 +2 +� +(Chp(E))2 − Chp2(E) +� +. +The equivariant index of the Dirac operator on the universal bundle takes the fol- +lowing form +Indq = +� +α +ϵαewa = +� +C2 Chq(E)Tdq +� +C2� +t . +Then, the contributions to the integrands of the 5d instanton partition function can +be read off by mapping +� +α +ϵαewα �→ +� +α +shϵϵα(wα) . +We refer the reader to [3, 11, 27] for the details. Note that we use the notation that +sh(x) := e +x +2 − e− x +2 , +ch(x) := e +x +2 + e− x +2 . +(2.1) +In this way, we can write down the contour integral expressions of the instanton +partition functions +Zrep +SU(N),k,κ = +1 +k!2k +� +JK +k +� +I=1 +dφI +2πi · eκ�k +I=1 φIIvec +SU(N),kIrep +SU(N),k +(2.2) +where the integrands are given by +Ivec +SU(N),k = +� +I̸=J sh(φI − φJ) · � +I,J sh(2ϵ+ − φI + φJ) +� +I,J sh(ϵ1,2 + φI − φJ) �k +I=1 +�N +s=1 sh(ϵ+ ± (φI − as)) +Ifund +SU(N),k(m) = +k +� +I=1 +sh(φI + m) +Isym +SU(N),k(m) = +k +� +I=1 +sh (2φI + m ± ϵ−) +N +� +s=1 +sh (φI + as + m) +k +� +I 0. +For n < 9, the full +partition function Zfull contains not only instanton contributions, but also spurious +contributions. +– 7 – + +n = 7, 8 +n > 8 +n < 7 +Figure 3. D1’-branes (blue) create hypermultiplet particles in the spinor representation. +As analyzed in [20], to extract genuine instanton contributions, we must remove +spurious contributions. Below, we focus only on the spinor hypermultiplet, but the +story is parallel for the conjugate spinor (just insert appropriate signs as in (3.2)). +For n ≤ 6, the partition function at k = 0 is given by +∞ +� +j=0 +e−jmZSO(n) +0,j,+ + ZSO(n) +0,j,− +2 += Zpert ≡ PE +� +�e−mZSO(n) +0,1,+ + ZSO(n) +0,1,− +2 +� +� +(3.3) +where PE is the plethystic exponent (A.1). Consequently, the full partition function +factorizes as +Zspinor +full,SO(n) = �Zspinor +SO(n)(q, ϵ1,2, m)Zpert(ϵ1,2, m) +(3.4) +where we extract the genuine instanton contribution +�Zspinor +SO(n)(q, ϵ1,2, m) = +∞ +� +k=0 +qk �Zspinor +SO(n),k(ϵ1,2, m) , +�Zspinor +SO(n),k(ϵ1,2, m) = +∞ +� +j=0 +e−jm �Zspinor +SO(n),k,j . +(3.5) +For the cases of n ≤ 6, the contributions from hypermultiplet particles more than +instanton numbers vanish as �Zk,j = 0 for j > k for both the spinor and the conjugate +spinor. +For n = 7, 8, the partition function at k = 0 is given by +∞ +� +j=0 +e−jmZSO(n) +0,j,+ + ZSO(n) +0,j,− +2 +=ZpertZextra +(3.6) +=PE +� +�e−mZSO(n) +0,1,+ + ZSO(n) +0,1,− +2 +� +� PE +� +−1 +2e−2mch 2ϵ+ +sh ϵ1,2 +� +, +where Zextra is the contribution from D1’-branes running away along the two parallel +fivebranes. (See the middle of Figure 3.) Thus, the full partition function factorizes +as +Zspinor +full,SO(n) = �Zspinor +SO(n),k(q, ϵ1,2, m)Zpert(ϵ1,2, m)Zextra (ϵ1,2, m) , +(3.7) +where �Zspinor +SO(n),k(q, ϵ1,2, m) takes the form (3.5). In the cases of n = 7, 8, for both +the spinor and the conjugate spinor, the instanton partition functions are subject to +�Zk,2k−n = �Zk,n so that �Zk,j = 0 for j > 2k. +– 8 – + +For n ≥ 9, we cannot express the contribution purely from the hypermultiplet in +the form of a plethystic exponent, unlike in previous examples (3.3) and (3.6). This +is because the leg fivebranes meet at a certain point in this case. Such a fivebrane +configuration has been studied in [24]. However, we believe that the full partition +functions (3.1) and (3.2) can still be applied to these fivebrane web configurations +even for n ≥ 9, although we leave a more detailed investigation of this issue for future +work. +As we learned in [8], the instanton partition function of pure Yang-Mills theory +for gauge groups of BCD type can be written as sums over 2d Young diagrams at +the unrefined level. Since the SO(n) theory with a spinor matter can be roughly +regarded as an “SO(n)-Sp(0) quiver” gauge theory, we expect that its instanton +partition function can also be expressed as a sum over 2d Young diagrams in the +unrefined limit. In the following, we will demonstrate that this is indeed the case. +Moreover, we can generalize the method in this section to the SO-Sp linear quiver +gauge theories. +3.1 +SO(2N + 1) with one spinor +The integral expressions for the partition functions are given in Appendix B.1. For a +gauge group of type B, the contributions from O(2ℓ)− and O(2ℓ+1)+ sectors vanish +due to fermionic zero modes. As a result, we have a single contribution, ZSO(2N+1) +k,j +, +for each hypermultiplet particle number j. This is another way to see that there is +no distinction between the spinor representation and its conjugate for SO(n) with +odd n. +As in [8] and we also discussed below (2.12), there are four effective Coulomb +branch parameters so that even the “Sp(0)” gauge theory contributes to the unrefined +partition function. Hence, for ZSO(2N+1) +k,2ℓ(+1) +, the non-trivial JK poles in the unrefined +limit are classified (N + 4)-tuples ⃗λ of 2d Young diagrams where the numbers of +boxes are subject to +N +� +s=1 +|λ(s)| = k , +N+4 +� +s=N+1 +|λ(s)| = ℓ . +(3.8) +The JK residue sums are expressed as +ZSO(2N+1) +k,2ℓ +(AN+1 = q +1 +2 , AN+2 = −q +1 +2 , AN+3 = 1, AN+4 = −1; q) = (−1)k+ℓ +� +⃗λ +CSp +⃗λ, ⃗A +N +� +s=1 +� +x∈λ(s) +sh4 2φs(x) +sh2(φs(x)) +N� +t=1 +sh2(Ns,t(x)) sh2(φs(x) + at) +N +� +s≤t +� +x∈λ(s) +y∈λ(t) +xJ +sh(2ϵ+ ± φI ± φJ) sh(±φI ± φJ) +sh(ϵ1,2 ± φI ± φJ) +1 +2ℓ ℓ! +shℓ(2ϵ+) +shℓ(ϵ1,2) +ℓ +� +i=1 +sh(±χi) �N +s=1 sh(±χi + as) +sh(ϵ1,2 ± 2χi) +� +i>j +sh(2ϵ+ ± χi ± χj) sh(±χi ± χj) +sh(ϵ1,2 ± χi ± χj) +k +� +I=1 +ℓ +� +i=1 +sh(ϵ− ± φI ± χi) +sh(−ϵ+ ± φI ± χi) . +(B.1) +Here φI (I = 1, · · · , k) are Sp(k) gauge fugacities, and χi (i = 1, · · · , ℓ) are O(2ℓ) +gauge fugacities so that they are integrated by the JK prescription. On the other +– 20 – + +hand, the integrand for k instantons and (2ℓ + 1) hypermultiplet particles is given +by +ISO(2N+1) +k,2ℓ+1 += +1 +2k k! +shk(2ϵ+) +shk(ϵ1,2) +k +� +I=1 +sh(2ϵ+ ± 2φI) sh(±2φI) +sh(ϵ+ ± φI) �N +s=1 sh(ϵ+ ± φI ± as) +� +I>J +sh(2ϵ+ ± φI ± φJ) sh(±φI ± φJ) +sh(ϵ1,2 ± φI ± φJ) +1 +2ℓ ℓ! +shℓ(2ϵ+) +shℓ+1(ϵ1,2) +ℓ +� +i=1 +ch(2ϵ+ ± χi) sh(±2χi) �N +s=1 sh(±χi + as) +ch(ϵ1,2 ± χi) sh(ϵ1,2 ± 2χi) +· +N +� +s=1 +ch(as) +� +i>j +sh(2ϵ+ ± χi ± χj) sh(±χi ± χj) +sh(ϵ1,2 ± χi ± χj) +k +� +I=1 +ch(ϵ− ± φI) +ch(−ϵ+ ± φI) +ℓ +� +i=1 +sh(ϵ− ± φI ± χi) +sh(−ϵ+ ± φI ± χi) . +(B.2) +B.2 +SO(2N) with (conjugate) spinor hypermultiplet +The brane system for SO(2N) gauge theory with (conjugate) spinor hypermultiplet +is shown in Figure 2. This brane system allows us to derive the instanton quantum +mechanics from open strings. In Figure 7, we see the N = 4 supersymmetric quantum +mechanics on k D1-branes and j D1’-branes. +In this case, there are non-trivial +contributions from the two connected components of the gauge group O(j). +SO(2N) +Sp(k) +O(j) +sym +antisym +Figure 7. The solid line represents a hypermultiplet while the dashed line represents a +Fermi multiplet. +To account for each contribution, we write down an integrand of the partition +– 21 – + +function for each one: +ISO(2N) +k,2ℓ,+ += +1 +2k k! +shk(2ϵ+) +shk(ϵ1,2) +k +� +I=1 +sh(2ϵ+ ± 2φI) sh(±2φI) +N� +s=1 +sh(ϵ+ ± φI ± as) +� +I>J +sh(2ϵ+ ± φI ± φJ) sh(±φI ± φJ) +sh(ϵ1,2 ± φI ± φJ) +2sign(ℓ) +2ℓ ℓ! +shℓ(2ϵ+) +shℓ(ϵ1,2) +ℓ +� +i=1 +N� +s=1 +sh(±χi + as) +sh(ϵ1,2 ± 2χi) +� +i>j +sh(2ϵ+ ± χi ± χj) sh(±χi ± χj) +sh(ϵ1,2 ± χi ± χj) +k +� +I=1 +ℓ +� +i=1 +sh(ϵ− ± φI ± χi) +sh(−ϵ+ ± φI ± χi) +(B.3) +ISO(2N) +k,2ℓ,− += +1 +2k k! +shk(2ϵ+) +shk(ϵ1,2) +k +� +I=1 +sh(2ϵ+ ± 2φI) sh(±2φI) +N� +s=1 +sh(ϵ+ ± φI ± as) +� +I>J +sh(2ϵ+ ± φI ± φJ) sh(±φI ± φJ) +sh(ϵ1,2 ± φI ± φJ) +� +N� +s=1 +sh(2as) +2ℓ−1 (ℓ − 1)! +ch(2ϵ+) +sh(2ϵ1,2) sh(ϵ1,2) +ℓ−1 +� +i=1 +sh(2ϵ+) sh(4ϵ+ ± 2χi) sh(±2χi) +N� +s=1 +sh(±χi + as) +sh(ϵ1,2) sh(2ϵ1,2 ± 2χi) sh(ϵ1,2 ± 2χi) +� +i>j +sh(2ϵ+ ± χi ± χj) sh(±χi ± χj) +sh(ϵ1,2 ± χi ± χj) +k +� +I=1 +sh(2ϵ− ± 2φI) +sh(−2ϵ+ ± 2φI) +ℓ−1 +� +i=1 +sh(ϵ− ± φI ± χi) +sh(−ϵ+ ± φI ± χi) +�sign(ℓ) +(B.4) +ISO(2N) +k,2ℓ+1,+ = +1 +2k k! +shk(2ϵ+) +shk(ϵ1,2) +k +� +I=1 +sh(2ϵ+ ± 2φI) sh(±2φI) +N� +s=1 +sh(ϵ+ ± φI ± as) +� +I>J +sh(2ϵ+ ± φI ± φJ) sh(±φI ± φJ) +sh(ϵ1,2 ± φI ± φJ) +N� +s=1 +sh(as) +2ℓ ℓ! +shℓ(2ϵ+) +shℓ+1(ϵ1,2) +ℓ +� +i=1 +sh(2ϵ+ ± χi) sh(±χi) +N� +s=1 +sh(±χi + as) +sh(ϵ1,2 ± χi) sh(ϵ1,2 ± 2χi) +� +i>j +sh(2ϵ+ ± χi ± χj) sh(±χi ± χj) +sh(ϵ1,2 ± χi ± χj) +k +� +I=1 +sh(ϵ− ± φI) +sh(−ϵ+ ± φI) +ℓ +� +i=1 +sh(ϵ− ± φI ± χi) +sh(−ϵ+ ± φI ± χi) +(B.5) +ISO(2N) +k,2ℓ+1,− = +1 +2k k! +shk(2ϵ+) +shk(ϵ1,2) +k +� +I=1 +sh(2ϵ+ ± 2φI) sh(±2φI) +N� +s=1 +sh(ϵ+ ± φI ± as) +� +I>J +sh(2ϵ+ ± φI ± φJ) sh(±φI ± φJ) +sh(ϵ1,2 ± φI ± φJ) +N� +s=1 +ch(as) +2ℓ ℓ! +shℓ(2ϵ+) +shℓ+1(ϵ1,2) +ℓ +� +i=1 +ch(2ϵ+ ± χi) ch(±χi) +N� +s=1 +sh(±χi + as) +ch(ϵ1,2 ± χi) sh(ϵ1,2 ± 2χi) +� +i>j +sh(2ϵ+ ± χi ± χj) sh(±χi ± χj) +sh(ϵ1,2 ± χi ± χj) +k +� +I=1 +ch(ϵ− ± φI) +ch(−ϵ+ ± φI) +ℓ +� +i=1 +sh(ϵ− ± φI ± χi) +sh(−ϵ+ ± φI ± χi) +(B.6) +– 22 – + +B.3 +Sp(N) with antisymmetric hypermultiplet +Let us recall the supersymmetric quantum mechanics of the k instanton moduli space +of pure Sp(N) gauge theory [10]. This system is described by an O(k) gauge theory +with a (rank-two) symmetric hypermultiplet and 2N fundamental half-hypermultiplets, +which have Sp(N) flavor symmetry. +The introduction of a 5d antisymmetric hypermultiplet modifies the field con- +tent of the instanton quantum mechanics. Specifically, it introduces an additional +symmetric hypermultiplet and 2N fundamental half-Fermi multiplets. (This is illus- +trated in Figure 8.) The equivariant index for the 5d antisymmetric hypermultiplet +has previously been calculated in [11, (5.14)]. For more details on the field content +of the instanton quantum mechanics, see [12, Appendix D]. +O(k) +sym +Sp(N) +Figure 8. The solid line represents a hypermultiplet while the dashed line represents a +Fermi multiplet. +The partition function receives two contributions due to the O(k) gauge group. +These contributions can be calculated by performing the JK residue integrals of the +following integrands: +I+ +k=2ℓ+χ += +� +1 +sh (ϵ1,2) �N +s=1 sh (±as + ϵ+) +· +ℓ +� +i=1 +sh (±φI) sh (±φI + 2ϵ+) +sh (±φI + ϵ1,2) +�χ +(B.7) +· +ℓ +� +i=1 +sh (2ϵ+) +sh (ϵ1,2) sh(±2φI + ϵ1,2) �N +s=1 sh (±φI ± as + ϵ+) +ℓ +� +I steam power +human manual labor -> use of steam-powered machines +Impact: Steam-powered machines significantly improve +production efficiency and release humans from heavy manual +labor. +2nd Industrial Revolution +− Electrification +Trigger: Electric power; assembling lines +Paradigm shift: Steam power -> electric power; +craft production, job production -> mass production +Impact: Electric power makes energy delivery highly +flexible and scalable and leads to the productive mass +production. +3rd Industrial Revolution +− Informatization +Trigger: Computer and electronics +Paradigm shift: information media -> digitized; +operation processes -> automated or computer-aided processes; +matter & energy centered -> information centered. +Impact: Humans started to use machine intelligence (MI) to explore +and leverage information power, which leads to new capabilities +and significantly improved efficiency and effectiveness. +4th Industrial Revolution +− Digitalization +Trigger: Digital technologies (IoT, 5/6G, AI, Cloud, blockchain, …) +Paradigm shift: digitizing information -> digitalizing everything of interest; +centralized and distributed computing & MI -> pervasive computing & MI; +information centered -> big data and machine intelligence centered. +Impact: The pervasive digitalization and ubiquitous machine intelligence +(integrated capabilities from IoT, 5G/6G, cloud, and AI) lead to many cyber- +physical-social smart systems (CPS3) with innovative applications. +Energy +(Physical Power) +Machine Intelligence +(Information Power) + +INDUSTRY 4.0 +NDUSTRY 3.0 +INDUSTRY 2.0 +INDUSTRY 1.0 +Mechanization,steam +Massproduction, +Automation,computers +Cyber Physical Systems, +power,weavingloom +assembly line, +and electronics +internet of things,networks +electrical energy +回4 +J Huang / Digital engineering transformation with trustworthy AI towards Industry 4.0: emerging paradigm shifts + +For this reason, the 3IR is also called the “information revolution,” and the period is called the +“information age.” Since the start of the 3IR, many digital technologies, such as artificial intelligence, +industrial robots, portable and then mobile computers, the Internet, wireless networks, mobile devices, and +others, were developed and profoundly changed human society regarding how we work and live already. +Those digital technologies prelude the next wave of technological revolution. +The 4IR can be characterized as “digitalization.” The convergence of transformative and disruptive +digital technologies, such as IoT, high-speed wireless communication, cloud computing, AI&ML, and +blockchain, among others, trigged the 4IR. +IoT, in a broader perspective, also named as “Internet of All Things” or “Internet of Everything,” has +four pillars: Internet of Things, Internet of Information, Internet of People, and Internet of Services +(Kirkpatrick et al., 2013). In the future internet infrastructure, the physical things of interest (such as various +machines, devices, buildings, roads, and other structures, even things of interest in the natural world), the +abstract things (things in the digital world, such as information contents), people, organizations, and various +processes of interest (examples include: service processes, or engineering processes, even social or natural +processes of interest, such as climate change, environmental change, living things status change, pandemic +monitoring, disease infection tracing, and others), anything of interest are interconnected, through future +internet, operating with supports from 5G/6G wireless communication networks, cloud computing, and AI. +In the rest of this paper, we refer term IoT in this broader sense of the Future Internet or Internet of +Everything. +IoT connects the physical world with the cyber world to form an integrated cyber-physical smart +environment. With IoT through sensors and actuators, smart things in the physical world can be remotely +located, monitored, and even controlled through the Internet. With mobile devices supported with 5G/6G +wireless networks, people will have ubiquitous access to the Internet, smart things in the physical world, +and the cloud services needed to handle those smart things. More importantly, people themselves become +nodes of this Internet. This interconnected world, facilitated by IoT, 5G/6G, cloud computing, and AI&ML, +is leading to numerous Cyber-Physical-Social Smart Systems (CPS3) (Huang, Seck, & Gheorghe, 2016), +such as autonomous vehicles, smart factories, smart stores, smart cities, and others. A necessary condition +to develop those smart systems is to digitalize things of interest to make them uniquely identifiable and +machine-understandable; then, we can leverage AI&ML to exploit the unprecedented richness of +information brought by the digital and connected world. +The 4IR with those disruptive digital technologies is “leading to unprecedented paradigm shifts in the +economy, business, society, and individually” (Schwab, 2017). The emerging paradigm shifts in the 4IR +can be viewed from several related perspectives as follows. +(1) Shift from digitizing information to digitalizing everything of interest, including artifacts (both +abstract and physical), natural things, organizations, and processes (such as production and/or +service processes, monitored natural processes, and others). +(2) Shift from centralized and distributed provisioning of computing services and machine intelligence +to pervasive provisioning of computing services and machine intelligence. +(3) Shift from information-centered processes (developed in 3IR) to big data and machine intelligence- +centered processes. + +About the first shift, different from the 3IR, where information was digitized, in the 4IR, not only +information (abstract artifacts) but also physical artifacts and natural living things of interest, as well as +processes of interest, could be digitalized. Everything of interest, even humans, could have their digital +counterparts (or “digital presence” as termed by (Schwab, 2017)). Such as, some devices or products will +have their digital twins, and some artifacts will have their digital augmentations (the associated metadata, +e.g., provenance). In addition to artifacts, some processes of interest (no matter about production, service, +or some natural processes such as climate change or regional ecosystem change, among others) could also +be digitalized. This pervasive digitalization could be enabled by semantic web technology (a branch of AI) +(Berners-Lee, Hall, Hendler, Shadbolt, & Weitzner, 2006; Berners-Lee, Hendler, & Lassila, 2001), RFID, +IoT, and blockchain. + + +J Huang / Digital Engineering Transformation with Trustworthy AI towards Industry 4.0: emerging paradigm shifts +5 + +The second shift can be seen from comparing how computing service was delivered before the 4IR. +From many people using a single large computer, each person using a computer and then a laptop, till each +computer was able to connect to the Internet with a wall jack, computing service was delivered in a +“centralized”, “decentralized”, and then “distributed” approach. In the 4IR, we can use computers and +interact with the world everywhere. +About thirty years ago, Mark Weiser and Nicholas Negroponte had a debate at MIT Media Lab. +Negroponte predicted that AI was going to be the leader for the next wave of computing, but Weiser argued +it would be Ubiquitous Computing, characterized as “invisible” and “connection” (Weiser, 1996). Weiser’s +vision, which can be found in a vivid illustration in his essay “Open House” (Weiser, 1996), is one of the +original ideas about the Internet of Things. Interestingly, IoT and AI together become the major drivers for +the 4IR. IoT extends the “nerves” of the Internet into the physical world, and AI is the “brain” of smart +things interacting with the digitalized connected world. In addition to IoT and AI, wireless networks and +cloud computing are critical contributors to the pervasive provisioning of computing and machine +intelligence. +First, of course, high-speed wireless communication is an essential infrastructure to facilitate the +mobility of communication, computing, sensing, and delivery of all types of digital services without or +significantly reducing the physical space constraints. Secondly, cloud computing enables ubiquitous on- +demand network access to a shared pool of computing resources, thus meeting scalable and elastic +computing needs (Armbrust et al., 2009; NIST, 2011). Cloud computing is essential for handling the big +data produced in the digital world and for the intensive computing required by AI applications, no matter +where we are. +This article uses term “ubiquitous machine intelligence” (uMI) to refer to the integrated capabilities +enabled by the convergence of IoT, 5G/6G, AI&ML, and cloud computing. Interestingly, the role +ubiquitous machine intelligence plays in meeting the needs of computing and machine intelligence is very +similar to the role of electricity in 2IR, where electricity enables the mobility of power delivery needed in +assembly lines and mass production. Here, ubiquitous machine intelligence makes computing services and +machine intelligence pervasively accessible and provides the essential technology for various smart systems. +For this reason, ubiquitous machine intelligence is regarded as the most fundamental change brought by the +4IR. +The third shift can be viewed through the following comparison. In the Industry 3.0, information flows +and material flows were still separated; thus, information is long-delayed. In the 4IR, the pervasive +digitalization and connectedness make the material and human flows associated with information flows; +thus, real-time information about the material and human flows can be used for decision-making. There +will be unprecedented rich information about the things of interest for decision making. Consequently, a +decisive success factor in the era of 4IR will be the capability of leveraging ubiquitous machine intelligence +to collect and extract information, discover knowledge from big data, and make smart decisions in the +digital smart and connected environment (Huang, 2017). +Finally, regarding the profound impact of the 4IR, similar to the 2IR that makes power (energy) easily +accessible and leads to mass production, the 4IR makes information power (computing and machine +intelligence) pervasively accessible and leads to various CPS3 with innovative applications. More +specifically, in the 4IR, the pervasive digitalization and pervasive connectedness make data (minerals of +information) available in an unprecedented scale and make computing services and machine intelligence +(brainpower or information power) ubiquitously accessible and scale up. Furthermore, they are triggering +and accommodating numerous innovative applications. Those disruptive technologies and their innovative +applications are “fundamentally changing the way we live, work, and relate to one another.” They lead to +“the transformation of entire systems, across (and within) countries, companies, industries and society as a +whole” (Schwab, 2017). + +6 +J Huang / Digital engineering transformation with trustworthy AI towards Industry 4.0: emerging paradigm shifts + +3. Digital engineering transformation +As discussed in the last section, digitalization is the central theme of the 4IR. As such, pervasive digital +transformations are ongoing in almost every aspect of human society. Given the essential role of +engineering in industries, the digitalization of engineering is at the core of 4IR. Digital engineering is the +digitalized engineering, or the targeted digital form of engineering to be realized by the digitalization of +engineering towards Industry 4.0. +The US Department of Defense (DoD) defines “digital engineering” as “an integrated digital approach +that uses the authoritative source of system data and models as a continuum across disciplines to support +lifecycle activities from concept through disposal” in their “Digital Engineering Strategy” (US DoD, 2018; +Zimmerman et al., 2019). As illustrated in Figure 2, the strategy targets five goals: (1) formalize the +development, integration, and use of models, leading to a continuous end-to-end digital representation of +the system of interest; (2) provide an enduring authoritative source of truth to share and exchange digital +models, data, and other digital artifacts across boundaries of organizations and the engineering lifecycle; +(3) incorporate technological innovation to improve the engineering practice; (4) establish a supporting +infrastructure and environments; (5) transform culture and workforce to adopt and support digital +engineering. Paper (Zimmerman et al., 2019) provides a comprehensive view of the US DoD’s efforts for +digital engineering transformation. The implementation of this strategy will significantly change how +engineering practice is conducted in the US DoD enterprise and propagate to industries in a broader range. + + +Figure 2. The US DoD Digital Engineering Strategy. Picture from (US DoD, 2018) (Fig.3&4) + +From the perspective of engineering processes, the transformation from conventional engineering to +digital engineering is illustrated in Figure 3. In the picture, from a general view, an engineering process is +illustrated as a process model with inputs (left), outputs (right), enablers (bottom), and control (top). +Digitalizing engineering artifacts, engineering processes, and engineering enterprises is the foundation for +digital engineering transformation. Without digitalized artifacts, the applications of emerging digital +technologies will be very limited, maybe in an ad hoc manner, or may need human or ad hoc devices as +intermediate agents. As a result, the level of automation and autonomous functioning will stay at a slightly +higher level than in Industry 3.0. On the other hand, with digitalized engineering artifacts and processes, +we can broadly leverage machine intelligence and other digital technologies to conduct engineering in a +digital, smart, and connected environment; thus achieving a new level of engineering, which is more +effective and efficient as well as more trustworthy. Some advantages of digital engineering include: +sharable knowledge and data across the engineering life cycle, increased explainability and transparency of +engineering processes and products, fast integration of distributed engineering resources for a given mission, +increased traceability and accountability, increased product traceability along supply chains, and quick +adaption to the changing environment, among others. + +? +P +and +rov +to +Use +Truth +DIGITAL +ENGINEERING +STRATEGY +pue +4SpecialtyEngineeringModels +Product +Management +Support +Models +Models +Authoritative +Source of +System +Design +Truth +Models +Models +Verificationand +Manufacturing +ValidationModels +Models +Key:Data +Figure 4: Examples of Models Connected +via the Authoritative Source of Truth +J Huang / Digital Engineering Transformation with Trustworthy AI towards Industry 4.0: emerging paradigm shifts +7 + + +Figure 3. Digital engineering transformation from a process perspective + + + +Figure 4. Model perspective: operations in the digital engineering lifecycle (Huang et al., 2020) + +In digital engineering as shown in Figure 3, the inputs, outputs, enablers, and control are different from +the conventional ones; correspondingly, the digital engineering operations will also significantly differ from +the conventional ones. In the digital engineering transformation, along with the fast-paced digital +technologies, “computational thinking” (Wing, 2006, 2012) is critical to the new engineering paradigms. +Knowledge and skills in applied computing and relevant digital technologies have become necessary +components in workforce development towards digital engineering. Figure 4 briefly describes operations +in the digital engineering lifecycle from a model perspective. Some further discussion of digital engineering +can be found in (Huang et al., 2020). +Finally, AI is a critical enabling technology to advance digital engineering (Huang, Beling, Freeman, & +Zeng, 2021). In addition to the broad applications of Machine Learning for modelling and prediction in +engineering, another AI branch, knowledge representation and reasoning, particularly semantics technology, +is a fundamental technology to support digitalization. For example, ontologies are used in MBSE (Madni +& Sievers, 2018). AI also plays a critical role in the logical formalization in building and operating digital +trust & security systems (Huang, 2018). We will discuss further in subsection 5.3. +Engineering +Process +Products (energy, materials, parts, +end products, structures, data, …) +Services +Natural resources +Products & services from +other engineering processes +Engineering standards and regulations +Domain-specific enabling technologies +Enterprise with workforce of specialty +Digital Engineering +Process +Digitalized Products +(product + digital augmentation) +Digitalized Services +Natural resources +Digitalized Products & Services +from digital environment +Domain-specific enabling technologies ++ Digital technologies +Digital enterprise/workforce of (specialty ++ Digital technologies skills) +Digital Transformation +Digitalizing engineering artifacts, processes and enterprises +Digitalized Engineering standards and regulations +Digitalizing engineering artifacts: +• Parts; Materials; Products; +• Devices; +• Models; Datasets; +• Software; +• Documents; +• … +Digitalizing engineering processes +• Actions; Functions; +• Manufacturing processes; +• Quality management processes; +• Environment management processes; +• … +Digitalizing enterprise +• Enterprise organizations; +• Job roles; Services; +• Policies; +• Supply-chain; +• Customer-relations; +• … +Concept +Stage +Development +Stage +Production +Stage +Utilization/support +Stage +Retirement +Stage +Model +creation. +Inputs: +- Digital artifacts (DAs) +in operating environment; +- Relevant data and +models from downstream +stages. +Inputs: +- Digital artifacts from +both upstream and +downstream; +- Digital artifacts of +external systems +Inputs: +- Digital artifacts from +both upstream and +downstream; +- and from production +environment. +Inputs: +- Digital artifacts from +both upstream and +downstream; +- and from operating +environment. +Inputs: +- Digital artifacts +from upstream; +- and from natural +environment. +Model learning +Apply AI&ML for model building from big data coming from upstream and downstream engineering stages and environment. +Model integration +Interaction between +digital models for both +SysCon and systems in +operating environment. +Interaction between +digital models for both +system components and +external systems. +Interaction between +digital models for both +the system and +systems in production +environment. +Interaction between digital +models for both the system +and external systems in +operating environment. +Interaction between +digital models for the +sys component and +DAs in natural +environment. +Model curation +Create model of models; maintain metadata for models; maintain model provenance; model update and propagation to downstream. +Model +sharing & use +Across lifecycle activities,; across the boundaries of disciplines and organizations +Model +Trustworthiness +Centralized standardization; decentralized standardization and mappings; distributed evolutionary fine-grained convergence; +model trustworthiness; model repeatability; Access Control; digital artifact intellectual property protection, … + +8 +J Huang / Digital engineering transformation with trustworthy AI towards Industry 4.0: emerging paradigm shifts + +4. Harness uncertainties in the 4IR – A key: trustworthy AI systems +The disruptive digital technologies in the 4IR not only offer transformative opportunities but also bring +in uncertainties and have triggered broad concerns and research on systems trustworthiness. The focused +concerns of trustworthiness have been shifting and expanding to cover new issues that pop up in the new +technologies used in each generation of engineering systems. This section discusses the trustworthiness +concerns of Industry 4.0 systems, starting from examining the evolution of the trustworthiness of +engineering systems, then focusing on trustworthy AI systems – the pivotal component of Industry 4.0 +systems. +Before we move to these topics, let us discuss the elementary concepts of trust first. We use trust +explicitly or implicitly, even unaware of using it, in our daily life and work. However, what is trust exactly? +There are many definitions of trust (Walterbusch, Gräuler, & Teuteberg, 2014). Our working definition of +trust used in this article is as follows (Huang & Fox, 2006) - Trust is a mental state comprising (1) +expectancy: the trustor expects a specific behaviour of the trustee, such as providing true information or +effectively performing cooperative actions; (2) belief: the trustor believes that the expected behaviour will +occur, based on the evidence of the trustee’s competence, good intention, and integrity; (3) willingness to +take the risk: the trustor is willing to take the risk for (or be vulnerable to) that belief. In brief, + +Trust = +Expectation + +Belief in that expectation + +Willingness to take the risk for that belief. + +Trust can be regarded as a relation between a “trustor” (trusting party) and a “trustee” (trusted party) +with respect to the properties of something provided/offered/conducted by the trustee. For example, a +passenger (trustor) trusts an autonomous car (trustee) regarding the safety of driving on the streets. What is +to be trusted is out of the control of the trustor. From the studies of trust, the factors leading to trust include +the trustee’s ability, good intention (or benevolence, as some researchers called it) towards the trustor, and +integrity (reflected by the principles and values guiding behaviours)(Mayer, Davis, & Schoorman, 1995). +The roles of ability and good intention in the concept of trust are apparent. Integrity is the need for trust +judgment with respect to the predictable behaviours of the trustee in situations that are out of watching by +the trustor. +According to Simon (Simon, 1997), a decision-making process in the real world is limited by “bounded +rationality,” i.e., the “rational choice that takes into account the cognitive limitations of the decision maker +- limitations of both knowledge and computational capacity.” In the real world, because we only have +limited information/knowledge, limited computing capacity, and limited time available for decision-making, +a decision has to be made partly based on bounded rational calculation and partly based on trust. As +Luhmann addressed (Luhmann, 1979), trust, as “a basic fact of social life,” functions as a “reduction of +complexity.” Without trust, social interactions, including the applications of technologies in engineering +and business, will be extremely difficult. Because of trust, we do not need to verify everything before a +decision making or action. Trust mechanisms (Huang & Nicol, 2013) play a critical role in harnessing the +disruptive digital technologies toward trustworthy utilizations of them. +4.1. +Trustworthiness of systems +Trustworthiness of a system is defined in (Schneider, 1999) as “assurance that a system deserves to be +trusted—that it will perform as expected despite environmental disruptions, human and operator error, +hostile attacks, and design and implementation errors.” In short (Avizienis, Laprie, Randell, & Landwehr, +2004), systems trustworthiness is the “assurance that a system will perform as expected.” Trustworthiness +is a relative concept and is dependent on what is expected. Trustworthiness is often characterized as a +collection of the expected properties of a system to be trusted, as illustrated in Figure 5. The “assurance” +can be achieved by various trust mechanisms implemented in engineering systems and in societal systems +such as audits, certifications, regulations, and law enforcement. + + +J Huang / Digital Engineering Transformation with Trustworthy AI towards Industry 4.0: emerging paradigm shifts +9 + + + + +Figure 5. The evolving trustworthiness properties as the evolution of systems with technologies +and the sources of concerns + +In the following, we discuss the expected properties of trustworthiness of different generations of +engineering systems and the sources of the concerns or threats to trustworthiness. Traditional systems are +typically stand-alone systems (a physical system, a computer system, or a system embedded with computer +systems) or just connected logically and internally within an organization rather than the Internet. As shown +in the bottom part of Figure 5, the trustworthiness properties concerned with those traditional systems are +mainly reliability, maintainability, availability, integrity, robustness, safety, and usability. Those properties +together are referred to as dependability. The sources of concerns or the threats to the trustworthiness +properties are mainly non-malicious faults caused by (1) errors introduced in systems design and +development; (2) errors introduced in manufacturing and maintenance; (3) errors happened in system +operations; (4) environmental disruptions, including natural disasters, environmental pollution, and gradual +but irreversible environment change associated with climate change. +In the middle part of Figure 5, enabled by Internet and wireless mobile communication, the networked +industrial systems evolve to cyber-physical systems (CPS) (Lee, 2008; NSF, 2008; Rajkumar, Lee, Sha, & +Stankovic, 2010), which use networked computing components to monitor, coordinate, control, and +integrate physical systems and processes. The connectedness of systems naturally led to great challenges +with respect to security and privacy, and correspondingly they become the central theme of systems +trustworthiness (Avizienis et al., 2004; Schneider, 1999). The threats to security and privacy are obviously +from various malicious attacks and incompetent security defence. +As illustrated in the upper part of Figure 5, trustworthiness for the systems in the 4IR involves concerns +with trustworthy AI, pervasive security, and technical dependability. The issues in an AI system could +trigger security issues and systems dependability issues; of course, security issues could also trigger systems +dependability issues. The sources causing the systems trustworthiness issues are in three folds: +Robustness +Reliability +Safety +Confidentiality +Usability +Resilience +Privacy +Ethicality +Fairness +Transparency +Maintainability +Accuracy +Explainability +Traceability +Reproducibility +Accountability +Malicious attacks +Cyber connectedness +Non-malicious faults +Trustworthiness(3) +Trustworthiness(2) +Trustworthiness(1) +AI Systems uncertainties +IoT Pervasive connectedness +Systems complexity & emergence +Dependability +Trustworthy AI +Security +Pervasive +Security +Cooperability +M.C. Escher, Convex and Concave (1955) + +10 +J Huang / Digital engineering transformation with trustworthy AI towards Industry 4.0: emerging paradigm shifts + +(1) The pervasive security issues caused by the pervasive connectedness facilitated by IoT: the +pervasive connectedness facilitated by IoT broadly extends the security and privacy concerns to +every corner of the world where humans work and live. Without assured technologies and societal +mechanisms, the Internet of Things can turn into “the Internet of Risky Things” (S. Smith, 2017), +even the Internet of dangerous things. The threats to security and privacy in 4IR include malicious +attacks and incompetent security defence as before but with a much larger attack/exposure surface. +Additionally, the threats also include certain relevant societal mechanisms. For example, as Smith +pointed out (S. Smith, 2017), some smart devices will last longer than their manufacturers and +suppliers, and who will keep updating the systems or their embedded computing components? +Without updates, a digital system will be vulnerable in the complex digital operating environment. +(2) The uncertainties caused by AI systems: the fast growth and broad applications of AI systems or +embedded AI components in various systems have triggered much concern in two aspects: technical +competence (such as reliability and security) and ethical concern (such as transparency, bias towards +groups of people, human-machine relations, and others). The next subsection will discuss this aspect. +(3) systems complexity & emergence - the aforementioned two primary two factors together bring in +high uncertainties and complexities into the systems in the 4IR. Complex systems typically exhibit +strong emergence (unexpected emergent system behaviour and properties that were not anticipated +in design and development, e.g., the US-Canada Northeast Blackout of 2003) (Adcock, Jackson, +Fairley, Singer, & Hybertson, 2021). +4.2. +Trustworthy AI +In the last decade, AI achieved remarkable advances. Just to name a few, AlphaGo (Silver et al., 2016, +2018), using deep neural networks-based reinforcement learning, won the world number one player of Go +game, which is regarded as the most challenging board game for computers to win. AlphaGo’s achievement +marks a new milestone of machine intelligence towards superhuman intelligence on specific tasks. +Although it is still a pilot project and has no broader adoption yet, Waymo One, the commercial taxi service +operating with level 4 autonomous vehicles, has been offered on the streets in a city (Waymo One, 2020). +This service marks a new level of integrated intelligent capabilities of an AI system on a type of complex +tasks which previously only humans could do. Another remarkable advance is Generative Adversarial +Networks (GANs) (Creswell et al., 2018; Goodfellow et al., 2014, 2020). A GAN consists of a pair of deep +neural networks, a generator, and a discriminator; together, they can generate realistic images and other +contents as requested (see examples at https://thispersondoesnotexist.com/). GANs lead to many potential +applications but also open the door for deepfake. +For many decades, the AI community's focus has been on extending machine intelligence capabilities. +Although the advances of AI today are still domain-specific (or narrow AI) rather than general AI, the +remarkable achievements, the disruptive impacts, and the fast growth of applications of AI systems +triggered concerns and research on trustworthy AI systems. The concerns are mainly from two perspectives. +(1) Ethical AI, focusing on social effects, mainly ethical considerations, essentially, the profound effects +AI systems bring to humans, human groups, and human society, and advocating using AI to benefit humans +as a central principle. (2) Reliable AI, focusing on the technical performance, or dependability, concerning +the competency of AI systems on technical matters. There is a broad range of expected properties of +trustworthiness in AI systems. +Ethical AI is an emerging interdisciplinary research field, gathering together researchers from a wide +range of areas, including computer science, philosophy, sociology, anthropology, public policy, law, and +others. Some thought leaders and researchers conducted pioneering work in the field. The Association for +the Advancement of Artificial Intelligence (AAAI) organized a panel of leading AI researchers that +conducted “Asilomar Study of Long-Term AI Futures” in 2008. The study explored a wide range of topics +on societal impacts and guidance of AI research (AAAI Presidential Panel on Long-Term AI Future, 2009). +The study assessed concerns and perspectives about disruptive outcomes of superhuman intelligence, +explored ethical and legal issues associated with autonomous systems, and identified near-term AI research + + +J Huang / Digital Engineering Transformation with Trustworthy AI towards Industry 4.0: emerging paradigm shifts +11 + +challenges and opportunities, including enhancing people’s privacy, enhancing human-AI collaboration +and interaction, making ML and reasoning transparent to people, and preventing using AI for malevolent +purpose. Continuing the effort to guide AI research for good, the AI100 2016 report (Stone et al., 2016) +addressed that AI research is “shifting from simply building systems that are intelligent to building +intelligent systems that are human-aware and trustworthy.” The report collectively presents the view and +prospects of a panel of experts on the opportunities and challenges of AI and policy recommendations in +eight selected AI application domains targeting people's lives in a typical North American city in 2030. +They addressed challenges regarding how to build safe and reliable systems, gain public trust concerning +safety, security, and privacy, make AI systems behave ethically and overcome bias, and how AI systems +smoothly interact with humans. In the long term, ethical AI concerns the fundamental relations between AI +systems and humans. There are concerns about “technological singularity” or “intelligence explosion,” +partially reflected by the dystopia depicted in fiction. Experts in the AI field believe those radical outcomes +remain fictional and are not immediate threats; well, some thought leaders suggest “avoid strong +assumptions regarding upper limits on future AI capabilities” (Future of Life Institute, 2017). Given AI’s +profound impacts on human society, it is necessary to review AI systems' purposes, usages, and impacts +and create principles and regulations for guiding AI research and development to avoid harming humans +and human society. The 23 Asilomar principles (Future of Life Institute, 2017) and the EU HELG ethical +AI guide (EU AI HLEG, 2019) reflect a broad range of concerns about ethical issues of AI systems. EU AI +HELG defined trustworthy AI as three components: lawful AI, ethical AI, and robust AI. The guide +proposed four ethical principles: (1) Respect for human autonomy; (2) Prevention of harm; (3) Fairness; (4) +Explicability, covering transparency, auditability, traceability, and explainability. On the ground of these +four principles, the guide further proposed a list of seven requirements: (1) Human agency and oversight; +(2) Technical robustness and safety, including security, accuracy, reliability, and reproducibility; (3) +Privacy and data governance; (4) Transparency, including traceability, explainability, and communication; +(5) Diversity, non-discrimination and fairness; (6) Societal and environmental wellbeing; (7) +Accountability. The EU guide also discussed technical and non-technical methods to implement the +requirements and proposed a list of questions used to assess the trustworthiness of AI systems. +The ethics of AI studies what is right/wrong regarding the purpose and usages of AI, based on the +profound effects AI systems bring to humans and human society. The central principle of ethical AI is about +using AI to benefit humans and human society. In the upper part of Figure 5, ethicality is about the extent +to which an AI system complies with this principle. +As addressed by the AI100 2021 report (Littman et al., 2021), “AI systems and humans have +complementary strengths;” thus, “combined, they can accomplish more than either alone.” Technically, it +remains a challenge regarding how to team up humans and AI systems effectively. For the fundamental +ethical principles of trustworthy AI (EU AI HLEG, 2019), no doubt human-AI teaming is the right direction +to go, not only for maximizing capability and performance but also for ethical consideration. Cooperability +is about the extent to which an AI system facilitates and supports human-machine teaming for complex +problem-solving, including the channels or methods to enhance interactions and collaborations between +humans and machines. Cooperability covers controllability; the latter is about the ability of an AI system +that allows humans to monitor and control the system. +Fairness is about whether an AI system fairly treats people of different groups regarding race, gender, +age, cultural background, and others. Fairness received much attention in recent years when AI systems +started being used for some life-changing scenarios, such as hiring decisions, financial credit evaluation, +and judicial decisions (Mehrabi, et al., 2021). Fairness is a highly challenging topic for complex human +societal reasons. Technically, fairness can be treated as bias-free. The recent deep learning progress +reflected by GANs can be used to create real-like samples for balanced data in ML. +Accountability is the availability and integrity of the identity of an entity that performed an operation in +the AI system of concern. In simple, it is who (human operators or autonomous components in an AI system) +did what and when and the responsibility for that. In the scenarios of a security incident, an accident, an +error, or a system failure, accountability helps to identify the causes, make responsibility clear, and avoid +future mistakes. + +12 +J Huang / Digital engineering transformation with trustworthy AI towards Industry 4.0: emerging paradigm shifts + +Transparency is about the extent to which how the AI system operates is transparent to various +stakeholders, such as operators, business partners, auditors, and users. Transparency is a basis for ensuring +ethicality, fairness, and privacy and facilitating controllability and cooperability. +Explainability is the ability to explain the outcomes and processes of an AI system to humans. The major +challenge is that in ML, neural networks have low-level coding for feature representation which is +inherently hard to be explained in high-level knowledge. A very large number of parameters and complex +structures of deep neural networks make the interpretability further harder. +Traceability is about the ability to collect and document the provenance of the data used and produced, +the models used and trained, and the operating processes of an AI system. Traceability is the basis for +transparency and supports explainability. +Reproducibility is about whether a model instance can be rebuilt with the same AI algorithm and data +and whether an experiment or, more generally, a process running with an AI model can be +reproduced. Reproducibility is essential to science and engineering. +Reliable AI reflects people’s concern about systems’ technical performance when more and more AI +systems are used or embedded in a large engineering system in the 4IR. Naturally, this aspect of concern +brings the focused attention partially back to more classical trustworthiness properties of engineering +systems but with a focus on AI systems or AI components and their impact on the larger systems. This +article uses the term “dependability” to cover all expected properties (for trustworthiness) on the technical +performance aspect. The concept of dependability here is broader than what was defined in (Avizienis et +al., 2004). Reliability is the probability of a system functioning without failure for a given period of time. +Maintainability is about how easy to make a system maintain healthy, updatable, and upgradeable. +Resilience is about the ability of a system to restore it to a working state when it is damaged in situations +such as natural disaster events or cyber-attacks. Safety is about how safe to humans a system is. The broad +use of AI components in engineering systems makes safety a significant concern. Accuracy is the measure +of errors made by a model or system; in the context of ML, it is critical to have high accuracy for new data +beyond the data used for training the model. In Systems Engineering, robustness is defined as “the degree +to which a system or component can function correctly in the presence of invalid inputs or stressful +environmental conditions”(ISO/IEC/IEEE, 2010). In ML, robustness is about the stable outcomes in the +presence of perturbations in inputs and could be measured with sensitivity. In deep learning, lack of +robustness is a critical cause for the possible deepfake by GANs. The general meaning of robustness is +similar to resilience. In robustness, the perturbations are on a small scale. Usability is how easy a system is +to be used, and usability is beyond human-machine interaction. Poor usability can lead to failures in many +other properties in modern systems, including security and safety. +Security is defined as the well-known definition of the CIA triad: Confidentiality (Prevention of +unauthorized access to the protected resources or disclosure of the protected information), Integrity +(absence of unauthorized alterations), and Availability (Readiness for correct services for authorized users). +Some other interesting security properties can be defined on top of the CIA triad. For example, authenticity +is the integrity of information content and its provenance (origin). Obviously, in the digital and connected +environment, failures in security have broad impacts and can compromise other trustworthiness properties. +Privacy is another big concern in AI. Since data is the fuel for AI systems, privacy concerns about whether +the data gathering, holding, processing, usage, sharing, and governance respect people's privacy. +As illustrated by Figure 5, the trustworthiness of AI systems is the new collection of concerns when the +systems evolve into Industry 4.0 systems. Interestingly, on the other hand, the digitalization of engineering +artifacts, processes, and enterprises in the 4IR could support achieving the expected trustworthiness +properties of AI systems. +In digital engineering transformation, it is an excellent opportunity for the engineering design +community at large to bring the new capabilities of AI and the trustworthy AI principles together in various +engineering systems design for human society to leverage the power of AI and at the same time to avoid or +minimize the potential negative impacts. + + +J Huang / Digital Engineering Transformation with Trustworthy AI towards Industry 4.0: emerging paradigm shifts +13 + +5. Open issues and discussion +In previous sections, we have discussed the characteristics of 4IR and their fundamental impacts, digital +engineering transformation - the manifestation of 4IR in engineering, and trustworthy AI - the leading +technology in the digital transformation for I4.0. This section continues the discussion on some open issues +in the direction of 4IR. +5.1. +What is the pattern of paradigm shifts? +As discussed in section 2, digitalization in the 4IR is leading to paradigm shifts in almost every aspect +of our society. What are the emerging new engineering paradigms? Engineering uses scientific principles +to design, build, and operate engineering systems for solving real-world problems or meeting humans’ +needs in a specific field, thus essentially sharing the paradigms in scientific research. A general pattern of +scientific paradigm shifts can be illustrated as the following Figure 6. + + +Figure 6. A view of paradigm shifts, based on Kuhn’s structure of scientific revolutions (Kuhn, 1962). +According to Kuhn’s structure of scientific revolutions (Kuhn, 1962), in the period of “normal research” +or “normal science,” a scientific community uses a dominant paradigm to conduct research and produce the +main body of knowledge in that field. Then, the dominant paradigm may have crises for its deficiencies or +limitations facing the new observations and/or new problems coming from the changing environment. To +battle the crises and meet the new need(s), the disciple enters a period of “extraordinary research” (as named +by Kuhn). In this period, new concepts, models, tools, methods, among others, will be created. If the new +disciplinary components are incremental and can be integrated into the current paradigm, the paradigm will +be updated, thus being a scientific evolution. Otherwise, the new components developed in “extraordinary +research,” possibly together with the components from other disciplines, will contribute to the development +of a new paradigm in a period of “pre-paradigm” (again, named by Kohn). In the “pre-paradigm” period, +one or multiple paradigms will be formed and compete. Finally, the most accepted paradigm(s) will become +the discipline's new dominant paradigm(s). This “paradigm shift” (Kohn) is a scientific revolution. After +the transformation, the discipline enters again “normal research” period and starts a new life cycle. +The above discussed pattern is revealed in the context of scientific research. It appears to be general, as +we treat a scientific paradigm as a knowledge system that evolves in the environment. +5.2. +What is the fourth paradigm of scientific research? +Jim Gray had a vision that scientific research is shifting to the fourth paradigm (data-intensive paradigm) +after the empirical, theoretical, and computational paradigms (Gray, 2007; Hey, Tansley, & Tolle, 2009), +as illustrated in Figure 7. The empirical paradigm is characterized as finding and describing patterns based +on the observations of real-world phenomena, and the theoretical paradigm is characterized as building +Normal Research +(Sci.&Engr.) +Pre-paradigm +Extraordinary Research +Dominant +paradigm +Crisis +Forming new paradigm; +competing among +multiple paradigms +Creating new +concepts, models, +methods, tools, … +Changing environment +New problems; +new opportunities +(technological and societal) +Paradigm shift +(scientific revolution) +Paradigm update +(scientific evolution) +New components +from other +disciplines + +14 +J Huang / Digital engineering transformation with trustworthy AI towards Industry 4.0: emerging paradigm shifts + +theories on mathematical models. The computational paradigm uses computer simulations to tackle the +difficulty when the theoretical models become too complex to derive and prove propositions. + + +Figure 7. Four Paradigms in Scientific Research, Slide from (Gray, 2007; Hey et al., 2009) + +The fourth paradigm reflects the scientific research in the new research environment with the big data +produced from instruments and simulations, high demands of computing for knowledge discovery from big +data, and emerging demands of data publishing and research work reproducibility. In the fourth paradigm, +scientific research significantly focuses on knowledge discovery from big data and reproducible models +and datasets sharing. +It is essential to point out that a new paradigm does not mean entirely replacing existing paradigms. +Instead, a new paradigm is established on the ground of existing paradigms. A new paradigm represents the +focus shift and the new approach to address new problems in a new environment. Theoretical models are +built on the ground of observations, and computational simulations are based on theoretical models. The +fourth paradigm enhances and unifies the empirical, theoretical, and computational paradigms, on the +ground of an unprecedented big data environment and the new technology for knowledge discovery from +big data. +Inspired by Jim Gray’s fourth paradigm, with the fast growth of big data in the last decade, data science +has emerged as a new discipline. A NIST publication (NIST Big Data Public Working Group, 2019) pointed +out that “data science is the fourth paradigm of science.” However, it has been heavily focused on +knowledge discovery from data. From the library and information science perspective, data curation and +knowledge curation are among data science topics. Still, the needs emerging from the fourth paradigm are +beyond just traditional curation and go much further into knowledge and data sharing with traceability, +explainability, accountability, reproducibility, and interoperability. +By the fourth paradigm, data-intensive science/engineering will emerge in various disciplines of +science/engineering. (Data-intensive engineering is the manifestation of data science in the engineering +field.) In this paradigm shift, data science is the common core, shared by many domain-specific +science/engineering disciplines that need knowledge discovery from big data and knowledge sharing. The +combination of data science (the fourth paradigm) with each specific discipline will form many domain- +specific data-intensive science/engineering disciplines. +5.3. +What are the emerging new engineering paradigms? +“Science is the systematic description of phenomena” (Richards, 1928). Science focuses on discovering +the essential laws of nature. Engineering is “the application of science to the optimum conversion of the +resources of nature to the uses of humankind” (R. J. Smith, 2022). On the one hand, there is an intersection +between science and engineering, as engineering includes scientific research for discovering applied + +Science Paradigms +Thousand years ago: +science was empirical +describing natural phenomena +Last few hundred years: +theoretical branch +using models, generalizations +4元Gp +Last few decades: +3 +a computational branch +simulating complex phenomena +Today: data exploration (eScience) +unify theory, experiment, and simulation + Data captured by instruments +or generated by simulator + Processed by software + Information/knowledge stored in computer +Scientist analyzes database/files +using data management and statistics +J Huang / Digital Engineering Transformation with Trustworthy AI towards Industry 4.0: emerging paradigm shifts +15 + +knowledge for engineering purposes. From this perspective, data-intensive engineering is the manifestation +of the data-intensive paradigm of science in the digital and connected operating environment of the 4IR. +On the other hand, engineering activities are beyond discovering new knowledge and focus more on the +design, manufacturing (or construction), operating, and support of useful systems for human society, thus +having much more complex interactions and relations with other systems in the operating environment, +including human stakeholders. Given this complexity of engineering, the disruptive digital technologies, +and the associated digital, connected, and smart environment in the 4IR, how should we conduct +engineering? What are the new engineering paradigms in the 4IR? +No doubt, the data-intensive paradigm of scientific research is a crucial one. It includes both aspects of +knowledge discovery from data and knowledge sharing (including data). The current “data science” central +theme has focused more on knowledge discovery from data. While “digital engineering” in the US DoD’s +vision is more from the perspectives of model-based engineering and sharing of models and data. +Interestingly, model-based engineering is an effort, like the development of theoretical and computational +paradigms in science, to introduce models in engineering workflows, including traditionally informal +activities such as requirement elicitation, requirement representation, system concept modelling, system +design, among others. Certainly, modelling needs to be based on scientific knowledge as much as possible. +Also, the availability and richness of data in the engineering environment and the fast progress of machine +learning make it a powerful way to build models from data, leading to data-intensive engineering. +However, given the complex interactions and relations between engineering, human society, and the +operating environment, the data-intensive paradigm (that focuses on scientific discovery) alone is +insufficient for tackling the complexity of engineering. The need for engineering paradigm shifts in the 4IR +is driven by the engineering environment, the disruptive digital technologies, the associated higher social- +economic needs, and the new challenging problems, such as the trust issues of AI systems (as discussed in +Section 4). In the landscape of the 4IR, the digital engineering transformation needs new concepts, models, +tools, methods, theories, methodologies, technologies, and standards. +As discussed in section 2, the 4IR has three aspects of paradigm shifts: digitalizing everything of interest, +provisioning ubiquitous machine intelligence, and big data and machine intelligence-centred business +processes. On the other hand, digitalization and ubiquitous machine intelligence also triggered broad +concerns and potential issues of trustworthiness, as discussed in section 4. To put the above-discussed +pieces together, we could have a relatively clear big picture in the direction of Industry 4.0, as illustrated in +Figure 8, in which we need three interdependent essential building blocks for the digital transformation of +engineering and industries: (1) digitalization of engineering; (2) leveraging ubiquitous machine intelligence; +(3) building digital trust and security. +Digitalization is a foundation to realize ubiquitous machine intelligence and needs to support digital +trust. For this consideration, to digitalize a thing of interest (which could be an object or a process), we need +the following four essential components: +(1) Digital representation of the thing of interest in a standard form with well-defined semantics to make +it universally accessible by different types of machines on different platforms. +(2) A unique identifier of the thing of interest, which is a necessary component for traceability, +verifiability, accountability, and explainability. +(3) Associated metadata, such as provenance, in a standard form with well-defined semantics to enable +the use of digital technologies to manipulate and operate the thing automatically. +(4) Verifiable association of the unique identifier, the digital representation, and the metadata with the +thing of interest. This association ensures the authenticity and integrity of the digital artifact. + +For the digitalization of engineering, basically, we need (i) to digitalize engineering artifacts, +engineering processes, and enterprises; (ii) enable the sharing and interoperability of digitalized artifacts +across the engineering lifecycle; (iii) to develop digital model-based engineering. Virtualization is a view +from the perspective of operating with digitalized artifacts and processes. + + +16 +J Huang / Digital engineering transformation with trustworthy AI towards Industry 4.0: emerging paradigm shifts + + +Figure 8. Three essential building blocks for digital engineering towards Industry 4.0: digitalization of +engineering, leveraging ubiquitous machine intelligence, and building digital trust & security. + +Digitalization needs to be realized with digital trust in design and supported by ubiquitous machine +intelligence among digital technologies. Naturally, in this direction, what are enabling technologies for +digitally modelling engineering artifacts, processes, and enterprises? Semantic web technology (a branch +of AI) (Berners-Lee et al., 2006, 2001), RFID, IoT, and blockchain are essential. Also, directly related, +intensive research on enterprise modeling and enterprise integration has been conducted since the 1990s +and can be used to support model sharing across boundaries (Chen, Doumeingts, & Vernadat, 2008; Fox +& Gruninger, 1998; Fox & Huang, 2005; Goranson, 2002; Vernadat, 2020). Enterprise modelling +combined with digital identity management could form a basis for digitalizing engineering artifacts. +Ontologies play an essential role in the digitalization of engineering with respect to artifacts, entities, +organizations, various engineering activities and processes, among others (Ahmed, Kim, & Wallace, +2006; Demoly, Kim, & Horváth, 2019; Dimassi et al., 2021; Fox, Barbuceanu, Gruninger, & Lin, 1998; +Gruninger & Fox, 1995; H. M. Kim, Fox, & Grüninger, 1999; K.-Y. Kim, Manley, & Yang, 2006; Sirin, +Coatanéa, Yannou, & Landel, 2013). +The realization of ubiquitous machine intelligence, the second aspect of paradigm shifts in the 4IR, is +on the top of digitalization. With digitalized engineering artifacts and processes, we can apply ubiquitous +machine intelligence in engineering to design, build, and support smart products, provide smart services, +build trustworthy supply chains, and conduct engineering intelligently. Consider the following examples. +Machine learning has been broadly applied in additive manufacturing (Z. Jin, Zhang, Demir, & Gu, 2020; +Meng et al., 2020; Wang, Tan, Tor, & Lim, 2020). In the digital, smart, and connected environment, +unprecedented big data provide rich information for using AI in engineering operations. Condition-based +maintenance or predictive maintenance are typical scenarios where AI&ML can be leveraged (Black, +Richmond, & Kolios, 2021; Carvalho et al., 2019). Engineering involves many sequential decision making; +with rich information about engineering systems’ status, reinforcement learning can be used to learn from +simulations, experiments, routine operations, and generally experience for optimization, such as in mesh +generation (Pan, Huang, Cheng, & Zeng, 2022), in manufacturing (Su, Huang, Adams, Chang, & Beling, +2022), for engineering design (Dworschak, Dietze, Wittmann, Schleich, & Wartzack, 2022). Machine +Digitalization of Engineering +Ubiquitous Machine Intelligence +Digital Trust & Security +• +Digital asset certification & verification +• +Digital asset access control +• +Digital trust mechanisms (Blockchain, IdM, digital C&V, +traceability, transparency, accountability, …) +• +Digital asset governance +• +… +Industry 4.0 +Engineering & Industries in the world +• +Systems Engineering +• +Manufacturing systems & networks +• +Supply-chain, Logistics, Transportation, … +• +Healthcare +• +… +• +Digitalization +• +Virtualization +• +Interoperability +• +Digitalizing engineering artifacts and processes, … +• +Digital model-based systems engineering +• +Digital models and data sharing in engineering lifecycle +• +… + +Smart factories +Autonomous vehicles +Smart cities +Smart homes +… +• +AI & machine learning +• +Internet of Things +• +Cloud computing & Big Data +• +Augmented/Virtual Realty +• +… + + +J Huang / Digital Engineering Transformation with Trustworthy AI towards Industry 4.0: emerging paradigm shifts +17 + +learning can also be applied to even traditionally labour-intensive and time-consuming requirement +elicitation process (Cheligeer et al., 2022; Mokammel et al., 2018). Essentially, engineering systems design +starts with the environment (Zeng, 2004, 2015, 2020). In the new engineering environment of the 4IR, +many engineering systems become cyber-physical-social smart systems, and it is a challenging issue +regarding how to design such systems (Horváth, 2022; Tavčar & Horvath, 2018). +Digitalization brings to us not only advantages but also new issues, most significantly, trust and security +issues with digital artifacts and those digital technologies. Some examples of those issues include how do +we ensure the authenticity and integrity of digital artifacts, what are policies regarding who can access what, +when to access and where to access, and how do we ensure traceability, transparency, accountability, and +reproducibility, among others. We must develop proper digital mechanisms to address those issues. How +could we achieve secure information sharing in digital engineering? As an example, a security policy +integrating role-based access control and attribute-based access control (including security classification- +based mandatory access control) (Huang, Nicol, Bobba, & Huh, 2012; X. Jin, Sandhu, & Krishnan, 2012; +Servos & Osborn, 2017) can help. Digital identity management and digital trust mechanisms (Huang & +Nicol, 2013), particularly blockchain (Y. Liu et al., 2020; Zheng et al., 2020), can support trust management +of digital artifacts. Digital archives are critical to ensure data integrity and proof of existence (Vigil, +Cabarcas, Buchmann, & Huang, 2013), and blockchain facilitates a new approach to distributed digital +archives. Scientific computing integrity (Huang, 2018; Peisert, Cybenko, & Jajodia, 2015) can be further +developed to support trusted engineering workflows. +Digital Engineering is an emerging form of engineering in the digital revolution. Many issues and +questions there need to be researched. Just name a few, in digitalization, should the standard forms for +digital representation and augmentation be supported by a centralized standardization (such as creating +international standards) or a distributed evolutionary standardization (such as many ontologies competing +to be standards at a fine-grained level and evolving gradually and naturally)? How do we achieve +trustworthy AI systems in 4IR? What are digital trust mechanisms for digital engineering? Many trust +mechanisms used in cloud service (Huang & Nicol, 2013) are applicable to digitalized products, systems, +and services, while still, what new mechanisms should be introduced? Should be the sharing of digital +engineering models and data in a centralized way or distributed way? There are many issues and challenges +ahead as we conduct engineering in a very different new digital environment (Coatanéa, Nagarajan, +Panicker, & Mokhtarian, 2022; Horváth, 2022; Huang et al., 2020). +5.4. +What is next? +Earlier in section 2, we discussed the journey of four industrial revolutions in human history. Potentially, +what is next? Given the broad and profound impacts of digital engineering transformation on human society, +a farther vision will help us to develop long-lasting engineering paradigm(s) and to better design and +develop digital engineering systems. +Fig. 9 illustrates the journey of industrial revolutions from passively following nature to actively +exploiting nature and possibly achieving harmony between human society and nature. Let us start the +discussion with the physical limits of computing. Computational power is the core of ubiquitous machine +intelligence, the defining power we gain from the 4IR. However, with Moore’s law reaching its limit, the +chips for CPUs have hit the ceilings of size and performance, particularly power consumption. The High- +Performance Computing (HPC) community has been striking hard to achieve an exascale computing system +with power consumption within 20MW (Lucas et al., 2014). In the most recent Top500 List issued in June +2022, the current fastest supercomputer frontier is the first exascale machine with 1.1 ExaFlop/s at 21MW. +What we hope to beat Moore’s law are new computational technologies emerging in the horizon, such as +new semiconductor materials, Quantum Computing, and DNA computing. +Data centres are the most significant contributors to today’s computational power, and data centres also +consume a tremendous amount of energy and have significant environmental footprints. For example, in +the US, data centres consumed 1.8% of electricity in 2014 and were responsible for 0.5% of greenhouse +emissions in 2018 (Siddik, Shehabi, & Marston, 2021). It has become more significant regarding how to + +18 +J Huang / Digital engineering transformation with trustworthy AI towards Industry 4.0: emerging paradigm shifts + +improve data centres’ sustainability. Microsoft’s Project Natick conducted a two years experiment and +found that subsea data centres are feasible, reliable, more energy-efficient, and environmentally sound +(Microsoft Research, 2020). + + +Figure 9. The journey of human society through the industrial revolutions + +In a global context, since the first industrial revolution, economic growth has been associated with high +environmental costs, including environmental pollution, overexploitation of natural resources, deterioration +of ecosystems, loss of biodiversity, and global climate change. The problem is challenging because entities +individually lack the motivation to behave environmental-friendly in the market economic mechanism due +to environmental externalities. “Sustainable development is development that meets the needs of the present +without compromising the ability of future generations to meet their own needs.” (UN WCED, 1987) It has +become a consensus of the international community (IPCC, 2022; United Nations, 1992). According to the +IPCC AR6 report (IPCC, 2022), global warming is speeding up - The increase in global surface temperature +in 2011–2020 is 1.09 [0.95 to 1.20] °C higher than in 1850–1900, compared to the increase of 0.19 [0.16 +to 0.22] °C in 2003–2012. The trend of increase will reach or exceed 1.5°C in the near term, even for the +very low greenhouse gas emissions scenario. Climate change has led to some irreversible impacts on the +earth and is approaching a tipping point. This cross-century challenge has become urgent and needs the +whole world to act collectively and immediately before it is too late. This global campaign appears geared +towards the next industrial revolution for renewable natural resources, particularly renewable energy, +reusable materials, and eco-environmental sound economies to achieve harmony between human activities +and nature. +In this direction, science, such as disciplines in biological and environmental sciences, energy science, +and material science, among others, will be a scientific foundation to reveal facts and provide knowledge +about sustainability. Engineering will bring scientific knowledge into the real world by developing +technologies and designing innovative solutions for sustainability. Sustainability is a highly complex +problem because it is across disciplinary domains, across industry sectors, across regional economies, +across cultures, across human groups, across human generations, and across human society and natural +systems. The solutions for sustainability need to be coordinated, comprehensive, and systematic. Systems +science and engineering will play a unique role in applying systems thinking and designing systematic +mechanisms and solutions to address the complex problem of sustainability. +With sustainability in mind, in today’s efforts for digital engineering transformation, we should take +into account the need for sustainability in digital engineering systems design and development. For example, +product provenance can help reuse and recycle after retirement. Particularly, it is critical to consider the +sustainability of new digital technologies (Colorado, Velásquez, & Monteiro, 2020; Kellens et al., 2017; +Paris, Mokhtarian, Coatanéa, Museau, & Ituarte, 2016). The consideration of sustainability in digitalization +will give the emerging digital engineering paradigm(s) a long vision and better address future needs. +2 +Pre-industrial +I1.0 +I2.0 +I3.0 +I4.0 +Mechanization +1IR +2IR +Electrification +3IR +Informatization +4IR +Digitalization +5IR +Sustainablization? +Assembling lines; +Mass Production +Mechanized factories +Automation; +Information-centered +I5.0 ? +Sustainable Development +Digitalized & connected; +Ubiquitous Intelligence; +Intelligence-centered + +INDUSTRY 4.0 +NDUSTRY 3.0 +INDUSTRY 2.0 +INDUSTRY 1.0 +Mechanization,steam +Massproduction, +Automation,computers +Cyber Physical Systems, +power,weavingloom +assembly line, +and electronics +internet of things,networks +electrical energy +回 +J Huang / Digital Engineering Transformation with Trustworthy AI towards Industry 4.0: emerging paradigm shifts +19 + +6. Concluding Remarks +This article investigated the opportunities and uncertainties in digital engineering transformation and +discussed the open issues about the emerging engineering paradigm shifts. +• +From investigating the pattern appeared in industrial revolutions, this article revealed that +ubiquitous machine intelligence is the defining power brought by the fourth industrial revolution. +Digitalization is commonly recognized as the main theme of the 4IR and is also the foundation and +necessary condition for the realization of ubiquitous machine intelligence. +• +Digital engineering, the digitalization of engineering, is at the core of the fourth industrial +revolution. With the broad applications of ubiquitous machine intelligence, many innovative cyber- +physical-social smart systems will appear, and traditional engineering systems are also becoming +digital smart and connected. The promising opportunities are also accompanied with uncertainties +represented by trustworthiness concerns. Given AI as a critical enabling technology in the 4IR, +trustworthy AI is a crucial field to ensure systems’ trustworthiness in Industry 4.0. +• +In the digital smart and connected environment, Digital Engineering needs new concepts, models, +tools, methods, theories, methodologies, technologies, and standards. The emerging engineering +paradigm shifts include but is beyond the data-intensive paradigm (or data-intensive engineering, +the counterpart of data science). Digital engineering transformation towards Industry 4.0 has three +essential building blocks: (1) digitalization, (2) leveraging ubiquitous machine intelligence, and (3) +building digital trust and security. +• +Digitalization is much beyond digitization. Digitization aims to make artifacts machine-readable; +digitalization aims to make the things of interest machine-understandable and virtually operatable. +Digitalization of a thing of interest should include: (1) digital representation, (2) a unique identifier, +(3) metadata including provenance, and (4) a verifiable association of the thing with the above three +components. +• +The remarkable progress, disruptive impacts, and fast growth of real-world AI applications in the +digital and connected environment have triggered concerns and research about the trustworthiness +of AI systems. The engineering design community at large is facing an excellent opportunity to +bring the new capabilities of ubiquitous machine intelligence and trustworthy AI principles, as well +as digital trust, together in various engineering systems design to ensure the trustworthiness of +systems in Industry 4.0. + +Digital engineering transformation is a crucial process for engineering paradigm shifts in the 4IR. Digital +engineering, at the core of the 4IR, is an exciting and broad field to explore and will further lead to “the +transformation of entire systems, across (and within) countries, companies, industries and society as a whole” +(Schwab, 2017). +References +AAAI Presidential Panel on Long-Term AI Future. (2009). Asilomar Study on Long-Term AI Features. Retrieved +from https://www.aaai.org/Organization/asilomar-study.pdf +Adcock, R., Jackson, S., Fairley, D., Singer, J., & Hybertson, D. (2021). Emergence. In Systems Engineering Body +of Knowledge (SEBoK). 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(2019). Digital engineering transformation across the Department of +Defense. The Journal of Defense Modeling and Simulation, 16(4), 325–338. + + diff --git a/vNAzT4oBgHgl3EQfB_pj/content/tmp_files/load_file.txt b/vNAzT4oBgHgl3EQfB_pj/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..cafbb935fa3c57e45f874653242b9c82e53cfde0 --- /dev/null +++ b/vNAzT4oBgHgl3EQfB_pj/content/tmp_files/load_file.txt @@ -0,0 +1,1642 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf,len=1641 +page_content='Transactions of the SDPS: Journal of Integrated Design and Process Science XXXX DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='3233/JIDXXXXXX http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='sdpsnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='org Digital Engineering Transformation with Trustworthy AI towards Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='0: Emerging Paradigm Shifts Jingwei Huang Department of Engineering Management and Systems Engineering, Old Dominion University Norfolk, VA 23529, USA j2huang@odu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='edu Abstract Digital engineering transformation is a crucial process for the engineering paradigm shifts in the fourth industrial revolution (4IR), and artificial intelligence (AI) is a critical enabling technology in digital engineering transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' This article discusses the following research questions: What are the fundamental changes in the 4IR?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' More specifically, what are the fundamental changes in engineering?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' What is digital engineering?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' What are the main uncertainties there?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' What is trustworthy AI?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Why is it important today?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' What are emerging engineering paradigm shifts in the 4IR?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' What is the relationship between the data-intensive paradigm and digital engineering transformation?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' What should we do for digitalization?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' From investigating the pattern of industrial revolutions, this article argues that ubiquitous machine intelligence (uMI) is the defining power brought by the 4IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Digitalization is a condition to leverage ubiquitous machine intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Digital engineering transformation towards Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='0 has three essential building blocks: digitalization of engineering, leveraging ubiquitous machine intelligence, and building digital trust and security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' The engineering design community at large is facing an excellent opportunity to bring the new capabilities of ubiquitous machine intelligence and trustworthy AI principles, as well as digital trust, together in various engineering systems design to ensure the trustworthiness of systems in Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Keywords Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='0, Fourth Industrial Revolution, digitalization, digital transformation, digital engineering, digital engineering transformation, paradigm shift, data science, digital technologies, uncertainties, trustworthiness, trustworthy AI systems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Introduction The convergence of disruptive digital technologies, such as artificial intelligence (AI), Internet of Things (IoT), cloud computing, high-speed wireless communication, and blockchain, among others, has driven the world into a new wave of a technology revolution, known as the “fourth industrial revolution” (4IR) (Schwab, 2017), also known as the “digital revolution” for the central role of digitalization in the 4IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' In 2011, Germany first launched their Industrie 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='0 initiative aiming at leveraging Cyber-Physical-Systems (CPS) to build smart factories and strengthen their leadership in manufacturing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' The US launched a national Advanced Manufacturing program in 2014 and declared manufacturing to be a national priority.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Recent Five-year Plans of China since the 2010s actively promote critical technologies towards 4IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='0 (I4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='0) is a term frequently used as an interchangeable term for 4IR;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' however, they have some subtle differences in their focuses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' In this article, we use the 4IR to address the revolutionary change in paradigms and tools, and we use Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='0 to address the targeted industrial systems (with manufacturing in the Accepted Version, To Be Published by IOS Press 2 J Huang / Digital engineering transformation with trustworthy AI towards Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='0: emerging paradigm shifts core as the fundamental component) in the 4IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Briefly, Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='0 is the target or goal, and the 4IR is the movement or process toward that goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' A defining feature of the fourth industrial revolution is digitalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Associated with this feature, a pervasive and profound digital transformation is ongoing in almost every aspect of human society globally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Digital Engineering is the digital transformation in the field of engineering, and digital engineering transformation is at the center of 4IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' In 2018, the US DoD launched their Digital Engineering Strategy (US DoD, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Zimmerman, Gilbert, & Salvatore, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' This strategy requires using and sharing formal models and digital data across engineering lifecycle and organizational boundaries through a trusted “authoritative source of truth”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' This move will impact the US defense industry and propagate to other industry sectors and change how engineering is conducted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Digital engineering transformation is about engineering paradigm shifts in the 4IR, and AI is a critical enabling technology in the digital engineering transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' AI has achieved remarkable progress in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' At the same time, the disruptive impacts and fast growth of applications of AI systems (particularly ML) also raise broad concerns about an AI system’s reliability, safety, security, privacy-preserving, fairness (or bias-free), explainability, traceability, transparency, and accountability, among other qualities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Research on those concerns has been ongoing under the umbrella of “Trustworthy AI” (AAAI Presidential Panel on Long-Term AI Future, 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' EU AI HLEG, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' NSF, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Stone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Wing, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Trustworthy AI has two primary aspects: societal effects (ethics) and technical performance (dependability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' The ethics of AI concerns the long-term impacts of AI on humans and human society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' The central principle is to use AI for good or for purposes beneficial to humans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Dependability concerns the competency of AI systems on technical matters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Given the critical role of AI in the 4IR, the recent concerns about trustworthy AI also propagate into Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='0 systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' The AI community has been striving for many decades to push the boundary of machine intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' The engineering community will bring AI technologies and ethical principles together to deliver trustworthy cyber-physical-social smart systems to human society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' This article will discuss the following research questions: (1) What are the fundamental changes brought by the fourth industrial revolution?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' (2) More specifically, what are the fundamental changes in engineering?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' What is digital engineering?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' (3) What are the main uncertainties associated with the 4IR?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' What is trustworthy AI?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' (4) What are emerging engineering paradigm shifts in the 4IR?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' The contents of this article are organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Section 2 discusses the answer to the first question by investigating the patterns of the four industrial revolutions, focusing on the ongoing fourth revolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Section 3 continues with the second question to discuss digital engineering transformation in the 4IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Section 4 discusses the uncertainties in the 4IR and trustworthy AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Section 5 discusses some open issues and focuses on the emerging engineering paradigm shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Finally, section 6 concludes the article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' The Fourth Industrial Revolution What are the fundamental changes (including paradigm shifts and their profound impacts) brought by the fourth industrial revolution?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Let us logically examine the patterns that emerged in industrial revolutions historically from the perspectives of technological triggers, paradigm shifts, and profound impacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' As illustrated in Figure 1, the fundamental changes of both the first industrial revolution (1IR) and the second industrial revolution (2IR) are the provision of energy (physical power) for industrial production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' The fundamental changes of the third industrial revolution (3IR) and the fourth industrial revolution (4IR) are the provision of machine intelligence (brainpower) to explore and leverage information power in industries and human society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Both 1IR and 3IR are fundamental for inventing revolutionary new tools (the steam engine and the computer) to deliver physical and information power, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Both 2IR and 4IR scale up the delivery of new powers (energy and information power) initiated in 1IR and 3IR, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' The 1IR is characterized as “mechanization.” The invention of the steam engine triggered the 1IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Steam engines lead to the industrial operations paradigm shift from human manual labor to the broad use of machines in production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' A representative example is frequently referred to as the first mechanical loom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' J Huang / Digital Engineering Transformation with Trustworthy AI towards Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='0: emerging paradigm shifts 3 The profound impact of the first industrial revolution is that the using steam-powered machines to replace human hands in production activities significantly improved the efficiency of production and released humans from heavy manual labor for more creative activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' The 2IR is characterized as “electrification.” The invention and use of electric power, together with the invention of assembly lines, triggered the second industrial revolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' The use of electricity makes power (energy) usage easy, flexible, free from the space-constraints, and easy to scale up or down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Without electricity, the idea of assembly lines could not have been realized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' With easy access to electric power, the invention of assembly lines leads to skill specialization-based mass production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' This change marks the industrial operations paradigm shift from early craft production and job production to much more productive mass production and the associated product standardization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' The profound impacts of the second industrial revolution are mass production and flexible and scalable access to power (energy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Four waves of Industrial Revolutions and their profound impacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Every wave was triggered by disruptive technologies and followed by paradigm shifts in industrial operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' The 3IR can be characterized as “informatization.” The invention and development of programmable general-purpose digital computers triggered the 3IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' The computer is the most powerful information processing tool in human history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' The availability of this critical tool allows humans to explore and leverage the power of information, along with using matter and energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' The paradigm shift brought by the 3IR can be viewed in three aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' First, with the broad applications of computers in control, such as the adoption of PLCs, DCS, and SCADA, many human-operated processes became automated or computer-aided processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Secondly, information media was digitized (or computerized).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Research reveals that 94% of information media became digital in 2007 (Hilbert & López, 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' The digital forms of information allow information to be stored and read with computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' This machine-readable (and later machine-processable) property makes information flow largely automated and fast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' The advantages of digital information make information a decisive success factor beyond other factors during the time of the 3IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' This change leads to the third aspect of the paradigm shift - the business process transition from material, energy, and product centered to information centered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' The fundamental impact of 3IR is the beginning of using machine intelligence to explore and leverage the power of information, which leads to new capabilities and significantly improved efficiency and effectiveness in industrial production and other human activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' 1st Industrial Revolution − Mechanization Increasing Accessibility, Flexibility, and Scalability Trigger: Steam Engine Paradigm shift: Human/animal power -> steam power human manual labor -> use of steam-powered machines Impact: Steam-powered machines significantly improve production efficiency and release humans from heavy manual labor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' 2nd Industrial Revolution − Electrification Trigger: Electric power;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' assembling lines Paradigm shift: Steam power -> electric power;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' craft production, job production -> mass production Impact: Electric power makes energy delivery highly flexible and scalable and leads to the productive mass production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' 3rd Industrial Revolution − Informatization Trigger: Computer and electronics Paradigm shift: information media -> digitized;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' operation processes -> automated or computer-aided processes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' matter & energy centered -> information centered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Impact: Humans started to use machine intelligence (MI) to explore and leverage information power, which leads to new capabilities and significantly improved efficiency and effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' 4th Industrial Revolution − Digitalization Trigger: Digital technologies (IoT, 5/6G, AI, Cloud, blockchain, …) Paradigm shift: digitizing information -> digitalizing everything of interest;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' centralized and distributed computing & MI -> pervasive computing & MI;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' information centered -> big data and machine intelligence centered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Impact: The pervasive digitalization and ubiquitous machine intelligence (integrated capabilities from IoT, 5G/6G, cloud, and AI) lead to many cyber- physical-social smart systems (CPS3) with innovative applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Energy (Physical Power) Machine Intelligence (Information Power) INDUSTRY 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='0 NDUSTRY 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='0 INDUSTRY 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='0 INDUSTRY 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='0 Mechanization,steam Massproduction, Automation,computers Cyber Physical Systems, power,weavingloom assembly line, and electronics internet of things,networks electrical energy 回4 J Huang / Digital engineering transformation with trustworthy AI towards Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='0: emerging paradigm shifts For this reason, the 3IR is also called the “information revolution,” and the period is called the “information age.” Since the start of the 3IR, many digital technologies, such as artificial intelligence, industrial robots, portable and then mobile computers, the Internet, wireless networks, mobile devices, and others, were developed and profoundly changed human society regarding how we work and live already.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Those digital technologies prelude the next wave of technological revolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' The 4IR can be characterized as “digitalization.” The convergence of transformative and disruptive digital technologies, such as IoT, high-speed wireless communication, cloud computing, AI&ML, and blockchain, among others, trigged the 4IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' IoT, in a broader perspective, also named as “Internet of All Things” or “Internet of Everything,” has four pillars: Internet of Things, Internet of Information, Internet of People, and Internet of Services (Kirkpatrick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' In the future internet infrastructure,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' the physical things of interest (such as various machines,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' devices,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' buildings,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' roads,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' and other structures,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' even things of interest in the natural world),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' the abstract things (things in the digital world,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' such as information contents),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' people,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' organizations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' and various processes of interest (examples include: service processes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' or engineering processes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' even social or natural processes of interest,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' such as climate change,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' environmental change,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' living things status change,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' pandemic monitoring,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' disease infection tracing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' and others),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' anything of interest are interconnected,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' through future internet,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' operating with supports from 5G/6G wireless communication networks,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' cloud computing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' and AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' In the rest of this paper, we refer term IoT in this broader sense of the Future Internet or Internet of Everything.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' IoT connects the physical world with the cyber world to form an integrated cyber-physical smart environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' With IoT through sensors and actuators, smart things in the physical world can be remotely located, monitored, and even controlled through the Internet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' With mobile devices supported with 5G/6G wireless networks, people will have ubiquitous access to the Internet, smart things in the physical world, and the cloud services needed to handle those smart things.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' More importantly, people themselves become nodes of this Internet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' This interconnected world, facilitated by IoT, 5G/6G, cloud computing, and AI&ML, is leading to numerous Cyber-Physical-Social Smart Systems (CPS3) (Huang, Seck, & Gheorghe, 2016), such as autonomous vehicles, smart factories, smart stores, smart cities, and others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' A necessary condition to develop those smart systems is to digitalize things of interest to make them uniquely identifiable and machine-understandable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' then, we can leverage AI&ML to exploit the unprecedented richness of information brought by the digital and connected world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' The 4IR with those disruptive digital technologies is “leading to unprecedented paradigm shifts in the economy, business, society, and individually” (Schwab, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' The emerging paradigm shifts in the 4IR can be viewed from several related perspectives as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' (1) Shift from digitizing information to digitalizing everything of interest, including artifacts (both abstract and physical), natural things, organizations, and processes (such as production and/or service processes, monitored natural processes, and others).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' (2) Shift from centralized and distributed provisioning of computing services and machine intelligence to pervasive provisioning of computing services and machine intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' (3) Shift from information-centered processes (developed in 3IR) to big data and machine intelligence- centered processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' About the first shift, different from the 3IR, where information was digitized, in the 4IR, not only information (abstract artifacts) but also physical artifacts and natural living things of interest, as well as processes of interest, could be digitalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Everything of interest, even humans, could have their digital counterparts (or “digital presence” as termed by (Schwab, 2017)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Such as, some devices or products will have their digital twins, and some artifacts will have their digital augmentations (the associated metadata, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=', provenance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' In addition to artifacts, some processes of interest (no matter about production, service, or some natural processes such as climate change or regional ecosystem change, among others) could also be digitalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' This pervasive digitalization could be enabled by semantic web technology (a branch of AI) (Berners-Lee, Hall, Hendler, Shadbolt, & Weitzner, 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Berners-Lee, Hendler, & Lassila, 2001), RFID, IoT, and blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' J Huang / Digital Engineering Transformation with Trustworthy AI towards Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='0: emerging paradigm shifts 5 The second shift can be seen from comparing how computing service was delivered before the 4IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' From many people using a single large computer, each person using a computer and then a laptop, till each computer was able to connect to the Internet with a wall jack, computing service was delivered in a “centralized”, “decentralized”, and then “distributed” approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' In the 4IR, we can use computers and interact with the world everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' About thirty years ago, Mark Weiser and Nicholas Negroponte had a debate at MIT Media Lab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Negroponte predicted that AI was going to be the leader for the next wave of computing, but Weiser argued it would be Ubiquitous Computing, characterized as “invisible” and “connection” (Weiser, 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Weiser’s vision, which can be found in a vivid illustration in his essay “Open House” (Weiser, 1996), is one of the original ideas about the Internet of Things.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Interestingly, IoT and AI together become the major drivers for the 4IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' IoT extends the “nerves” of the Internet into the physical world, and AI is the “brain” of smart things interacting with the digitalized connected world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' In addition to IoT and AI, wireless networks and cloud computing are critical contributors to the pervasive provisioning of computing and machine intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' First, of course, high-speed wireless communication is an essential infrastructure to facilitate the mobility of communication, computing, sensing, and delivery of all types of digital services without or significantly reducing the physical space constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Secondly, cloud computing enables ubiquitous on- demand network access to a shared pool of computing resources, thus meeting scalable and elastic computing needs (Armbrust et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=', 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' NIST, 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Cloud computing is essential for handling the big data produced in the digital world and for the intensive computing required by AI applications, no matter where we are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' This article uses term “ubiquitous machine intelligence” (uMI) to refer to the integrated capabilities enabled by the convergence of IoT, 5G/6G, AI&ML, and cloud computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Interestingly, the role ubiquitous machine intelligence plays in meeting the needs of computing and machine intelligence is very similar to the role of electricity in 2IR, where electricity enables the mobility of power delivery needed in assembly lines and mass production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Here, ubiquitous machine intelligence makes computing services and machine intelligence pervasively accessible and provides the essential technology for various smart systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' For this reason, ubiquitous machine intelligence is regarded as the most fundamental change brought by the 4IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' The third shift can be viewed through the following comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' In the Industry 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='0, information flows and material flows were still separated;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' thus, information is long-delayed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' In the 4IR, the pervasive digitalization and connectedness make the material and human flows associated with information flows;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' thus, real-time information about the material and human flows can be used for decision-making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' There will be unprecedented rich information about the things of interest for decision making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Consequently, a decisive success factor in the era of 4IR will be the capability of leveraging ubiquitous machine intelligence to collect and extract information, discover knowledge from big data, and make smart decisions in the digital smart and connected environment (Huang, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Finally, regarding the profound impact of the 4IR, similar to the 2IR that makes power (energy) easily accessible and leads to mass production, the 4IR makes information power (computing and machine intelligence) pervasively accessible and leads to various CPS3 with innovative applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' More specifically, in the 4IR, the pervasive digitalization and pervasive connectedness make data (minerals of information) available in an unprecedented scale and make computing services and machine intelligence (brainpower or information power) ubiquitously accessible and scale up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Furthermore, they are triggering and accommodating numerous innovative applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Those disruptive technologies and their innovative applications are “fundamentally changing the way we live, work, and relate to one another.” They lead to “the transformation of entire systems, across (and within) countries, companies, industries and society as a whole” (Schwab, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' 6 J Huang / Digital engineering transformation with trustworthy AI towards Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='0: emerging paradigm shifts 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Digital engineering transformation As discussed in the last section, digitalization is the central theme of the 4IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' As such, pervasive digital transformations are ongoing in almost every aspect of human society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Given the essential role of engineering in industries, the digitalization of engineering is at the core of 4IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Digital engineering is the digitalized engineering, or the targeted digital form of engineering to be realized by the digitalization of engineering towards Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' The US Department of Defense (DoD) defines “digital engineering” as “an integrated digital approach that uses the authoritative source of system data and models as a continuum across disciplines to support lifecycle activities from concept through disposal” in their “Digital Engineering Strategy” (US DoD, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Zimmerman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' As illustrated in Figure 2, the strategy targets five goals: (1) formalize the development, integration, and use of models, leading to a continuous end-to-end digital representation of the system of interest;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' (2) provide an enduring authoritative source of truth to share and exchange digital models, data, and other digital artifacts across boundaries of organizations and the engineering lifecycle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' (3) incorporate technological innovation to improve the engineering practice;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' (4) establish a supporting infrastructure and environments;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' (5) transform culture and workforce to adopt and support digital engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Paper (Zimmerman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=', 2019) provides a comprehensive view of the US DoD’s efforts for digital engineering transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' The implementation of this strategy will significantly change how engineering practice is conducted in the US DoD enterprise and propagate to industries in a broader range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' The US DoD Digital Engineering Strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Picture from (US DoD, 2018) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='3&4) From the perspective of engineering processes, the transformation from conventional engineering to digital engineering is illustrated in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' In the picture, from a general view, an engineering process is illustrated as a process model with inputs (left), outputs (right), enablers (bottom), and control (top).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Digitalizing engineering artifacts, engineering processes, and engineering enterprises is the foundation for digital engineering transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Without digitalized artifacts, the applications of emerging digital technologies will be very limited, maybe in an ad hoc manner, or may need human or ad hoc devices as intermediate agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' As a result, the level of automation and autonomous functioning will stay at a slightly higher level than in Industry 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' On the other hand, with digitalized engineering artifacts and processes, we can broadly leverage machine intelligence and other digital technologies to conduct engineering in a digital, smart, and connected environment;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' thus achieving a new level of engineering, which is more effective and efficient as well as more trustworthy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Some advantages of digital engineering include: sharable knowledge and data across the engineering life cycle, increased explainability and transparency of engineering processes and products, fast integration of distributed engineering resources for a given mission, increased traceability and accountability, increased product traceability along supply chains, and quick adaption to the changing environment, among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' P and rov to Use Truth DIGITAL ENGINEERING STRATEGY pue 4SpecialtyEngineeringModels Product Management Support Models Models Authoritative Source of System Design Truth Models Models Verificationand Manufacturing ValidationModels Models Key:Data Figure 4: Examples of Models Connected via the Authoritative Source of Truth J Huang / Digital Engineering Transformation with Trustworthy AI towards Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='0: emerging paradigm shifts 7 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Digital engineering transformation from a process perspective Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Model perspective: operations in the digital engineering lifecycle (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=', 2020) In digital engineering as shown in Figure 3, the inputs, outputs, enablers, and control are different from the conventional ones;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' correspondingly, the digital engineering operations will also significantly differ from the conventional ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' In the digital engineering transformation, along with the fast-paced digital technologies, “computational thinking” (Wing, 2006, 2012) is critical to the new engineering paradigms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Knowledge and skills in applied computing and relevant digital technologies have become necessary components in workforce development towards digital engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Figure 4 briefly describes operations in the digital engineering lifecycle from a model perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Some further discussion of digital engineering can be found in (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Finally, AI is a critical enabling technology to advance digital engineering (Huang, Beling, Freeman, & Zeng, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' In addition to the broad applications of Machine Learning for modelling and prediction in engineering, another AI branch, knowledge representation and reasoning, particularly semantics technology, is a fundamental technology to support digitalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' For example, ontologies are used in MBSE (Madni & Sievers, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' AI also plays a critical role in the logical formalization in building and operating digital trust & security systems (Huang, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' We will discuss further in subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Engineering Process Products (energy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' materials,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' parts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' end products,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' structures,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' data,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' …) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='Services ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='Natural resources ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='Products & services from ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='other engineering processes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='Engineering standards and regulations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='Domain-specific enabling technologies ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='Enterprise with workforce of specialty ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='Digital Engineering ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='Process ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='Digitalized Products ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='(product + digital augmentation) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='Digitalized Services ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='Natural resources ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='Digitalized Products & Services ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='from digital environment ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='Domain-specific enabling technologies ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='+ Digital technologies ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='Digital enterprise/workforce of (specialty ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='+ Digital technologies skills) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='Digital Transformation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='Digitalizing engineering artifacts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' processes and enterprises Digitalized Engineering standards and regulations Digitalizing engineering artifacts: Parts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Materials;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Products;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Devices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Models;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Datasets;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Software;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Documents;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' … Digitalizing engineering processes Actions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Functions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Manufacturing processes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Quality management processes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Environment management processes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' … Digitalizing enterprise Enterprise organizations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Job roles;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Services;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Policies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Supply-chain;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Customer-relations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' … Concept Stage Development Stage Production Stage Utilization/support Stage Retirement Stage Model creation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Inputs: Digital artifacts (DAs) in operating environment;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Relevant data and models from downstream stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Inputs: Digital artifacts from both upstream and downstream;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Digital artifacts of external systems Inputs: Digital artifacts from both upstream and downstream;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' and from production environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Inputs: Digital artifacts from both upstream and downstream;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' and from operating environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Inputs: Digital artifacts from upstream;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' and from natural environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Model learning Apply AI&ML for model building from big data coming from upstream and downstream engineering stages and environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Model integration Interaction between digital models for both SysCon and systems in operating environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Interaction between digital models for both system components and external systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Interaction between digital models for both the system and systems in production environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Interaction between digital models for both the system and external systems in operating environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Interaction between digital models for the sys component and DAs in natural environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Model curation Create model of models;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' maintain metadata for models;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' maintain model provenance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' model update and propagation to downstream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Model sharing & use Across lifecycle activities,;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' across the boundaries of disciplines and organizations Model Trustworthiness Centralized standardization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' decentralized standardization and mappings;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' distributed evolutionary fine-grained convergence;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' model trustworthiness;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' model repeatability;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Access Control;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' digital artifact intellectual property protection, … 8 J Huang / Digital engineering transformation with trustworthy AI towards Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='0: emerging paradigm shifts 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Harness uncertainties in the 4IR – A key: trustworthy AI systems The disruptive digital technologies in the 4IR not only offer transformative opportunities but also bring in uncertainties and have triggered broad concerns and research on systems trustworthiness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' The focused concerns of trustworthiness have been shifting and expanding to cover new issues that pop up in the new technologies used in each generation of engineering systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' This section discusses the trustworthiness concerns of Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='0 systems, starting from examining the evolution of the trustworthiness of engineering systems, then focusing on trustworthy AI systems – the pivotal component of Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='0 systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Before we move to these topics, let us discuss the elementary concepts of trust first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' We use trust explicitly or implicitly, even unaware of using it, in our daily life and work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' However, what is trust exactly?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' There are many definitions of trust (Walterbusch, Gräuler, & Teuteberg, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Our working definition of trust used in this article is as follows (Huang & Fox, 2006) - Trust is a mental state comprising (1) expectancy: the trustor expects a specific behaviour of the trustee, such as providing true information or effectively performing cooperative actions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' (2) belief: the trustor believes that the expected behaviour will occur, based on the evidence of the trustee’s competence, good intention, and integrity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' (3) willingness to take the risk: the trustor is willing to take the risk for (or be vulnerable to) that belief.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' In brief, Trust = Expectation + Belief in that expectation + Willingness to take the risk for that belief.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Trust can be regarded as a relation between a “trustor” (trusting party) and a “trustee” (trusted party) with respect to the properties of something provided/offered/conducted by the trustee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' For example, a passenger (trustor) trusts an autonomous car (trustee) regarding the safety of driving on the streets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' What is to be trusted is out of the control of the trustor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' From the studies of trust, the factors leading to trust include the trustee’s ability, good intention (or benevolence, as some researchers called it) towards the trustor, and integrity (reflected by the principles and values guiding behaviours)(Mayer, Davis, & Schoorman, 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' The roles of ability and good intention in the concept of trust are apparent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Integrity is the need for trust judgment with respect to the predictable behaviours of the trustee in situations that are out of watching by the trustor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' According to Simon (Simon, 1997), a decision-making process in the real world is limited by “bounded rationality,” i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=', the “rational choice that takes into account the cognitive limitations of the decision maker limitations of both knowledge and computational capacity.” In the real world, because we only have limited information/knowledge, limited computing capacity, and limited time available for decision-making, a decision has to be made partly based on bounded rational calculation and partly based on trust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' As Luhmann addressed (Luhmann, 1979), trust, as “a basic fact of social life,” functions as a “reduction of complexity.” Without trust, social interactions, including the applications of technologies in engineering and business, will be extremely difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Because of trust, we do not need to verify everything before a decision making or action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Trust mechanisms (Huang & Nicol, 2013) play a critical role in harnessing the disruptive digital technologies toward trustworthy utilizations of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Trustworthiness of systems Trustworthiness of a system is defined in (Schneider,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' 1999) as “assurance that a system deserves to be trusted—that it will perform as expected despite environmental disruptions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' human and operator error,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' hostile attacks,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' and design and implementation errors.”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' In short (Avizienis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Laprie,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Randell,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' & Landwehr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' 2004),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' systems trustworthiness is the “assurance that a system will perform as expected.”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Trustworthiness is a relative concept and is dependent on what is expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Trustworthiness is often characterized as a collection of the expected properties of a system to be trusted, as illustrated in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' The “assurance” can be achieved by various trust mechanisms implemented in engineering systems and in societal systems such as audits, certifications, regulations, and law enforcement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' J Huang / Digital Engineering Transformation with Trustworthy AI towards Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='0: emerging paradigm shifts 9 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' The evolving trustworthiness properties as the evolution of systems with technologies and the sources of concerns In the following, we discuss the expected properties of trustworthiness of different generations of engineering systems and the sources of the concerns or threats to trustworthiness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Traditional systems are typically stand-alone systems (a physical system, a computer system, or a system embedded with computer systems) or just connected logically and internally within an organization rather than the Internet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' As shown in the bottom part of Figure 5, the trustworthiness properties concerned with those traditional systems are mainly reliability, maintainability, availability, integrity, robustness, safety, and usability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Those properties together are referred to as dependability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' The sources of concerns or the threats to the trustworthiness properties are mainly non-malicious faults caused by (1) errors introduced in systems design and development;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' (2) errors introduced in manufacturing and maintenance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' (3) errors happened in system operations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' (4) environmental disruptions, including natural disasters, environmental pollution, and gradual but irreversible environment change associated with climate change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' In the middle part of Figure 5, enabled by Internet and wireless mobile communication, the networked industrial systems evolve to cyber-physical systems (CPS) (Lee, 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' NSF, 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Rajkumar, Lee, Sha, & Stankovic, 2010), which use networked computing components to monitor, coordinate, control, and integrate physical systems and processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' The connectedness of systems naturally led to great challenges with respect to security and privacy, and correspondingly they become the central theme of systems trustworthiness (Avizienis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=', 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Schneider, 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' The threats to security and privacy are obviously from various malicious attacks and incompetent security defence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' As illustrated in the upper part of Figure 5, trustworthiness for the systems in the 4IR involves concerns with trustworthy AI, pervasive security, and technical dependability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' The issues in an AI system could trigger security issues and systems dependability issues;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' of course, security issues could also trigger systems dependability issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' The sources causing the systems trustworthiness issues are in three folds: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='Robustness ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='Reliability ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='Safety ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='Confidentiality ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='Usability ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='Resilience ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='Privacy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='Ethicality ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='Fairness ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='Transparency ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='Maintainability ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='Accuracy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='Explainability ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='Traceability ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='Reproducibility ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='Accountability ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='Malicious attacks ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='Cyber connectedness ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='Non-malicious faults ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='Trustworthiness(3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='Trustworthiness(2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='Trustworthiness(1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='AI Systems uncertainties ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='IoT Pervasive connectedness ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='Systems complexity & emergence ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='Dependability ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='Trustworthy AI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='Security ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='Pervasive ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='Security ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='Cooperability ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Escher, Convex and Concave (1955) 10 J Huang / Digital engineering transformation with trustworthy AI towards Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='0: emerging paradigm shifts (1) The pervasive security issues caused by the pervasive connectedness facilitated by IoT: the pervasive connectedness facilitated by IoT broadly extends the security and privacy concerns to every corner of the world where humans work and live.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Without assured technologies and societal mechanisms, the Internet of Things can turn into “the Internet of Risky Things” (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Smith, 2017), even the Internet of dangerous things.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' The threats to security and privacy in 4IR include malicious attacks and incompetent security defence as before but with a much larger attack/exposure surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Additionally, the threats also include certain relevant societal mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' For example, as Smith pointed out (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Smith, 2017), some smart devices will last longer than their manufacturers and suppliers, and who will keep updating the systems or their embedded computing components?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Without updates, a digital system will be vulnerable in the complex digital operating environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' (2) The uncertainties caused by AI systems: the fast growth and broad applications of AI systems or embedded AI components in various systems have triggered much concern in two aspects: technical competence (such as reliability and security) and ethical concern (such as transparency, bias towards groups of people, human-machine relations, and others).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' The next subsection will discuss this aspect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' (3) systems complexity & emergence - the aforementioned two primary two factors together bring in high uncertainties and complexities into the systems in the 4IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Complex systems typically exhibit strong emergence (unexpected emergent system behaviour and properties that were not anticipated in design and development, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=', the US-Canada Northeast Blackout of 2003) (Adcock, Jackson, Fairley, Singer, & Hybertson, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Trustworthy AI In the last decade, AI achieved remarkable advances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Just to name a few, AlphaGo (Silver et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=', 2016, 2018), using deep neural networks-based reinforcement learning, won the world number one player of Go game, which is regarded as the most challenging board game for computers to win.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' AlphaGo’s achievement marks a new milestone of machine intelligence towards superhuman intelligence on specific tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Although it is still a pilot project and has no broader adoption yet, Waymo One, the commercial taxi service operating with level 4 autonomous vehicles, has been offered on the streets in a city (Waymo One, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' This service marks a new level of integrated intelligent capabilities of an AI system on a type of complex tasks which previously only humans could do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Another remarkable advance is Generative Adversarial Networks (GANs) (Creswell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Goodfellow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=', 2014, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' A GAN consists of a pair of deep neural networks, a generator, and a discriminator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' together, they can generate realistic images and other contents as requested (see examples at https://thispersondoesnotexist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='com/).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' GANs lead to many potential applications but also open the door for deepfake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=" For many decades, the AI community's focus has been on extending machine intelligence capabilities." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Although the advances of AI today are still domain-specific (or narrow AI) rather than general AI, the remarkable achievements, the disruptive impacts, and the fast growth of applications of AI systems triggered concerns and research on trustworthy AI systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' The concerns are mainly from two perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' (1) Ethical AI, focusing on social effects, mainly ethical considerations, essentially, the profound effects AI systems bring to humans, human groups, and human society, and advocating using AI to benefit humans as a central principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' (2) Reliable AI, focusing on the technical performance, or dependability, concerning the competency of AI systems on technical matters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' There is a broad range of expected properties of trustworthiness in AI systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Ethical AI is an emerging interdisciplinary research field, gathering together researchers from a wide range of areas, including computer science, philosophy, sociology, anthropology, public policy, law, and others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Some thought leaders and researchers conducted pioneering work in the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' The Association for the Advancement of Artificial Intelligence (AAAI) organized a panel of leading AI researchers that conducted “Asilomar Study of Long-Term AI Futures” in 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' The study explored a wide range of topics on societal impacts and guidance of AI research (AAAI Presidential Panel on Long-Term AI Future, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' The study assessed concerns and perspectives about disruptive outcomes of superhuman intelligence, explored ethical and legal issues associated with autonomous systems, and identified near-term AI research J Huang / Digital Engineering Transformation with Trustworthy AI towards Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='0: emerging paradigm shifts 11 challenges and opportunities, including enhancing people’s privacy, enhancing human-AI collaboration and interaction, making ML and reasoning transparent to people, and preventing using AI for malevolent purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Continuing the effort to guide AI research for good, the AI100 2016 report (Stone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=", 2016) addressed that AI research is “shifting from simply building systems that are intelligent to building intelligent systems that are human-aware and trustworthy.” The report collectively presents the view and prospects of a panel of experts on the opportunities and challenges of AI and policy recommendations in eight selected AI application domains targeting people's lives in a typical North American city in 2030." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' They addressed challenges regarding how to build safe and reliable systems, gain public trust concerning safety, security, and privacy, make AI systems behave ethically and overcome bias, and how AI systems smoothly interact with humans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' In the long term, ethical AI concerns the fundamental relations between AI systems and humans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' There are concerns about “technological singularity” or “intelligence explosion,” partially reflected by the dystopia depicted in fiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Experts in the AI field believe those radical outcomes remain fictional and are not immediate threats;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' well, some thought leaders suggest “avoid strong assumptions regarding upper limits on future AI capabilities” (Future of Life Institute, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=" Given AI’s profound impacts on human society, it is necessary to review AI systems' purposes, usages, and impacts and create principles and regulations for guiding AI research and development to avoid harming humans and human society." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' The 23 Asilomar principles (Future of Life Institute, 2017) and the EU HELG ethical AI guide (EU AI HLEG, 2019) reflect a broad range of concerns about ethical issues of AI systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' EU AI HELG defined trustworthy AI as three components: lawful AI, ethical AI, and robust AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' The guide proposed four ethical principles: (1) Respect for human autonomy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' (2) Prevention of harm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' (3) Fairness;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' (4) Explicability, covering transparency, auditability, traceability, and explainability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' On the ground of these four principles, the guide further proposed a list of seven requirements: (1) Human agency and oversight;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' (2) Technical robustness and safety, including security, accuracy, reliability, and reproducibility;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' (3) Privacy and data governance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' (4) Transparency, including traceability, explainability, and communication;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' (5) Diversity, non-discrimination and fairness;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' (6) Societal and environmental wellbeing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' (7) Accountability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' The EU guide also discussed technical and non-technical methods to implement the requirements and proposed a list of questions used to assess the trustworthiness of AI systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' The ethics of AI studies what is right/wrong regarding the purpose and usages of AI, based on the profound effects AI systems bring to humans and human society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' The central principle of ethical AI is about using AI to benefit humans and human society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' In the upper part of Figure 5, ethicality is about the extent to which an AI system complies with this principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' As addressed by the AI100 2021 report (Littman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=', 2021), “AI systems and humans have complementary strengths;”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' thus, “combined, they can accomplish more than either alone.” Technically, it remains a challenge regarding how to team up humans and AI systems effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' For the fundamental ethical principles of trustworthy AI (EU AI HLEG, 2019), no doubt human-AI teaming is the right direction to go, not only for maximizing capability and performance but also for ethical consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Cooperability is about the extent to which an AI system facilitates and supports human-machine teaming for complex problem-solving, including the channels or methods to enhance interactions and collaborations between humans and machines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Cooperability covers controllability;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' the latter is about the ability of an AI system that allows humans to monitor and control the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Fairness is about whether an AI system fairly treats people of different groups regarding race, gender, age, cultural background, and others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Fairness received much attention in recent years when AI systems started being used for some life-changing scenarios, such as hiring decisions, financial credit evaluation, and judicial decisions (Mehrabi, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Fairness is a highly challenging topic for complex human societal reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Technically, fairness can be treated as bias-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' The recent deep learning progress reflected by GANs can be used to create real-like samples for balanced data in ML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Accountability is the availability and integrity of the identity of an entity that performed an operation in the AI system of concern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' In simple, it is who (human operators or autonomous components in an AI system) did what and when and the responsibility for that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' In the scenarios of a security incident, an accident, an error, or a system failure, accountability helps to identify the causes, make responsibility clear, and avoid future mistakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' 12 J Huang / Digital engineering transformation with trustworthy AI towards Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='0: emerging paradigm shifts Transparency is about the extent to which how the AI system operates is transparent to various stakeholders, such as operators, business partners, auditors, and users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Transparency is a basis for ensuring ethicality, fairness, and privacy and facilitating controllability and cooperability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Explainability is the ability to explain the outcomes and processes of an AI system to humans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' The major challenge is that in ML, neural networks have low-level coding for feature representation which is inherently hard to be explained in high-level knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' A very large number of parameters and complex structures of deep neural networks make the interpretability further harder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Traceability is about the ability to collect and document the provenance of the data used and produced, the models used and trained, and the operating processes of an AI system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Traceability is the basis for transparency and supports explainability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Reproducibility is about whether a model instance can be rebuilt with the same AI algorithm and data and whether an experiment or, more generally, a process running with an AI model can be reproduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Reproducibility is essential to science and engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Reliable AI reflects people’s concern about systems’ technical performance when more and more AI systems are used or embedded in a large engineering system in the 4IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Naturally, this aspect of concern brings the focused attention partially back to more classical trustworthiness properties of engineering systems but with a focus on AI systems or AI components and their impact on the larger systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' This article uses the term “dependability” to cover all expected properties (for trustworthiness) on the technical performance aspect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' The concept of dependability here is broader than what was defined in (Avizienis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=', 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Reliability is the probability of a system functioning without failure for a given period of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Maintainability is about how easy to make a system maintain healthy, updatable, and upgradeable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Resilience is about the ability of a system to restore it to a working state when it is damaged in situations such as natural disaster events or cyber-attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Safety is about how safe to humans a system is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' The broad use of AI components in engineering systems makes safety a significant concern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Accuracy is the measure of errors made by a model or system;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' in the context of ML, it is critical to have high accuracy for new data beyond the data used for training the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' In Systems Engineering, robustness is defined as “the degree to which a system or component can function correctly in the presence of invalid inputs or stressful environmental conditions”(ISO/IEC/IEEE, 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' In ML, robustness is about the stable outcomes in the presence of perturbations in inputs and could be measured with sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' In deep learning, lack of robustness is a critical cause for the possible deepfake by GANs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' The general meaning of robustness is similar to resilience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' In robustness, the perturbations are on a small scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Usability is how easy a system is to be used, and usability is beyond human-machine interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Poor usability can lead to failures in many other properties in modern systems, including security and safety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Security is defined as the well-known definition of the CIA triad: Confidentiality (Prevention of unauthorized access to the protected resources or disclosure of the protected information), Integrity (absence of unauthorized alterations), and Availability (Readiness for correct services for authorized users).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Some other interesting security properties can be defined on top of the CIA triad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' For example, authenticity is the integrity of information content and its provenance (origin).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Obviously, in the digital and connected environment, failures in security have broad impacts and can compromise other trustworthiness properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Privacy is another big concern in AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=" Since data is the fuel for AI systems, privacy concerns about whether the data gathering, holding, processing, usage, sharing, and governance respect people's privacy." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' As illustrated by Figure 5, the trustworthiness of AI systems is the new collection of concerns when the systems evolve into Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='0 systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Interestingly, on the other hand, the digitalization of engineering artifacts, processes, and enterprises in the 4IR could support achieving the expected trustworthiness properties of AI systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' In digital engineering transformation, it is an excellent opportunity for the engineering design community at large to bring the new capabilities of AI and the trustworthy AI principles together in various engineering systems design for human society to leverage the power of AI and at the same time to avoid or minimize the potential negative impacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' J Huang / Digital Engineering Transformation with Trustworthy AI towards Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='0: emerging paradigm shifts 13 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Open issues and discussion In previous sections, we have discussed the characteristics of 4IR and their fundamental impacts, digital engineering transformation - the manifestation of 4IR in engineering, and trustworthy AI - the leading technology in the digital transformation for I4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' This section continues the discussion on some open issues in the direction of 4IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' What is the pattern of paradigm shifts?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' As discussed in section 2, digitalization in the 4IR is leading to paradigm shifts in almost every aspect of our society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' What are the emerging new engineering paradigms?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Engineering uses scientific principles to design, build, and operate engineering systems for solving real-world problems or meeting humans’ needs in a specific field, thus essentially sharing the paradigms in scientific research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' A general pattern of scientific paradigm shifts can be illustrated as the following Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' A view of paradigm shifts, based on Kuhn’s structure of scientific revolutions (Kuhn, 1962).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' According to Kuhn’s structure of scientific revolutions (Kuhn, 1962), in the period of “normal research” or “normal science,” a scientific community uses a dominant paradigm to conduct research and produce the main body of knowledge in that field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Then, the dominant paradigm may have crises for its deficiencies or limitations facing the new observations and/or new problems coming from the changing environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' To battle the crises and meet the new need(s), the disciple enters a period of “extraordinary research” (as named by Kuhn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' In this period, new concepts, models, tools, methods, among others, will be created.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' If the new disciplinary components are incremental and can be integrated into the current paradigm, the paradigm will be updated, thus being a scientific evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Otherwise, the new components developed in “extraordinary research,” possibly together with the components from other disciplines, will contribute to the development of a new paradigm in a period of “pre-paradigm” (again, named by Kohn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' In the “pre-paradigm” period, one or multiple paradigms will be formed and compete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=" Finally, the most accepted paradigm(s) will become the discipline's new dominant paradigm(s)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' This “paradigm shift” (Kohn) is a scientific revolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' After the transformation, the discipline enters again “normal research” period and starts a new life cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' The above discussed pattern is revealed in the context of scientific research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' It appears to be general, as we treat a scientific paradigm as a knowledge system that evolves in the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' What is the fourth paradigm of scientific research?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Jim Gray had a vision that scientific research is shifting to the fourth paradigm (data-intensive paradigm) after the empirical, theoretical, and computational paradigms (Gray, 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Hey, Tansley, & Tolle, 2009), as illustrated in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' The empirical paradigm is characterized as finding and describing patterns based on the observations of real-world phenomena, and the theoretical paradigm is characterized as building Normal Research (Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='&Engr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=') Pre-paradigm Extraordinary Research Dominant paradigm Crisis Forming new paradigm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' competing among multiple paradigms Creating new concepts, models, methods, tools, … Changing environment New problems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' new opportunities (technological and societal) Paradigm shift (scientific revolution) Paradigm update (scientific evolution) New components from other disciplines 14 J Huang / Digital engineering transformation with trustworthy AI towards Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='0: emerging paradigm shifts theories on mathematical models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' The computational paradigm uses computer simulations to tackle the difficulty when the theoretical models become too complex to derive and prove propositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Four Paradigms in Scientific Research, Slide from (Gray, 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Hey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=', 2009) The fourth paradigm reflects the scientific research in the new research environment with the big data produced from instruments and simulations, high demands of computing for knowledge discovery from big data, and emerging demands of data publishing and research work reproducibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' In the fourth paradigm, scientific research significantly focuses on knowledge discovery from big data and reproducible models and datasets sharing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' It is essential to point out that a new paradigm does not mean entirely replacing existing paradigms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Instead, a new paradigm is established on the ground of existing paradigms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' A new paradigm represents the focus shift and the new approach to address new problems in a new environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Theoretical models are built on the ground of observations, and computational simulations are based on theoretical models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' The fourth paradigm enhances and unifies the empirical, theoretical, and computational paradigms, on the ground of an unprecedented big data environment and the new technology for knowledge discovery from big data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Inspired by Jim Gray’s fourth paradigm, with the fast growth of big data in the last decade, data science has emerged as a new discipline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' A NIST publication (NIST Big Data Public Working Group, 2019) pointed out that “data science is the fourth paradigm of science.” However, it has been heavily focused on knowledge discovery from data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' From the library and information science perspective, data curation and knowledge curation are among data science topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Still, the needs emerging from the fourth paradigm are beyond just traditional curation and go much further into knowledge and data sharing with traceability, explainability, accountability, reproducibility, and interoperability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' By the fourth paradigm, data-intensive science/engineering will emerge in various disciplines of science/engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' (Data-intensive engineering is the manifestation of data science in the engineering field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=') In this paradigm shift, data science is the common core, shared by many domain-specific science/engineering disciplines that need knowledge discovery from big data and knowledge sharing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' The combination of data science (the fourth paradigm) with each specific discipline will form many domain- specific data-intensive science/engineering disciplines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' What are the emerging new engineering paradigms?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' “Science is the systematic description of phenomena” (Richards, 1928).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Science focuses on discovering the essential laws of nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Engineering is “the application of science to the optimum conversion of the resources of nature to the uses of humankind” (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Smith, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' On the one hand,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' there is an intersection between science and engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' as engineering includes scientific research for discovering applied Science Paradigms Thousand years ago: science was empirical describing natural phenomena Last few hundred years: theoretical branch using models,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' generalizations 4元Gp Last few decades: 3 a computational branch simulating complex phenomena Today: data exploration (eScience) unify theory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' experiment,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' and simulation Data captured by instruments or generated by simulator Processed by software Information/knowledge stored in computer Scientist analyzes database/files using data management and statistics J Huang / Digital Engineering Transformation with Trustworthy AI towards Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='0: emerging paradigm shifts 15 knowledge for engineering purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' From this perspective, data-intensive engineering is the manifestation of the data-intensive paradigm of science in the digital and connected operating environment of the 4IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' On the other hand, engineering activities are beyond discovering new knowledge and focus more on the design, manufacturing (or construction), operating, and support of useful systems for human society, thus having much more complex interactions and relations with other systems in the operating environment, including human stakeholders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Given this complexity of engineering, the disruptive digital technologies, and the associated digital, connected, and smart environment in the 4IR, how should we conduct engineering?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' What are the new engineering paradigms in the 4IR?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' No doubt, the data-intensive paradigm of scientific research is a crucial one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' It includes both aspects of knowledge discovery from data and knowledge sharing (including data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' The current “data science” central theme has focused more on knowledge discovery from data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' While “digital engineering” in the US DoD’s vision is more from the perspectives of model-based engineering and sharing of models and data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Interestingly, model-based engineering is an effort, like the development of theoretical and computational paradigms in science, to introduce models in engineering workflows, including traditionally informal activities such as requirement elicitation, requirement representation, system concept modelling, system design, among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Certainly, modelling needs to be based on scientific knowledge as much as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Also, the availability and richness of data in the engineering environment and the fast progress of machine learning make it a powerful way to build models from data, leading to data-intensive engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' However, given the complex interactions and relations between engineering, human society, and the operating environment, the data-intensive paradigm (that focuses on scientific discovery) alone is insufficient for tackling the complexity of engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' The need for engineering paradigm shifts in the 4IR is driven by the engineering environment, the disruptive digital technologies, the associated higher social- economic needs, and the new challenging problems, such as the trust issues of AI systems (as discussed in Section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' In the landscape of the 4IR, the digital engineering transformation needs new concepts, models, tools, methods, theories, methodologies, technologies, and standards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' As discussed in section 2, the 4IR has three aspects of paradigm shifts: digitalizing everything of interest, provisioning ubiquitous machine intelligence, and big data and machine intelligence-centred business processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' On the other hand, digitalization and ubiquitous machine intelligence also triggered broad concerns and potential issues of trustworthiness, as discussed in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' To put the above-discussed pieces together, we could have a relatively clear big picture in the direction of Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='0, as illustrated in Figure 8, in which we need three interdependent essential building blocks for the digital transformation of engineering and industries: (1) digitalization of engineering;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' (2) leveraging ubiquitous machine intelligence;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' (3) building digital trust and security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Digitalization is a foundation to realize ubiquitous machine intelligence and needs to support digital trust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' For this consideration, to digitalize a thing of interest (which could be an object or a process), we need the following four essential components: (1) Digital representation of the thing of interest in a standard form with well-defined semantics to make it universally accessible by different types of machines on different platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' (2) A unique identifier of the thing of interest, which is a necessary component for traceability, verifiability, accountability, and explainability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' (3) Associated metadata, such as provenance, in a standard form with well-defined semantics to enable the use of digital technologies to manipulate and operate the thing automatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' (4) Verifiable association of the unique identifier, the digital representation, and the metadata with the thing of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' This association ensures the authenticity and integrity of the digital artifact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' For the digitalization of engineering, basically, we need (i) to digitalize engineering artifacts, engineering processes, and enterprises;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' (ii) enable the sharing and interoperability of digitalized artifacts across the engineering lifecycle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' (iii) to develop digital model-based engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Virtualization is a view from the perspective of operating with digitalized artifacts and processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' 16 J Huang / Digital engineering transformation with trustworthy AI towards Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='0: emerging paradigm shifts Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Three essential building blocks for digital engineering towards Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='0: digitalization of engineering, leveraging ubiquitous machine intelligence, and building digital trust & security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Digitalization needs to be realized with digital trust in design and supported by ubiquitous machine intelligence among digital technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Naturally, in this direction, what are enabling technologies for digitally modelling engineering artifacts, processes, and enterprises?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Semantic web technology (a branch of AI) (Berners-Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=', 2006, 2001), RFID, IoT, and blockchain are essential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Also, directly related, intensive research on enterprise modeling and enterprise integration has been conducted since the 1990s and can be used to support model sharing across boundaries (Chen, Doumeingts, & Vernadat, 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Fox & Gruninger, 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Fox & Huang, 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Goranson, 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Vernadat, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Enterprise modelling combined with digital identity management could form a basis for digitalizing engineering artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Ontologies play an essential role in the digitalization of engineering with respect to artifacts, entities, organizations, various engineering activities and processes, among others (Ahmed, Kim, & Wallace, 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Demoly, Kim, & Horváth, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Dimassi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Fox, Barbuceanu, Gruninger, & Lin, 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Gruninger & Fox, 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Kim, Fox, & Grüninger, 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Kim, Manley, & Yang, 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Sirin, Coatanéa, Yannou, & Landel, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' The realization of ubiquitous machine intelligence, the second aspect of paradigm shifts in the 4IR, is on the top of digitalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' With digitalized engineering artifacts and processes, we can apply ubiquitous machine intelligence in engineering to design, build, and support smart products, provide smart services, build trustworthy supply chains, and conduct engineering intelligently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Consider the following examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Machine learning has been broadly applied in additive manufacturing (Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Jin, Zhang, Demir, & Gu, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Meng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Wang, Tan, Tor, & Lim, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' In the digital, smart, and connected environment, unprecedented big data provide rich information for using AI in engineering operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Condition-based maintenance or predictive maintenance are typical scenarios where AI&ML can be leveraged (Black, Richmond, & Kolios, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Carvalho et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Engineering involves many sequential decision making;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' with rich information about engineering systems’ status, reinforcement learning can be used to learn from simulations, experiments, routine operations, and generally experience for optimization, such as in mesh generation (Pan, Huang, Cheng, & Zeng, 2022), in manufacturing (Su, Huang, Adams, Chang, & Beling, 2022), for engineering design (Dworschak, Dietze, Wittmann, Schleich, & Wartzack, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Machine Digitalization of Engineering Ubiquitous Machine Intelligence Digital Trust & Security Digital asset certification & verification Digital asset access control Digital trust mechanisms (Blockchain, IdM, digital C&V, traceability, transparency, accountability, …) Digital asset governance … Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='0 Engineering & Industries in the world Systems Engineering Manufacturing systems & networks Supply-chain,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Logistics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Transportation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' … Healthcare … Digitalization Virtualization Interoperability Digitalizing engineering artifacts and processes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' … Digital model-based systems engineering Digital models and data sharing in engineering lifecycle … Smart factories Autonomous vehicles Smart cities Smart homes … AI & machine learning Internet of Things Cloud computing & Big Data Augmented/Virtual Realty … J Huang / Digital Engineering Transformation with Trustworthy AI towards Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='0: emerging paradigm shifts 17 learning can also be applied to even traditionally labour-intensive and time-consuming requirement elicitation process (Cheligeer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Mokammel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Essentially, engineering systems design starts with the environment (Zeng, 2004, 2015, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' In the new engineering environment of the 4IR, many engineering systems become cyber-physical-social smart systems, and it is a challenging issue regarding how to design such systems (Horváth, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Tavčar & Horvath, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Digitalization brings to us not only advantages but also new issues, most significantly, trust and security issues with digital artifacts and those digital technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Some examples of those issues include how do we ensure the authenticity and integrity of digital artifacts, what are policies regarding who can access what, when to access and where to access, and how do we ensure traceability, transparency, accountability, and reproducibility, among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' We must develop proper digital mechanisms to address those issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' How could we achieve secure information sharing in digital engineering?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' As an example, a security policy integrating role-based access control and attribute-based access control (including security classification- based mandatory access control) (Huang, Nicol, Bobba, & Huh, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Jin, Sandhu, & Krishnan, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Servos & Osborn, 2017) can help.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Digital identity management and digital trust mechanisms (Huang & Nicol, 2013), particularly blockchain (Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=', 2020), can support trust management of digital artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Digital archives are critical to ensure data integrity and proof of existence (Vigil, Cabarcas, Buchmann, & Huang, 2013), and blockchain facilitates a new approach to distributed digital archives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Scientific computing integrity (Huang, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Peisert, Cybenko, & Jajodia, 2015) can be further developed to support trusted engineering workflows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Digital Engineering is an emerging form of engineering in the digital revolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Many issues and questions there need to be researched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Just name a few, in digitalization, should the standard forms for digital representation and augmentation be supported by a centralized standardization (such as creating international standards) or a distributed evolutionary standardization (such as many ontologies competing to be standards at a fine-grained level and evolving gradually and naturally)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' How do we achieve trustworthy AI systems in 4IR?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' What are digital trust mechanisms for digital engineering?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Many trust mechanisms used in cloud service (Huang & Nicol, 2013) are applicable to digitalized products, systems, and services, while still, what new mechanisms should be introduced?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Should be the sharing of digital engineering models and data in a centralized way or distributed way?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' There are many issues and challenges ahead as we conduct engineering in a very different new digital environment (Coatanéa, Nagarajan, Panicker, & Mokhtarian, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Horváth, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' What is next?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Earlier in section 2, we discussed the journey of four industrial revolutions in human history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Potentially, what is next?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Given the broad and profound impacts of digital engineering transformation on human society, a farther vision will help us to develop long-lasting engineering paradigm(s) and to better design and develop digital engineering systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' 9 illustrates the journey of industrial revolutions from passively following nature to actively exploiting nature and possibly achieving harmony between human society and nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Let us start the discussion with the physical limits of computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Computational power is the core of ubiquitous machine intelligence, the defining power we gain from the 4IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' However, with Moore’s law reaching its limit, the chips for CPUs have hit the ceilings of size and performance, particularly power consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' The High- Performance Computing (HPC) community has been striking hard to achieve an exascale computing system with power consumption within 20MW (Lucas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' In the most recent Top500 List issued in June 2022, the current fastest supercomputer frontier is the first exascale machine with 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='1 ExaFlop/s at 21MW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' What we hope to beat Moore’s law are new computational technologies emerging in the horizon, such as new semiconductor materials, Quantum Computing, and DNA computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Data centres are the most significant contributors to today’s computational power, and data centres also consume a tremendous amount of energy and have significant environmental footprints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' For example, in the US, data centres consumed 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='8% of electricity in 2014 and were responsible for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='5% of greenhouse emissions in 2018 (Siddik, Shehabi, & Marston, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' It has become more significant regarding how to 18 J Huang / Digital engineering transformation with trustworthy AI towards Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='0: emerging paradigm shifts improve data centres’ sustainability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Microsoft’s Project Natick conducted a two years experiment and found that subsea data centres are feasible, reliable, more energy-efficient, and environmentally sound (Microsoft Research, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' The journey of human society through the industrial revolutions In a global context, since the first industrial revolution, economic growth has been associated with high environmental costs, including environmental pollution, overexploitation of natural resources, deterioration of ecosystems, loss of biodiversity, and global climate change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' The problem is challenging because entities individually lack the motivation to behave environmental-friendly in the market economic mechanism due to environmental externalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' “Sustainable development is development that meets the needs of the present without compromising the ability of future generations to meet their own needs.” (UN WCED, 1987) It has become a consensus of the international community (IPCC, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' United Nations, 1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' According to the IPCC AR6 report (IPCC, 2022), global warming is speeding up - The increase in global surface temperature in 2011–2020 is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='09 [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='95 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='20] °C higher than in 1850–1900, compared to the increase of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='19 [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='16 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='22] °C in 2003–2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' The trend of increase will reach or exceed 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='5°C in the near term, even for the very low greenhouse gas emissions scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Climate change has led to some irreversible impacts on the earth and is approaching a tipping point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' This cross-century challenge has become urgent and needs the whole world to act collectively and immediately before it is too late.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' This global campaign appears geared towards the next industrial revolution for renewable natural resources, particularly renewable energy, reusable materials, and eco-environmental sound economies to achieve harmony between human activities and nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' In this direction, science, such as disciplines in biological and environmental sciences, energy science, and material science, among others, will be a scientific foundation to reveal facts and provide knowledge about sustainability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Engineering will bring scientific knowledge into the real world by developing technologies and designing innovative solutions for sustainability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Sustainability is a highly complex problem because it is across disciplinary domains, across industry sectors, across regional economies, across cultures, across human groups, across human generations, and across human society and natural systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' The solutions for sustainability need to be coordinated, comprehensive, and systematic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Systems science and engineering will play a unique role in applying systems thinking and designing systematic mechanisms and solutions to address the complex problem of sustainability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' With sustainability in mind, in today’s efforts for digital engineering transformation, we should take into account the need for sustainability in digital engineering systems design and development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' For example, product provenance can help reuse and recycle after retirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Particularly, it is critical to consider the sustainability of new digital technologies (Colorado, Velásquez, & Monteiro, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Kellens et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Paris, Mokhtarian, Coatanéa, Museau, & Ituarte, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' The consideration of sustainability in digitalization will give the emerging digital engineering paradigm(s) a long vision and better address future needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' 2 Pre-industrial I1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='0 I2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='0 I3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='0 I4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='0 Mechanization 1IR 2IR Electrification 3IR Informatization 4IR Digitalization 5IR Sustainablization?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Assembling lines;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Mass Production Mechanized factories Automation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Information-centered I5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Sustainable Development Digitalized & connected;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Ubiquitous Intelligence;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Intelligence-centered INDUSTRY 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='0 NDUSTRY 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='0 INDUSTRY 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='0 INDUSTRY 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='0 Mechanization,steam Massproduction, Automation,computers Cyber Physical Systems, power,weavingloom assembly line, and electronics internet of things,networks electrical energy 回 J Huang / Digital Engineering Transformation with Trustworthy AI towards Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='0: emerging paradigm shifts 19 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Concluding Remarks This article investigated the opportunities and uncertainties in digital engineering transformation and discussed the open issues about the emerging engineering paradigm shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' From investigating the pattern appeared in industrial revolutions, this article revealed that ubiquitous machine intelligence is the defining power brought by the fourth industrial revolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Digitalization is commonly recognized as the main theme of the 4IR and is also the foundation and necessary condition for the realization of ubiquitous machine intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Digital engineering, the digitalization of engineering, is at the core of the fourth industrial revolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' With the broad applications of ubiquitous machine intelligence, many innovative cyber- physical-social smart systems will appear, and traditional engineering systems are also becoming digital smart and connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' The promising opportunities are also accompanied with uncertainties represented by trustworthiness concerns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Given AI as a critical enabling technology in the 4IR, trustworthy AI is a crucial field to ensure systems’ trustworthiness in Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' In the digital smart and connected environment, Digital Engineering needs new concepts, models, tools, methods, theories, methodologies, technologies, and standards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' The emerging engineering paradigm shifts include but is beyond the data-intensive paradigm (or data-intensive engineering, the counterpart of data science).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Digital engineering transformation towards Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='0 has three essential building blocks: (1) digitalization, (2) leveraging ubiquitous machine intelligence, and (3) building digital trust and security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Digitalization is much beyond digitization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Digitization aims to make artifacts machine-readable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' digitalization aims to make the things of interest machine-understandable and virtually operatable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Digitalization of a thing of interest should include: (1) digital representation, (2) a unique identifier, (3) metadata including provenance, and (4) a verifiable association of the thing with the above three components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' The remarkable progress, disruptive impacts, and fast growth of real-world AI applications in the digital and connected environment have triggered concerns and research about the trustworthiness of AI systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' The engineering design community at large is facing an excellent opportunity to bring the new capabilities of ubiquitous machine intelligence and trustworthy AI principles, as well as digital trust, together in various engineering systems design to ensure the trustworthiness of systems in Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Digital engineering transformation is a crucial process for engineering paradigm shifts in the 4IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' Digital engineering, at the core of the 4IR, is an exciting and broad field to explore and will further lead to “the transformation of entire systems, across (and within) countries, companies, industries and society as a whole” (Schwab, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfB_pj/content/2301.00951v1.pdf'} +page_content=' References AAAI Presidential Panel on Long-Term AI Future.' metadata={'source': 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N. Caballero1,3, Ce Cai14, Mao-Zheng Chen15, Zi-Gao +Dai16, A. Esamdin15, Heng-Qian Gan2, Jin-Lin Han2, Long-Fei Hao17, Yu-Xiang Huang17, +Peng Jiang2, Cheng-Kui Li14, Di Li2,18, Hui Li2, Xin-Qiao Li14, Zhi-Xuan Li17, Zhi-Yong +Liu15, Rui Luo19, Yun-Peng Men20, Chen-Hui Niu2, Wen-Xi Peng14, Lei Qian2, Li-Ming +Song14, Jing-Hai Sun2, Fa-Yin Wang21, Min Wang17, Na Wang15, Wei-Yang Wang3, Xue-Feng +Wu10, Shuo Xiao14, Shao-Lin Xiong14, Yong-Hua Xu17, Ren-Xin Xu1,3,22, Jun Yang21, Xuan +Yang10, Rui Yao2, Qi-Bin Yi14, You-Ling Yue2, Dong-Jun Yu2, Wen-Fei Yu23, Jian-Ping +Yuan15, Bin-Bin Zhang21,24, Song-Bo Zhang10, Shuang-Nan Zhang14, Yi Zhao14, Wei-Kang +Zheng12, Yan Zhu2, Jin-Hang Zou21,25 +1 Kavli Institute for Astronomy and Astrophysics,Peking University, Beijing 100871, China, +2 National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100101, China, +3 Department of Astronomy, Peking University, Beijing 100871, China, +4 University of Chinese Academy of Sciences, Chinese Academy of Sciences, Beijing 100049, China, +5 Department of Particle Physics and Astrophysics, Weizmann Institute of Science, Rehovot 76100, Israel, +6 Nevada Center for Astrophysics, University of Nevada, Las Vegas, NV 89154, USA, +7 Department of Physics and Astronomy, University of Nevada, Las Vegas, NV 89154, USA, +8 Department of Physics & Astronomy, University of Iowa, Iowa City, IA 52242, USA, +9 Zhejiang Lab, Hangzhou, Zhejiang 311121, China, +10 Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing 210008, China, +11 TAPIR, Walter Burke Institute for Theoretical Physics, Mail Code 350-17, Caltech, Pasadena, CA 91125, USA, +12 Department of Astronomy, University of California at Berkeley,Berkeley, CA 94720, USA, +13 South-Western Institute For Astronomy Research, Yunnan University, Yunnan 650504, China, +14 Key laboratory of Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences, Beijing +100049, China, +15 Xinjiang Astronomical Observatory, Chinese Academy of Sciences, Urumqi 830011, China, +16 University of Science and Technology of China, Anhui 230026, China, +17 Yunnan Observatories, Chinese Academy of Sciences, Kunming 650216, China, +18 Guizhou Normal University, Guiyang 550001, China, +19 CSIRO Space and Astronomy, Epping, NSW 1710, Australia, +20 Max-Planck institut f¨𝑢r Radioastronomie, Auf Dem H¨𝑢gel, Bonn, 53121, Germany, +21 School of Astronomy and Space Science, Nanjing University, Nanjing 210093, China, +1 +arXiv:2301.01429v1 [astro-ph.HE] 4 Jan 2023 + +22 State Key Laboratory of Nuclear Physics and Technology, School of Physics, Peking University, +Beijing 100871, China, +23Shanghai Astronomical Observatory, Chinese Academy of Sciences, Shanaghai 200030, China, +24Key Laboratory of Modern Astronomy and Astrophysics (Nanjing University), Ministry of Education, China, +25 College of Physics, Hebei Normal University, Shijiazhuang 050024, China, +January 5, 2023 +Abstract +Fast radio bursts (FRBs) are highly dispersed millisecond-duration radio bursts [1,2], of which the physical +origin is still not fully understood. FRB 20201124A is one of the most actively repeating FRBs. In this paper, +we present the collection of 1863 burst dynamic spectra of FRB 20201124A measured with the Five-hundred- +meter Aperture Spherical radio Telescope (FAST). The current collection, taken from the observation during +the FRB active phase from April to June 2021, is the largest burst sample detected in any FRB so far. The +standard PSRFITs format is adopted, including dynamic spectra of the burst, and the time information of +the dynamic spectra, in addition, mask files help readers to identify the pulse positions are also provided. +The dataset is available in Science Data Bank, with the link https://www.doi.org/10.57760/sciencedb. +j00113.00076. +Keywords: Fast radio burst, FAST +1. Introduction +Fast radio bursts (FRBs) are bright, millisecond duration pulses with dispersion measures (DM) mostly +well in excess of Galactic values, since first discovered in 2007 [1], more than 800 FRBs have been detected so far +and 27 of them can emit repeating bursts [3,4] (https://www.herta-experiment.org/frbstats/catalogue). +Currently, 19 FRBs have been localized to host galaxies (https://frbhosts.org/). Although the physical +mechanism of FRBs still remains unknown, FRB 200428 [5–8] produced by Galactic magnetar SGR J1935+2154 +suggests that some of the FRBs can be emitted by magnetars [2,9]. Among all the FRBs, FRB 20201124A, which +was discovered by CHIME [10], has been frequently studied recently. Its radio bursts show rich pulse structures +[11,12]. Through dynamic spectra, researchers investigated the scintillation time-scale of FRB 20201124A [13]. +Efforts had also been made to localize its host galaxy [14–17]. +Dynamic spectra record the FRB intensity as a function of time and frequency. Dynamic spectra contain +information of FRB intrinsic emission properties as well as density fluctuation of interstellar and inter-galactic +medium. We noted that there is lack of a systematic collection of dynamic spectra for FRBs. In this paper, we +present the dynamic spectra data of FRB 20201124A which covers 1863 pulses detected by our team. +∗E-mail: kjlee@pku.edu.cn +†Email: bing.zhang@unlv.edu +‡Email: zhuww@nao.cas.cn +2 + +2. Observation, data acquisition, and analysis +We used the Five-hundred-meter Aperture Spherical radio Telescope (FAST) [18] to monitor FRB 20201124A +from April to June in 2021. The FAST 19-beam Pulsar back-end covers 1.0-1.5 GHz in frequency band and has +𝑎 system temperature about 20 to 25 K [19]. The data were recorded using the digital back-end based on the +Re-configurable Open Architecture Computing Hardware-2 (Roach2) board [20] with temporal resolution of +49.152 𝜇s or 196.608 𝜇s and frequency resolutions of 122.07 kHz. +Our data processing contains two major steps, searching for single pulses and post processing to form the +dynamic spectra. Firstly, we searched for the FRB candidates offline with software package TransientX. +Frequency channels affected by radio frequency interference (RFI) were removed. The data were de-dispersed +in the dispersion measure (DM) range of 380-440 cm−3 pc with a step of 0.1 cm−3 pc since FRB 20201124A is a +known repeater. The pulse width is searched from 0.1 ms to 100 ms in the box-car-shaped matched filter. 3364 +candidates with a S/N threshold larger than 7 were plotted and visually inspected. +In the post processing phase, we used the software package DMPhase to further refine the DM. The +DMPhase use the Fourier-domain method, where DM is found by maximising the time derivative of normalized +”intensity”. To measure the intensity, the polarisation calibration is then performed with software package +PSRCHIVE [21]. We adopted the single axis model in polarisation calibration, where the differential gain +and phase between the two polarisation channels are calibrated with the injected noise signal. To reduce the +dynamic spectra to a manageable size, we integrate over time and frequency to reduce the resolution. The +frequency and time resolutions of the final dynamic spectra are ≈ 1.0 MHz and ≈ 0.2 ms, respectively. We +store the data in the PSRFITs [21] format, which is widely used in the community of pulsar astronomy. +3. Data format and contents of the library +The psrfits format is based on the Flexible Image Transport System (FITS)(https://fits.gsfc.nasa. +gov/) [22]. According to FITS standards, a psrfits file consists of a primary header-data unit (HDU) followed +by a series of extension HDUs [21]. As for our data, the primary HDU contains basic information such as +telescope name and its location, source location, observation time and etc. Four extension HDUs, which are +in a binary table format, contain specific information related to the observation: processing history, pulsar +ephemeris, tempo2 predictor and the pulsar data. Notice that there are several psrfits files contain more than +one burst because the interval between their TOAs (time of arrivals) is quiet small. +We associate each pulse with a mask file. The mask file, formatted in plain ascii file, contains two rows +of data. The first row consists two integer numbers corresponding to the boundary of pulse on-phase in the +profile. The second row of mask file shows where the baseline lies in. Figure 1 shows the dynamic spectrum of +pulse No.12 as an example. +4. Statistics of data properties +Our detection threshold was a signal-to-noise ratio 𝑆/𝑁 > 7, and 1103 bright bursts reached 𝑆/𝑁 > 30 +among a total of 1863 detected bursts. The left panel of Figure 2 shows the 𝑆/𝑁 distribution of all detected +3 + +Figure 1: Example of pulse profile (upper panel) and dynamic spectra (lower panel). Purple box and blue area +in the pulse profile show the baseline and pulse on-phase definition. White strips in the dynamical spectra +indicate the removed channels due to the RFI. +4 + +Normalized Flux +1.5 +1.4 +Frequency (GHz) +1.3 +1.2 +1.1 +1.0 +0 +20 +40 +60 +80 +100 +Time (ms)Figure 2: Left: The 𝑆/𝑁 distribution of 1863 pulses. Right: The distribution of RFI zapping for all data. +bursts. The right panel of Figure 2 shows the distribution of removed channel in frequency band. Usually, a +few percent frequency channels had been removed due to the RFI. +The sample completeness was determined with the following method. We simulated 10,000 mock bursts +with Gaussian profile and bandpass matching the detected distributions. We then randomly injected the mock +bursts into the original FAST data when no FRB was detected. The mock burst injected data are then fed to +our burst-searching pipeline to compute the detection rate. The procedure shows that the fluence threshold +achieving the 95% detection probability with 𝑆/𝑁 ≥ 7 is 53 mJy ms [11]. +Parameters of each burst (including burst MJD, 𝑆/𝑁, DM, etc) are available in the section Data avail- +ability of Ref [11]. +5. Conclusion +In this work, we present a collection of dynamic spectra for 1863 FAST-detected radio bursts of FRB 20201124A +during April to June in 2021. This is the largest burst sample detected in any FRB so far. +The signal of FRB 20201124A is highly polarised [11]. Our dynamic spectra is polarisation calibrated. +Previous study shows that 0.5% polarisation fidelity can be achieved with the current calibration method [3]. +The current data set is of high S/N, where 5%, 30% and 67% data had S/N≥ 560.63, 116.02, and 23.85, +respectively. Simulation is used to determine the completeness of burst detection, where 95% completeness +fluence threshold is 53 mJy ms. +For each burst, we provide one PSRFITs file and one mask file. We provide the total intensity data in +PSRFITs format, and mask file in ascii format which labels the burst. +5 + +1.5 +103 +1.4 +102 , + +Count +1.2 +101 +1.1 +100 , +1.0 +500 +1000 +1500 +2000 +2500 +0 +250 +500 +750 +1000 +1250 +1500 +1750 +S/N +Pulse Number6. Data availability and related softwares +The data that support the findings of this study are openly available in Science Data Bank at https: +//www.doi.org/10.57760/sciencedb.j00113.00076. The related software can be found the corresponding +repositories. +TransientX: https://github.com/ypmen/TransientX +DMPhase: https://www.github.com/DanieleMichilli/DM˙phase +psrchive: http://psrchive.sourceforge.net +Acknowledgments +This work made use of data from the FAST. FAST is a Chinesenational megascience facility, built and +operated by the NationalAstronomical Observatories, Chinese Academy of Sciences. We acknowledge the use of +public data from the Fermi Science Support Center (FSSC). This work is supported by National SKA Program +of China (2020SKA0120100, 2020SKA0120200), Natural Science Foundation of China (12041304, 11873067, +11988101, 12041303, 11725313, 11725314, 11833003, 12003028, 12041306, 12103089, U2031209, U2038105, +U1831207), National Program on Key Research and Development Project (2019YFA0405100, 2017YFA0402602, +2018YFA0404204, 2016YFA0400801), Key Research Program of the CAS (QYZDJ-SSW-SLH021), Natural Sci- +ence Foundation of Jiangsu Province (BK20211000), Cultivation Project for FAST Scientific Payoff and Re- +search Achievement of CAMS-CAS, the Strategic Priority Research Program on Space Science, the Western +Light Youth Project of Chinese Academy of Sciences, the Chinese Academy of Sciences (grants XDA15360000, +XDA15052700, XDB23040400), funding from the Max-Planck Partner Group, the science research grants from +the China Manned Space Project (CMS-CSST-2021-B11,NO. CMS-CSST-2021-A11), and PKU development +grant 7101502590. KJL acknowledge support from the XPLORER PRIZE. BBZ is supported by Fundamental +Research Funds for the Central Universities (14380046), and the Program for Innovative Talents, Entrepreneur +in Jiangsu. +6 + +References +[1] Lorimer D R, Bailes M,McLaughlin M A, Narkevic D J andCrawford F 2007Science 318,777–780. +[2] Zhang B 2020Nature 587,45–53. +[3] Luo R, Wang B J, Men Y P, et al. 2020Nature 586,693–696. +[4] Niu C H, Aggarwal K, Li D, et al. 2022Nature 606,873–877. +[5] Scholz P and Chime/Frb Collaboration2020The Astronomer’s Telegram13681, 1. +[6] CHIME/FRB Collaboration, Andersen B C, Bandura K M, et al.2020Nature 587,54–58. +[7] Bochenek C, Kulkarni S, Ravi V, McKenna D, Hallinan G and Belov K 2020 The Astronomer’s Tele- +gram13684, 1. +[8] Bochenek C D, Ravi V, Belov K V, et al. 2020Nature 587,59–62. +[9] Lin L, Zhang C F, Wang P, et al. 2020Nature 587,63–65. +[10] Chime/FRB Collabortion 2021The Astronomer’s Telegram14497, 1. +[11] Xu H, Niu J R, Chen P, et al. 2022Nature 609,685. +[12] Marthi V R, Bethapudi S, Main R A, et al. 2022MNRAS 509,2209–2219. +[13] Main R A, Hilmarsson G H, Marthi V R, et al. 2022MNRAS 509,3172–3180. +[14] Day C K, Bhandari S,Deller A T, Shannon R M and Moss V A 2021The Astronomer’s Telegram14515, 1. +[15] Kilpatrick C D, Fong W, Prochaska J X, et al. 2021The Astronomer’s Telegram14516, 1. +[16] Xu H, Niu J R, Lee K J, et al. 2021The Astronomer’s Telegram14518, 1. +[17] Wharton R, Bethapudi S, Marthi, V, et al. 2021The Astronomer’s Telegram14538, 1. +[18] Jiang P, Yue Y L, Gan H Q, et al. 2019Science China Physics, Mechanics, andAstronomy 62, 959502. +[19] Jiang P, Tang N Y, Hou L G, et al. 2020Research in Astronomy and Astrophysics20, 064. +[20] Hickish J, Abdurashidova Z, Ali Z, et al. 2016Journal of Astronomical Instrumentation05, 1641001. +[21] Hotan A W, van Straten W and Manchester R N 2004Publ. 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' Shijiazhuang 050024,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' China,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' January 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' 2023 Abstract Fast radio bursts (FRBs) are highly dispersed millisecond-duration radio bursts [1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content='2],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' of which the physical origin is still not fully understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' FRB 20201124A is one of the most actively repeating FRBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' In this paper, we present the collection of 1863 burst dynamic spectra of FRB 20201124A measured with the Five-hundred- meter Aperture Spherical radio Telescope (FAST).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' The current collection, taken from the observation during the FRB active phase from April to June 2021, is the largest burst sample detected in any FRB so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' The standard PSRFITs format is adopted, including dynamic spectra of the burst, and the time information of the dynamic spectra, in addition, mask files help readers to identify the pulse positions are also provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' The dataset is available in Science Data Bank, with the link https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content='doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content='57760/sciencedb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' j00113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content='00076.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' Keywords: Fast radio burst, FAST 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' Introduction Fast radio bursts (FRBs) are bright, millisecond duration pulses with dispersion measures (DM) mostly well in excess of Galactic values, since first discovered in 2007 [1], more than 800 FRBs have been detected so far and 27 of them can emit repeating bursts [3,4] (https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content='herta-experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content='org/frbstats/catalogue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' Currently, 19 FRBs have been localized to host galaxies (https://frbhosts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content='org/).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' Although the physical mechanism of FRBs still remains unknown, FRB 200428 [5–8] produced by Galactic magnetar SGR J1935+2154 suggests that some of the FRBs can be emitted by magnetars [2,9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' Among all the FRBs, FRB 20201124A, which was discovered by CHIME [10], has been frequently studied recently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' Its radio bursts show rich pulse structures [11,12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' Through dynamic spectra, researchers investigated the scintillation time-scale of FRB 20201124A [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' Efforts had also been made to localize its host galaxy [14–17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' Dynamic spectra record the FRB intensity as a function of time and frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' Dynamic spectra contain information of FRB intrinsic emission properties as well as density fluctuation of interstellar and inter-galactic medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' We noted that there is lack of a systematic collection of dynamic spectra for FRBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' In this paper, we present the dynamic spectra data of FRB 20201124A which covers 1863 pulses detected by our team.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' ∗E-mail: kjlee@pku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content='cn †Email: bing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content='zhang@unlv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content='edu ‡Email: zhuww@nao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content='cas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content='cn 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' Observation, data acquisition, and analysis We used the Five-hundred-meter Aperture Spherical radio Telescope (FAST) [18] to monitor FRB 20201124A from April to June in 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' The FAST 19-beam Pulsar back-end covers 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content='0-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content='5 GHz in frequency band and has 𝑎 system temperature about 20 to 25 K [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' The data were recorded using the digital back-end based on the Re-configurable Open Architecture Computing Hardware-2 (Roach2) board [20] with temporal resolution of 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content='152 𝜇s or 196.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content='608 𝜇s and frequency resolutions of 122.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content='07 kHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' Our data processing contains two major steps, searching for single pulses and post processing to form the dynamic spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' Firstly, we searched for the FRB candidates offline with software package TransientX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' Frequency channels affected by radio frequency interference (RFI) were removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' The data were de-dispersed in the dispersion measure (DM) range of 380-440 cm−3 pc with a step of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content='1 cm−3 pc since FRB 20201124A is a known repeater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' The pulse width is searched from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content='1 ms to 100 ms in the box-car-shaped matched filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' 3364 candidates with a S/N threshold larger than 7 were plotted and visually inspected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' In the post processing phase, we used the software package DMPhase to further refine the DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' The DMPhase use the Fourier-domain method, where DM is found by maximising the time derivative of normalized ”intensity”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' To measure the intensity, the polarisation calibration is then performed with software package PSRCHIVE [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' We adopted the single axis model in polarisation calibration, where the differential gain and phase between the two polarisation channels are calibrated with the injected noise signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' To reduce the dynamic spectra to a manageable size, we integrate over time and frequency to reduce the resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' The frequency and time resolutions of the final dynamic spectra are ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content='0 MHz and ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content='2 ms, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' We store the data in the PSRFITs [21] format, which is widely used in the community of pulsar astronomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' Data format and contents of the library The psrfits format is based on the Flexible Image Transport System (FITS)(https://fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content='gsfc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content='nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' gov/) [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' According to FITS standards, a psrfits file consists of a primary header-data unit (HDU) followed by a series of extension HDUs [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' As for our data, the primary HDU contains basic information such as telescope name and its location, source location, observation time and etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' Four extension HDUs, which are in a binary table format, contain specific information related to the observation: processing history, pulsar ephemeris, tempo2 predictor and the pulsar data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' Notice that there are several psrfits files contain more than one burst because the interval between their TOAs (time of arrivals) is quiet small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' We associate each pulse with a mask file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' The mask file, formatted in plain ascii file, contains two rows of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' The first row consists two integer numbers corresponding to the boundary of pulse on-phase in the profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' The second row of mask file shows where the baseline lies in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' Figure 1 shows the dynamic spectrum of pulse No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content='12 as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' Statistics of data properties Our detection threshold was a signal-to-noise ratio 𝑆/𝑁 > 7, and 1103 bright bursts reached 𝑆/𝑁 > 30 among a total of 1863 detected bursts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' The left panel of Figure 2 shows the 𝑆/𝑁 distribution of all detected 3 Figure 1: Example of pulse profile (upper panel) and dynamic spectra (lower panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' Purple box and blue area in the pulse profile show the baseline and pulse on-phase definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' White strips in the dynamical spectra indicate the removed channels due to the RFI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' 4 Normalized Flux 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content='4 Frequency (GHz) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content='0 0 20 40 60 80 100 Time (ms)Figure 2: Left: The 𝑆/𝑁 distribution of 1863 pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' Right: The distribution of RFI zapping for all data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' bursts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' The right panel of Figure 2 shows the distribution of removed channel in frequency band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' Usually, a few percent frequency channels had been removed due to the RFI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' The sample completeness was determined with the following method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' We simulated 10,000 mock bursts with Gaussian profile and bandpass matching the detected distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' We then randomly injected the mock bursts into the original FAST data when no FRB was detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' The mock burst injected data are then fed to our burst-searching pipeline to compute the detection rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' The procedure shows that the fluence threshold achieving the 95% detection probability with 𝑆/𝑁 ≥ 7 is 53 mJy ms [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' Parameters of each burst (including burst MJD, 𝑆/𝑁, DM, etc) are available in the section Data avail- ability of Ref [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' Conclusion In this work, we present a collection of dynamic spectra for 1863 FAST-detected radio bursts of FRB 20201124A during April to June in 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' This is the largest burst sample detected in any FRB so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' The signal of FRB 20201124A is highly polarised [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' Our dynamic spectra is polarisation calibrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' Previous study shows that 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content='5% polarisation fidelity can be achieved with the current calibration method [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' The current data set is of high S/N, where 5%, 30% and 67% data had S/N≥ 560.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content='63, 116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content='02, and 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content='85, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' Simulation is used to determine the completeness of burst detection, where 95% completeness fluence threshold is 53 mJy ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' For each burst, we provide one PSRFITs file and one mask file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' We provide the total intensity data in PSRFITs format, and mask file in ascii format which labels the burst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content='5 103 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content='4 102 , Count 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content='2 101 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content='1 100 , 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content='0 500 1000 1500 2000 2500 0 250 500 750 1000 1250 1500 1750 S/N Pulse Number6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' Data availability and related softwares The data that support the findings of this study are openly available in Science Data Bank at https: //www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content='doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content='57760/sciencedb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content='j00113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content='00076.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' The related software can be found the corresponding repositories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' TransientX: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content='com/ypmen/TransientX DMPhase: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content='com/DanieleMichilli/DM˙phase psrchive: http://psrchive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content='sourceforge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content='net Acknowledgments This work made use of data from the FAST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' FAST is a Chinesenational megascience facility, built and operated by the NationalAstronomical Observatories, Chinese Academy of Sciences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' We acknowledge the use of public data from the Fermi Science Support Center (FSSC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' This work is supported by National SKA Program of China (2020SKA0120100,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' 2020SKA0120200),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' Natural Science Foundation of China (12041304,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' 11873067,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' the Strategic Priority Research Program on Space Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' the Western Light Youth Project of Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' the Chinese Academy of Sciences (grants XDA15360000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' XDA15052700,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' XDB23040400),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' funding from the Max-Planck Partner Group,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' the science research grants from the China Manned Space Project (CMS-CSST-2021-B11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content='NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' CMS-CSST-2021-A11), and PKU development grant 7101502590.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' KJL acknowledge support from the XPLORER PRIZE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' BBZ is supported by Fundamental Research Funds for the Central Universities (14380046), and the Program for 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+page_content='21, 302–309.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' [22] Hanisch R J, Farris A, Greisen E W, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' 2001A&A 376,359–380.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} +page_content=' 7' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfd_xv/content/2301.01429v1.pdf'} diff --git a/xtE0T4oBgHgl3EQf-gJ8/content/2301.02815v1.pdf b/xtE0T4oBgHgl3EQf-gJ8/content/2301.02815v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..d40fcd669c415e43e331d1590d5fb5f6e4564a59 --- /dev/null +++ b/xtE0T4oBgHgl3EQf-gJ8/content/2301.02815v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid 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Edwards2, Maged Abdelsamie3, Peter Brown4, David +Webster4, Arvydas Ruseckas4, Gopika Rajan5, Ana I. S. Neves5, Robert W. +Martin2, Carolin M. Sutter-Fella6, Graham A. Turnbull4, Ifor D. W. Samuel4 +and Lethy Krishnan Jagadamma1* +1Energy Harvesting Research Group, School of Physics & Astronomy, SUPA, University of +St Andrews, North Haugh, St Andrews, Fife KY16 9SS, United Kingdom +2Department of Physics, SUPA, University of Strathclyde, Glasgow G4 0NG, United +Kingdom +3 Materials Sciences Division, Lawrence Berkeley National Laboratory, California 94720, +USA, United States +4Organic Semiconductor Centre, SUPA, School of Physics and Astronomy, University of St +Andrews, North Haugh, St Andrews Fife, KY16 9SS, United Kingdom +5Department of Engineering, University of Exeter, North Park Road, EX4 4QF Exeter, +United Kingdom +6Molecular Foundry, Lawrence Berkeley National Laboratory, California 94720, USA, +United States + +*Email: lkj2@st-andrews.ac.uk +Keywords: Internet of Things, WDX, transient photocurrent, chlorine incorporation, in situ +GIWAXS +Abstract: Indoor photovoltaics are receiving tremendous attention due to the continuous +development of the Internet of Things (IoT). Here we report a triple anion (TA) perovskite +CH3NH3PbI2.6(BrCl)0.2 with a tailored bandgap suitable for maximizing indoor light harvesting +compared to methyl ammonium lead iodide CH3NH3PbI3. The best-performing TA perovskite + +2 + +indoor-photovoltaic device achieved a steady-state power conversion efficiency (PCE) of +25.1% with an output power density of ~ 75 µW/cm2 under 1000 lux indoor illumination + (0.3 mW/cm2 irradiance). This PCE is almost 40% higher than that of equivalent CH3NH3PbI3- +based devices (PCE of 17.9%). Longer carrier lifetime, reduced density of trap states and +improved crystalline quality were achieved by the triple anion alloying method. The decisive +role of chlorine (Cl) in the better performance of TA-based indoor photovoltaic devices was +further investigated by successively reducing the Cl content and correlating it with the +corresponding photovoltaic device performance. Replacing the commonly used hole +transporting layer of Spiro-MeOTAD with undoped P3HT was found to significantly reduce +the current-voltage hysteresis under indoor lighting conditions. A graphene-coated textile +fiber-based temperature sensor was successfully powered by the triple anion perovskite indoor +photovoltaic devices. The results from the present study demonstrate a novel route to maximize +the PCE of halide perovskite indoor photovoltaic devices and their potential for application in +the IoT industry. +1. Introduction +Indoor photovoltaic (indoor PV) technology is receiving rejuvenated research attention +due to its potential for self-powering the innumerable wireless sensors in the huge technology +field of the Internet of Things1,2. More than half of these wireless sensors are going to be inside +buildings due to the anticipated radical changes in the built environment to realize smart and +energy-secure buildings. The power requirements of the IoT components have been +continuously decreasing in recent years: nowadays IoT wireless sensors need only a micro- to +milliwatt range of power to operate, and efficient indoor photovoltaic cells are promising +candidates to self-power them1. This autonomous powering of the IoT wireless sensors would +reduce the dependence of this emerging technology on batteries and make it more +environmentally sustainable and widely deployable. Among the various photovoltaic materials +available today, hybrid halide perovskites are very promising for indoor light harvesting due +to their various outstanding optoelectronic properties including tunable bandgap (≈1.2–3.1 +eV)3, high absorption coefficient (absorption length of 200–300 nm)4, long carrier diffusion +length (>1000 nm)5 and high defect tolerance6. These promising properties have already +triggered extensive research in perovskite solar cells and resulted in rapid development before +their implementation in indoor PVs. + +3 + +Developing efficient indoor PV starts with an understanding of the difference between +indoor artificial light sources and outdoor sunlight. These light sources differ in spectrum and +illumination intensity. As shown in Figure 1(a), the illumination spectrum of modern indoor +artificial light is much narrower than that of the sun. It emits only in the visible spectral range +whereas the solar spectrum spans from the near- UV to mid IR. The visible emission of the +indoor light source means that the optimal bandgap for a single junction solar cell is 1.9 eV +(compared to 1.4 eV for 1 sun illumination) to maximize the PCE of indoor PV7. The wider +bandgap of the photoactive layer can increase open circuit voltage (VOC). In addition, the wider +bandgap can also lead to the strong absorption of the narrow emission spectrum of the indoor +light sources, increasing the short circuit current density (JSC)7,8. A further important difference +is that indoor light intensity is much lower than solar irradiance. The standard irradiance level +for sunlight is defined as AM 1.5G which represents 100 mW/cm2 (1 sun), while for indoor +light, which is dominated by white light-emitting diodes (LEDs) and fluorescent lamps, the +irradiance level is 0.05–0.5 mW/cm2 and thus 100–1000 times lower than that of 1 sun +illumination9. This dramatically lower light intensity makes defect control an important topic +for perovskite-based indoor PVs since the beneficial effect of trap filling at the higher +excitation density is no longer present. Therefore, there is greater possibility for trap-assisted +recombination losses of the photogenerated charge carriers in the case of indoor PVs 10. +In ABX3 perovskites, the valence band consists of a hybrid mixture of B site metal +orbitals ns2 and X site halide orbitals np6, with the major contribution from the latter. The +conduction band is formed by a hybrid mixture of B site metal orbitals np6 and X site halide +orbitals np6, with the major contribution from the former. The commonly used approach to +widen the bandgap is to adjust the valence band-edge by compositional tuning of the halide +ions. For the most widely investigated halide perovskite of methylammonium lead iodide +(CH3NH3PbI3), which has a bandgap of 1.56 eV, iodine ions locate at the halide (X-) site. When +iodine ions are substituted by halides with lower energy p orbitals, such as bromine ion and +chlorine ion, the valence band is lowered significantly by 0.60 eV11. This makes mixed halide +perovskite composition tuning a viable route to widen the bandgap in order to maximize the +indoor light harvesting properties8,10,12,13. Following this approach, iodide-bromide alloying +has successfully shifted the bandgap from 1.6 to 1.75 eV with a 40% Br ratio14. However, it is +reported that a larger amount of Br incorporation (greater than 20% halide mole fraction) into +CH3NH3PbI3 results in phase segregation under illumination or during aging15. Upon phase +segregation, the perovskite phase is transformed into I-rich and Br-rich domains; the excited + +4 + +electrons will relax down to the lower bandgap of the I-rich domains, making the bandgap shift +ineffective and reducing the VOC. Hence, the addition of Cl to give the triple anion (TA) system +is presented as an effective method to suppress phase segregation by modifying the morphology +and surface passivation and to realize efficient bandgap tuning16. +The wide bandgap TA perovskites have been successfully used in perovskite-silicon +tandem solar cells16. However, little is known yet regarding their suitability for indoor +photovoltaic applications. Our recent study on halide perovskite indoor PV has demonstrated +the necessity of optimizing the device architecture and photoactive layers separately for indoor +and 1 sun illumination conditions17. Kim et al. found that mixed cation-triple anion perovskite +[FA0.963MA0.037PbI2.813Br0.037Cl0.15] is beneficial for suppressing ion migration and non- +radiative recombination and reported a PCE of ~20% and output power density of 35.25 +µW/cm2 (based on transient J-V measurements) under 800 lux LED illumination10. Cheng et +al. reported that triple-anion perovskite (CH3NH3PbI2−xBrClx) can have enhanced charge +carrier lifetime and suppressed light-induced phase segregation up to 100 suns compared to +CH3NH3PbI3 and CH3NH3PbI2Br, reporting a higher PCE of 36.2% under 1000 lux fluorescent +light (275.4 µW/cm2)8. These results indicate that triple anion alloying could be a promising +method to develop efficient indoor PVs8,10,18. However, the above reports on triple anion indoor +PVs employ a two-step fabrication process for the incorporation of chloride ions (CH3NH3Cl, +CH3NH3Br or HC(NH2)2I deposition after the CH3NH3PbI3 film formation) which increases +the complexity of device fabrication, and they lack discussion of J-V hysteresis and the steady- +state power output from these devices. Our recent study has shown that, under indoor lighting +conditions, the J-V hysteresis effects can become more significant than for 1 sun illumination +and hence steady-state power output measurements should be prioritized over the conventional +transient J-V scan17. +In the present work, we report a facile single-step fabrication of TA halide perovskite +composition of CH3NH3PbI2.6(BrCl)0.2 with a tailored bandgap of 1.69 eV and compare its +indoor light harvesting and J-V hysteresis properties with that of CH3NH3PbI3. To gain more +insight into the benefits of the triple anion alloying method in indoor photovoltaics, we +systematically investigated the microstructural, photophysical and optoelectronic properties of +CH3NH3PbI2.6(BrCl)0.2 films, partial heterostructures and completed devices and compared +them to those of CH3NH3PbI3. Our study revealed that the triple anion composition of +CH3NH3PbI2.6(BrCl)0.2 suffers significantly fewer defect-related recombination losses, exhibits + +5 + +enhanced charge carrier lifetime, and possesses better crystalline properties in comparison to +CH3NH3PbI3 resulting in improved power conversion efficiency and suppressed hysteresis +effects under indoor lighting conditions. The decisive role of Cl in the better performance of +CH3NH3PbI2.6(BrCl)0.2 based indoor photovoltaic devices is further verified by successively +reducing the Cl content and correlating it with the corresponding photovoltaic device +performance. +2. Results and Discussion +2.1. +Microstructural characterization of triple anion perovskite films +To obtain triple anion CH3NH3PbI2.6(BrCl)0.2, equimolar amounts of PbBr2 and PbCl2 +were added into the PbI2 stoichiometry-adjusted CH3NH3PbIx precursor solution (with the +nominal perovskite composition then assumed to follow that of the precursor stoichiometry). +The standard CH3NH3PbI3 perovskite is used as the control sample in the study. During spin +coating, both films were washed by anti-solvent diethyl ether followed by a thermal annealing +treatment for 2 minutes on a hotplate at 100 °C. A detailed description of the preparation +method is given in the experimental section in the supporting information. The UV-vis +spectroscopy measurement was performed initially on TA films and CH3NH3PbI3 control +samples to characterize the bandgap properties. As shown in Figure 1 (b) the absorption edge +for the TA perovskite composition is blue-shifted from ~775 nm (CH3NH3PbI3) to 750 nm +(TA). This corresponds to an increase of the bandgap from 1.61 eV for CH3NH3PbI3 to 1.69 +eV for the TA films (Figure S1). The valence band (VB) position was characterized by ambient +photoemission spectroscopy (APS) as shown in Figure 1 (c). The TA films showed a VB edge +of 5.46 eV, deeper than that of the CH3NH3PbI3 VB at 5.39 eV as expected due to the +incorporation of Br− and Cl− anions. +To understand the crystalline properties, X-ray diffraction (XRD) characterization was +performed (Figure S2). Peaks at 14.18° and 28.6° can be indexed to the (110) and (220) +diffraction peaks of the tetragonal perovskite phase. For both diffraction peaks, the TA sample +shows significantly higher peak intensity indicating its enhanced crystallinity compared to +CH3NH3PbI3. Also, the TA sample revealed the existence of PbI2 with the appearance of a +small peak at 12.65°. This observation of PbI2 agrees with the previous research on perovskite +compositions containing bromide and chloride anions16,19. Figure 1 (d), shows a typical peak +shifting from 14.11° for CH3NH3PbI3 to 14.20° for the TA sample as expected due to shrinkage + +6 + +of the perovskite lattice due to incorporation of Cl− (1.81 Å) and Br− (1.96 Å) with smaller +ionic radii compared to I− (2.20 Å).8 In addition, a new peak is observed at 15.49°, which is +considered to be a characteristic peak of CH3NH3PbCl3 as per previous reports16,19,20. The main +challenge with the triple anion alloying compared to the iodide-bromide double halide system +is to confirm the presence of Cl in the perovskite lattice, because it can volatilize during the +thermal annealing process as CH3NH3Cl18. The existence of the CH3NH3PbCl3 peak provides +primary proof that Cl is incorporated and remains within the perovskite active layer instead of +volatilizing. Previous research has pointed out the difficulty in detecting chlorine within the +perovskite layer after the thermal annealing process although it was present in the precursor +solution21. In the present study, the existence of chlorine in the CH3NH3PbI2.6(BrCl)0.2 films +were further confirmed using wavelength dispersive X-ray (WDX) spectroscopy. By +contrasting the WDX results from the CH3NH3PbI3 and TA samples respectively, it is evident +that Cl is incorporated within the TA samples. Figure 1 (e) compares the WDX counts from +the Cl content of CH3NH3PbI3 and TA samples; the CH3NH3PbI3 sample has a bremsstrahlung +(continuum) background of ~300 counts, while the TA sample shows a clear characteristic X- +ray peak (~1600 counts) at the Cl Kα energy, confirming the presence of Cl within the +perovskite active layer. Monitoring the WDX peak count rates over 10 minutes verified that +there was minimal dissociation/volatilisation caused by the incident 8 keV electron beam +[Figure S3 (a–c)]. This precaution was taken based on the previous studies, where it has been +shown that under the high dose electron beam, methylamine and hydrogen iodide (hydrogen +halide) can escape from the halide perovskite samples due to electron beam induced damage22– +24. Figure 1 (f) shows the SEM images of CH3NH3PbI3 and TA thin film samples. For both +films, the surface morphology was compact and dense whereas, in the case of the TA sample, +dark and bright contrast domains were observed. According to previous research, the brighter +domains may be PbI225. Although the XRD spectra of the TA samples indicated the presence +of PbI2, its presence has been further verified using low electron beam voltage (and hence +surface-sensitive) cathodoluminescence (CL) spectroscopy. As shown in Figure 1(g), the CL +emission spectrum of the TA sample shows two emission peaks; one corresponding to its near +band-edge emission and the other at ~2.5 eV arising from the PbI225. Also, the near band-edge +CL emission of the TA sample is blue-shifted compared to the control CH3NH3PbI3 sample as +expected. The peak of the near band-edge CL emission energy from both the CH3NH3PbI3 +(1.63 eV) and TA (1.73 eV) sample is found to be slightly higher than the band gap energy +estimated using the UV-vis absorption spectra. This could be due to the bandgap energy +estimated from the absorption edge being lower than the actual bandgap due to the existence + +7 + +of tail states in the CH3NH3PbI3 and TA films26. Also, the UV-vis absorption measurements +probe the full thickness (350 nm) of the sample, whereas the CL analysis depth is limited by +the beam energy; with the 5 keV electron beam used, the penetration depth and hence the signal +generation depth is estimated as  240 nm using a Monte-Carlo simulation method27. This +would mean that any vertical compositional heterogeneity, as the authors have previously noted +in all-inorganic mixed halide perovskites28, can also contribute to this slightly higher CL +emission energy compared to the band gap energy estimated from the UV-vis measurements. + +Figure 1. (a) Comparison of the 1 sun spectrum with the warm white LED indoor light source +used in this study (b) Absorbance spectra of triple anion perovskite film and CH3NH3PbI3 +control film from UV-Vis spectroscopy. (c) Ambient photoemission spectra of TA and +CH3NH3PbI3 films. (d) X-ray diffraction pattern of TA and CH3NH3PbI3 films (e) Wavelength- +dispersive X-ray spectra of TA and CH3NH3PbI3 films in the region of the Cl Kα X-ray line +showing incorporation of Cl in the TA film. (f) Scanning electron microscopy images of TA and +CH3NH3PbI3 films. (g) Room temperature CL spectra of TA and CH3NH3PbI3 films showing +the blue-shifted near bandage emission from the TA films and the appearance of PbI2 emission. +(h) Time-resolved photoluminescence spectra of TA and CH3NH3PbI3 films. Excitation was at +515 nm + + +HoMQ 5.46 eV +CH,NH,Pbl3 +t = 27 ns +FA +1.00 um +1.00um8 + +The PbI2 induced by the triple anion alloying method and the incorporated Cl has been +reported to effectively passivate the defects in the perovskite layer and reduce non-radiative +recombination25,29–31. We further tested this in our samples by making measurements of time- +resolved photoluminescence (TRPL) measurements shown in Figure 1(h). For the TRPL +measurements, the perovskite layers were deposited onto bare ITO substrates. The measured +TRPL data clearly show a much slower decay for TA films than for CH3NH3PbI3 films. The +fitted decay time τ is 27 ns for TA films and 9 ns for CH3NH3PbI3 control films, indicating +fewer trap states and non-radiative recombination losses for triple anion films. After confirming +the wider bandgap, better crystalline quality, compact surface morphology, enhanced PL +lifetime and the presence of Cl in the CH3NH3PbI2.6(BrCl)0.2 films, the photovoltaic properties +of the TA films were characterized and compared to those of CH3NH3PbI3. +2.2. +Photovoltaic properties +To investigate the photovoltaic device performance of the TA perovskites, devices were +fabricated +in +typical +n-i-p +planar +architecture +with +a +layer +structure +of +glass/ITO/SnO2/perovskite/Spiro-OMeTAD/Au. The J-V scan measurements were performed +for the CH3NH3PbI3 and TA devices under indoor warm white LED illumination of 1000 lux +(0.3 mW/cm2) and 1 sun. The J-V characteristics and the PCE distribution from these +measurements are shown in Figure 2. The corresponding photovoltaic performance parameters +are shown in Figure S4, Table 1 and Table S1. The box plots in Figures 2 and S4 present the +distribution and average PCE of more than 20 photovoltaic devices and performance +parameters of VOC, FF and JSC. The J-V characteristics of the champion devices under indoor +illumination are given in Figure 2(a). The TA device shows a maximum PCE of 26.1% for the +forward scan and 33.6% for the reverse scan. The CH3NH3PbI3 control devices present a +forward scan PCE of 21.0% and reverse scan PCE of 30.1% for their best performance. As +shown in the box plots in Figure 2(b) and Table 1, the average PCE of TA devices reaches +22.6% for the forward scan and 30.1% for the reverse scan while for CH3NH3PbI3 devices it is +only 17.8% and 27.8% respectively, which shows a substantial enhancement of indoor light +harvesting by the TA films. From Figure 2 (a) & (c), and Table 1 it can be seen that a +significant part of the enhancement in PCE of the TA device under indoor lighting is due to the +higher VOC which is 0.86 V and 0.89 V for different scan directions. The corresponding VOC +for the CH3NH3PbI3 control sample is only 0.78 V and 0.84 V, respectively. The fill factor of +TA devices is also consistently improved particularly for the forward scan, the average FF of + +9 + +TA devices is 57.2% while that of CH3NH3PbI3 devices is only 46.5%. The significant +improvement of FF indicates better charge extraction and collection in the TA films and can +attribute to the better crystalline quality of TA films compared to CH3NH3PbI3. On the other +hand, the JSC of both types of devices are comparable under indoor illumination as shown in +Figure S4 (c) & Table 1 (forward scan: 0.15 mA/cm2 for CH3NH3PbI3 vs 0.14 mA/cm2 for TA; +reverse scan: 0.14 mA/cm2 for CH3NH3PbI3 vs 0.13mA/cm2 for TA). Figure 2 (f) shows the +external quantum efficiency (EQE) spectra. The blue shift of the TA absorption edge, compared +to CH3NH3PbI3 is evident and supports the UV-vis absorption measurements. Figure 2(f) also +shows the emission spectrum of the indoor warm white LED source used in the present study +to explore the indoor photovoltaic properties. The larger bandgap of TA composition reduces +the thermalization losses under the indoor light illumination, resulting in higher VOC without +sacrificing JSC. +As for the device performance under 1 sun, from the J-V curves in Figure 2 (d), TA +devices reach a maximum PCE of 14.6% for forward scan, and 16.5% for reverse scan, while +the PCE of the CH3NH3PbI3 control device is only 12.8% for the forward scan and 13.4% for +the reverse scan for its champion device. From Figure 2 (e), it is noticed that the overall PCE +of TA devices is slightly improved compared to CH3NH3PbI3 under 1 sun. A comparison of +the photovoltaic performance parameters in Figure S4 (d) shows that the average VOC of TA +devices is improved to 1.06 V compared to that of 0.9 V for CH3NH3PbI3. This observation of +higher VOC for TA devices is consistent with their larger bandgap compared to CH3NH3PbI3, +as revealed by UV-vis spectroscopy and APS measurements [Figure 1 (b) & (c)]. +The JSC from both types of devices are broadly comparable (forward scan: 19.0 mA/cm2 +for TA vs 19.2 mA/cm2 for CH3NH3PbI3; reverse scan: 17.9 mA/cm2 for TA vs 18.0 mA/cm2 +for CH3NH3PbI3 from Figure S4 (f) and Table S1). The fill factors of these devices are also +within a similar range, being 61.3% and 56.9% for forward, and 70.1% and 71.6% for reverse +scans for CH3NH3PbI3 and TA devices, respectively. These results show that the triple anion +alloying method is effective in boosting the VOC without compromising JSC and FF thus +yielding higher PCE. + + + + +10 + + + + + + + + + + +Figure 2. (a) J-V curves of TA and CH3NH3PbI3 based devices under warm white LED +illumination. (b) The statistical distribution of PCE values of TA and CH3NH3PbI3 devices +under warm white LED illumination. (c) The distribution of VOC of two types of photovoltaic +devices under warm white LED illumination. (d) J-V curves of TA and CH3NH3PbI3 under 1 +sun illumination. (e) The statistical distribution of PCE values of TA and CH3NH3PbI3 devices +under 1 sun illumination. (f) EQE spectra of the TA and CH3NH3PbI3 devices. The irradiance +spectrum of the warm white LED illumination used in the present study is also shown. +Table 1. Photovoltaic performance parameters of CH3NH3PbI3 and TA device under 1000 lux +warm white LED illumination +Device type + +Average +PCE (%) +Average +FF (%) +Average +Jsc (mA/cm2) +Average Voc (V) +CH3NH3PbI3 +FW +17.8 ± 2.0 +46.5 ± 4.1 +0.147 ± 0.004 +0.779 ± 0.039 +RV +27.8 ± 1.7 +75.4 ± 3.7 +0.131 ± 0.004 +0.844 ± 0.017 +TA +FW +22.6 ± 2.8 +57.2 ± 5.6 +0.138 ± 0.009 +0.860 ± 0.022 +RV +30.1 ± 2.3 +78.4 ± 5.0 +0.129 ± 0.007 +0.892 ± 0.036 +FW +RV +FW +RV +CH3NH3PbI3 +TA +0 +4 +8 +12 +16 +20 +PCE (%) +1 sun +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +-25 +-20 +-15 +-10 +-5 +0 +5 +10 +Current Density (mA cm-2) +Voltage (V) + CH3NH3PbI3 + TA +1 sun +PCE +CH3NH3PbI3 +FW +RV +12.8% +13.4% +TA +FW +RV +14.6% +16.5% +FW +RV +FW +RV +MAPI +TA +0.5 +0.6 +0.7 +0.8 +0.9 +1 +VOC (V) +Warm white LED +(a) +(b) +(c) +(d) +(e) +FW +RV +FW +RV +MAPI +TA +0 +10 +20 +30 +40 +PCE (%) +Warm white LED +0 +0.2 +0.4 +0.6 +0.8 +1 +-0.1 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +Current Density (mA cm-2) +Voltage (V) + CH3NH3PbI3 + TA +Warm white LED +PCE +CH3NH3PbI3 +FW +RV +21.0% +30.1% +TA +FW +RV +26.1% +33.6% +(f) +400 +500 +600 +700 +800 +900 +0 +20 +40 +60 +80 +100 + MAPI + TA +Wavelength (nm) +EQE (%) +LED +0 +0.5 +1 +1.5 +2 +2.5 +Irradiance (mW/cm2/nm) + +11 + +Since J-V hysteresis exists for the photovoltaic devices under both 1 sun and indoor +illuminations, the PCE of the devices was also measured using the steady-state method of +maximum power point tracking (MPPT). Steady-state MPPT measurement holds the device at +the highest power output point for a period, which simulates the scenario where the +photovoltaic device used in the real industrial or domestic application of powering the load. +The PCE obtained from the MPPT measurements is shown in Figure 3 (a). Under 1000 lux +warm white LED illumination, the steady-state PCE of TA device reaches as high as 25.1%, +which is significantly higher than 17.9% for CH3NH3PbI3 devices. Thus under 1000 lux +illuminance conditions, the TA devices can deliver a steady-state power output of 75.4 µW/cm2 +compared to the 53.7 µW/cm2 from the CH3NH3PbI3 devices. These results are summarized in +Table 2. Regarding the current IoT wireless protocols industry, the 75.4 µW/cm2 power density +would allow a 20 cm2 TA perovskite indoor photovoltaic device to power most of the RFID, +LoTA Backscatter, Passive Wi-Fi, BLE, ANT and ZigBee nodes2. The excellent steady-state +PCE of the TA device further proves the superiority of the triple anion alloying method. The +steady-state PCE of the devices under 1 sun illumination condition is shown in Figure 3 (b); +the steady-state PCE of TA devices is 13.5%, slightly higher (13.1%) than that of CH3NH3PbI3, +which still emphasizes the better quality of TA devices. + + Table 2. Summary of steady state PCE and output power density for CH3NH3PbI3 and TA +devices + +To gain more insight into the enhanced photovoltaic properties of TA devices, light +intensity-dependent J-V characterization, transient photocurrent (TPC) and photovoltage +(TPV) measurements were carried out. Under the low-intensity indoor lighting conditions +Light source +Device type +Steady +state PCE (%) +Output +Power Density +(µW/cm2) +Warm +white LED + (1000 lux) +CH3NH3PbI3 +17.9 +53.7 +TA +25.1 +75.4 + +12 + +suppression of trap states is very crucial to maximize the power output and therefore non- +radiative recombination losses were studied by plotting VOC vs light intensity (𝐿)32. The +relationship between light intensity (𝐿) and VOC is given as33: +𝑉𝑂𝐶 = +𝑛𝑖𝑑𝑘𝐵𝑇 +𝑞 +ln⁡( +𝐼𝐿 +𝐼0 + 1) = +𝑛𝑖𝑑𝑘𝐵𝑇 +𝑞 +ln𝐿, (1) +where kB is the Boltzmann factor, nid is an ideality factor, 𝐼𝐿 is the total solar cell current under +illumination (photocurrent), and I0 is the dark saturation current. Since the current under +illumination is much higher than the dark current and photocurrent is linearly related to light +intensity, L can be approximated as the ratio of IL and I0. +In general, an ideality factor of 1 (or close to 1) indicates the dominant recombination +mechanism is bimolecular (radiative) recombination, whereas values closer to 2 indicate +pronounced trap-assisted Shockley-Read-Hall (SRH) recombination34,35. From Figure 3(c), the +ideality factor calculated for CH3NH3PbI3 is 2.03. while for TA, it is only 1.41, indicating that +the non-radiative recombination is significantly reduced for the TA devices, which is consistent +with the TRPL measurement shown in Figure 1(h). In addition to TRPL characterization of TA +and CH3NH3PbI3 films on bare ITO substrates, TRPL was also carried out for perovskite films +deposited on SnO2/ITO films (SnO2 is the ETL in the device architecture considered here). By +taking the TRPL ratio of perovskite films on electron transport layer SnO2 to the perovskite +films on bare ITO substrate, we account for the natural PL decay in perovskite films and get +the decay which is caused solely by electron extraction. The PL ratio shows very similar decay +time of ~40 ns for both perovskites which indicates that the electron extraction rate is the same +for CH3NH3PbI3 and TA (Figure S5)36. Hence, the longer lifetime of free carriers from TA +perovskite film provides more efficient charge extraction which significantly enhances FF. +TPV and TPC measurements were used to further investigate the improvements of carrier +lifetime. The advantage of these measurements is that they can be made on completed devices +under the conditions of practical photovoltaic operation. Figure 3(d) shows carrier lifetime +variation as a function of VOC (light intensity). The steep decrease of carrier lifetime at higher +VOC is attributed to faster carrier recombination owing to higher light intensity. The results in +Figure 3(d) reveal that the carrier lifetime of TA devices is consistently higher than that of +CH3NH3PbI3 devices, supporting their improved photovoltaic performance as shown in Figure +2, and implying their reduced trap states and better interface conditions. The equation of carrier +lifetime versus VOC used to extract 𝛽 is given as below (extracted from TPV measurements) 37: + +13 + +𝜏 = 𝜏0𝑒−𝛽𝑉𝑂𝐶, (2) +where τ is the carrier lifetime and β is the decay constant obtained by fitting the TPV data. +From Figure 3(d), β is 17.7 μs V-1 and 12.2 μs V-1 for TA and CH3NH3PbI3 devices, +respectively. The enhanced voltage decay constant of TA devices also suggests a longer carrier +lifetime and better device quality. Figure 3(e) shows the TPC charge extraction results for the +devices. The equation used for the fitting is: +𝑛 = 𝑛0𝑒𝛾𝑉𝑂𝐶, (3) +TA devices exhibit a higher γ parameter (10.5 cm-3V-1) for the rate of increase of charge +density n compared to that of CH3NH3PbI3 (8.4 cm-3V-1). The γ parameter is the rate of increase +of n with respect to bias which has a value of 19 V-1 for an ideal semiconductor (γ ≈ e/2kBT)37. +The larger deviation from ideality of the γ parameter of the CH3NH3PbI3 device indicates more +non-radiative recombination, suggesting a higher density trap states for CH3NH3PbI3 devices +compared to TA devices, and agrees with the findings of VOC vs light intensity measurements. +The higher γ parameter of TA devices implies better interfacial conditions which also agrees +with the improved decay lifetime τ from the TRPL measurements. The perturbed lifetime τ∆n +can be related to the total charge density with a power dependence in η through the following +equation37: +𝜏∆𝑛 = 𝜏∆𝑛0( +𝑛0 +𝑛 )𝜂, (4) +The exponent η can be obtained by fitting τ∆n vs the charge density 𝑛; it can also be +obtained by rearranging equation (2) and (3) (η = β/γ). η for TA and CH3NH3PbI3 devices is +1.68 and 1.44, respectively. The η exponent can be used to calculate the total lifetime for +devices from the perturbed lifetime using equation (5). With the higher η value, the calculated +total lifetime will be higher for TA devices with higher intrinsic perturbed lifetime, which +further confirms reduced trap states from triple anion perovskites. +𝜏𝑛 = (𝜂 + 1)𝜏∆𝑛. (5) + +14 + +The reduced non-radiative recombination, prolonged lifetime and better interfacial +properties can be linked to the improved device performance and emphasize the importance of +controlling trap for maximizing the efficiency of indoor photovoltaic devices. + +Figure 3. (a) MPPT PCE comparison of TA and CH3NH3PbI3 devices under warm white LED +illumination. (b) MPPT PCE comparison of TA and CH3NH3PbI3 devices 1 sun illumination. +(c) VOC variation versus light intensity. (d) Transient photovoltage characterization of TA and +CH3NH3PbI3 devices. (e) Transient photocurrent characterization of TA and CH3NH3PbI3 +devices. +2.3. +Role of Cl in the enhanced photovoltaic properties of TA devices +To obtain deeper insight into the better indoor photovoltaic properties of the TA +composition, the influence of the halide content needs to be investigated. Since bromine is +relatively stable, it is particularly important to investigate how the photovoltaic device +properties are influenced by chlorine content. Previously, thermal annealing process has been +successfully used to vary the Cl content in a double halide CH3NH3PbI3−xClx perovskite layer19. +This was made possible because of the volatile nature of Cl, allowing the incorporated chlorine + +TA MPP PCE = 25.1% +1.41 KT/g +TA MPP PCE = 13.5% +B= +17.7 μs/V +: 10.5 cm3V-15 + +to be released in the form of CH3NH3Cl during thermal annealing. In the present investigation, +we used a similar thermal annealing method to control the content of chlorine in the TA +perovskite layer. We selected the thermal annealing steps of 2 min, 10 min, 30 min, 45 min and +60 min with the same temperature 100 °C to investigate the effect of chlorine content. We used +WDX spectroscopy to estimate the Cl content in the resultant TA perovskite layers. The iodine +and bromine contents are relatively stable during the different duration of the thermal annealing +process as shown in Figure S6. Figure 4 (a) shows WDX spectra in the region of the chlorine +Kα characteristic X-ray line, after background correcting by subtracting the corresponding +spectrum for Cl-free CH3NH3PbI3. This background correction accurately accounts for both +the bremsstrahlung continuum radiation and the tail of the nearby Pb Mγ line at 2.653 keV (see +Fig. 2e). The resultant net X-ray peak intensity shows the trend of chlorine content decline on +increasing the annealing duration. The 2 min-annealed sample has the highest chlorine content +with a peak of over 1200 X-ray counts, reducing to a near-negligible signal after 60 min +annealing. In the absence of readily available WDX standards of a suitably close composition, +it is adequate for the purposes of this work to estimate relative compositions based on the +assumptions that (i) the 2-minute annealed TA sample is close to its nominal +CH3NH3PbI2.6Br0.2Cl0.2 stoichiometry, and (ii) any atomic number, X-ray absorption and +secondary fluorescence (“ZAF”) effects are minimal. The estimated Cl wt % is thus 1.2%, 0.74 +%, 0.13 %, 0.03 % and 0.003% respectively for the 2-, 10-, 30-, 45- and 60-minute annealed +TA samples. This WDX result matches the results from the Sun et al. study, which constructs +a Cl content gradient to investigate the effect of chlorine19. It is noteworthy that, except for the +2 minute thermally annealed sample, the Cl content in different TA samples is relatively stable +with respect to the WDX electron beam irradiation duration of 10 minutes, as shown in Figure +S6(f). While some slight reduction in counts is seen in the Cl peak intensity (strongest, as +expected, in the 2 minutes thermally annealed TA sample which retains the most Cl), the +magnitude is not sufficient to have a significant effect on the measured WDX compositions. + +16 + + +Figure 4. (a) Background-corrected wavelength dispersive X-ray spectra of the Cl Kα line as a +function of thermal annealing time. (b) Absorbance spectra of triple anion incorporated +perovskite film based on different thermal annealing times. (c) X-ray diffraction pattern +showing the effect of thermal annealing of the TA films. (d) secondary electron (SE) images of +annealed films as a function of different thermal annealing duration. The contrast domains +represent PbI2. (e) Cathodoluminescence spectra of TA films showing the enhanced CL +emission with the increase in thermal annealing duration. +Figure 4 (b) shows the UV-vis absorption spectra of the TA films with different Cl +content, revealing a redshift with the decreasing chlorine content. The absorption edge of the +2 min annealed sample is at ~1.69 eV and is gradually shifted to ~1.64 eV for the 60 min +annealed sample, in which the chlorine content is negligibly small. The redshift in the +absorption edge with a longer annealing time is consistent with the WDX characterization that +chlorine is constantly reduced in the thermal annealing process. The crystalline properties of +the TA perovskite layer as a function of thermal annealing (and hence as a function of chlorine +content) were investigated by XRD. The peaks at 14.18° and 28.6° which index to (110) and +(220) planes of the tetragonal perovskite phase remain. Another significant change can be + +10min +30min +2 min +45 min +60 min +1.00 um +1.00 um +1.00um +1.00um +1.00 um17 + +noticed from the CH3NH3PbCl3 characteristic peak at 15.49°. Though the CH3NH3PbCl3 peak +can be identified from the 2 min-annealed samples (with the highest chlorine content as per the +WDX data), it is then reduced dramatically in the 10 min-annealed sample and completely +disappeared in other samples, which further implies the escape of chlorine during the thermal +annealing process. In addition, along with decreasing chlorine, the intensity of the PbI2 +characteristic peak at 12.65° is enhanced as the thermal annealing time increases. +Figure 4(d) shows the secondary electron SEM images of the TA films as a function +of different thermal annealing duration. From Figure 4 (d), with longer thermal annealing time, +the density of white domain-like features most likely related to (PbI2 phase) is increased, which +is consistent with the XRD results. For the 45 min and 60 min annealed samples, which only +have a trace amount of chlorine, the PbI2 features dominate the film surface. The increased PbI2 +further matches the CL emission results shown in Figure 4 (e). Besides the TA perovskite near +band-edge emission peak, the PbI2 peak at 2.4 eV is significantly enhanced for samples +annealed for 45 and 60 minutes. +The removal of the chlorine content from the TA samples is further verified using +grazing +incidence +wide-angle +X-ray +scattering +(GIWAXS) +experiments. +From +Figure 5(a)-(e), the GIWAXS diffraction peaks located at q ~ 1.0 and ~ 2.0 Å-1 correspond to +the (110) and (220) lattice planes, in line with the XRD peaks at 14.18° and 28.6°38,39as shown +in Figure S2. The PbI2 diffraction peak at q ~ 0.9 Å-1 appears with arc-like scattering with +preferred out-of-plane orientation40. The PbI2 peak intensity increases with the longer +annealing time as evidenced by the XRD peaks at 12.65° also suggest. Notably, the +CH3NH3PbCl3 crystals exhibit preferred out-of-plane orientation for the 2 min and 10 min +annealed TA films, as evidenced by the peak at q ~ 1.1 Å-1, which further emphasizes the +presence of chlorine within the TA samples. Consistent with the XRD and WDX results, the 2 +min annealed films have the strongest diffraction intensity (at q ~ 1.1 Å-1) corresponding to +CH3NH3PbCl3 crystals. The CH3NH3PbCl3 phase then gradually decreases and vanishes +completely for the 30 minutes and longer thermally annealed samples. To probe the evolution +of phases during thermal annealing, in-situ GIWAX measurements were performed based on +prepared 2-minute annealed TA perovskite films as a function of thermal annealing time as +shown in Figure 5(f). The PbI2 gradually increases with a longer annealing process while the +CH3NH3PbCl3 phase gradually decreases and vanishes at around ~ 100 seconds of the 100 °C +thermal annealing process. + +18 + +The evolution of the q position of the TA perovskite (110) peak during thermal +annealing can give information about the changes in the composition of TA perovskite phase +(i.e. incorporation or removal of Cl), as shown in Figure S7. During the initial ramping of +temperature from 25 °C to 100 °C, a change in the q position to lower values (increase in (110) +d-spacing) can be attributed to lattice expansion dominated by thermal expansion. The +subsequent changes in q position occur at a constant temperature (100 °C) and hence can be +attributed to changes in the composition of the TA perovskite phase. Upon reaching 100 °C, +the evolution of the q position indicates a shrinkage of lattice constant for the initial 5 min of +annealing followed by an expansion of lattice constant for a longer annealing time. We recall +that CH3NH3PbCl3 phase dissociates during the initial annealing time giving rise to the release +of Cl ions that can be incorporated into the TA perovskite phase. Therefore, we hypothesize +that the initial shrinkage of the lattice constant of TA perovskite phase results from the +incorporation of smaller ions (such as Cl) into the TA perovskite phase. These Cl ions are +perhaps supplied from CH3NH3PbCl3 phase dissociation. However, for longer annealing times, +some of these Cl leave the film due to the high volatility of Cl, in agreement with previous +studies. These results reveal that careful optimization of annealing time can be used to control +the Cl content in TA perovskite films. +The indoor photovoltaic properties of the devices with different thermal annealing times +are characterized to investigate the effect of chlorine release and PbI2 growth. In terms of +Figure 6, Figure S8 and Table S2, the optimized annealing time is 2 minutes, with these devices +reaching the highest PCE of 26.6% and 30.1% for forward and reverse bias, respectively. The +lowest PCE is obtained from 30 min-annealed devices (forward: 18.4%; reverse: 26.8%), +followed by 60 min-annealed devices (forward: 19.0%; reverse: 28.9%). Regarding the +variation of photovoltaic parameters, 2 min-annealed samples hold the highest VOC of 0.87 V +and 0.92 V among the five conditions, with VOC is continuously decreasing during the annealing +process, indicating the gradual reduction of chlorine content and narrowing of the bandgap. +Notably, 2 min-annealed devices gave the highest FF of 57.2% and 78.4%. This drop in FF +with longer annealing suggests the worsening of interface conditions and transport properties. +The 2 min-annealed TA samples show the highest MPPT PCE as well [Figure 6(b)]. The +increase in the PbI2 phase with an increase in thermal annealing time can hinder the charge + +19 + +transport, deteriorating the PV properties. The thermal annealing study thus shows that the +presence of chlorine in the TA films is contributing to better device properties. + +Figure 5. (a) – (e) Grazing incidence wide-angle X-ray scattering diffraction patterns of TA +films thermally annealed at 100 °C for different durations. With a longer thermal annealing +time, the PbI2 diffraction peaks increase whereas the CH3NH3PbCl3 diffraction peaks decrease +(f) False-color plot of in-situ GIWAXS pattern during the thermal annealing process for the +pre-prepared 2 min-annealed TA film. + + + + + +2 min +10 min +30 min +(220) +CH,NH,PbCl3 +(110) +Pbl2 + 45 min +60 min +CH3NH3PbC13 +Pbl220 + + +Figure 6. (a) The statistics of PCE of devices as a function of different thermal annealing time +under 1000 lux warm white LED illumination. The 2 min-annealed samples hold the highest +average PCE of 26.6% and 30.1%. (b) The comparison of MPPT PCE of devices as a function +of different thermal annealing time +2.4. +Hysteresis properties +Our recent investigation has shown that compared to 1 sun illumination, the halide +perovskite photovoltaic devices demonstrate a completely different J-V hysteresis behaviour +under indoor lighting conditions, depending on the selection of the device architecture and the +photoactive layers17. Addressing J-V hysteresis is a critical aspect since reliable PCE/power +output is required to self-power an external circuit using a photovoltaic device. Compared to +CH3NH3PbI3, TA devices show slightly greater J-V hysteresis under 1 sun illumination, [Figure +2(e)]. However, under indoor lighting illumination, J-V hysteresis of TA devices is suppressed +in comparison to the control CH3NH3PbI3 devices [Figure 2(b)]. The more pronounced J-V +hysteresis of TA devices under 1 sun compared to indoor illumination can be related to the +higher possibility of light-induced ion migration effects under 1 sun light intensity and the +presence of different types of halide ions in the TA composition41–45. Previously we have shown +that the SnO2/perovskite interface can contribute to J-V hysteresis in halide perovskite indoor +photovoltaic devices17. However, the origin of the hysteresis effect can also be due to the +interfacial defects existing at the perovskite/Spiro-OMeTAD interface46. In this case, we +employed P3HT as the hole transport layer to replace Spiro-OMeTAD. Under indoor lighting, +the PCE of P3HT-based devices reached 20.4% and 20.6% for forward and reverse bias. +FW +RV +FW +RV +FW +RV +FW +RV +FW +RV +2 min +10 min +30 min +45 min +60 min +10 +20 +30 +40 +PCE (%) +Average forward PCE +Average reverse PCE +0 +20 +40 +60 +80 +100 +120 +140 +160 +180 +0 +5 +10 +15 +20 +25 +30 +PCE (%) +Time (s) + 2 min + 10 min + 30 min + 45 min + 60 min +(a) +(b) + +21 + +Notably, the hysteresis of P3HT-based devices is reduced drastically compared to Spiro- +OMeTAD based devices, as shown in Figure S9, indicating that the Spiro-OMeTAD interface +as well as the SnO2/perovskite interface are contributing to the J-V hysteresis17. However, the +photovoltaic performances of P3HT-based devices are consistently lower than Spiro-OMeTAD +devices and one contributing factor could be the low conductivity of the undoped P3HT films +used as HTL. More optimization is needed at the charge transporting layer/perovskite buried +interfaces to eliminate the J-V hysteresis without compromising the PCE of the TA based +indoor photovoltaic devices. +2.5. +Powering a temperature sensor +To investigate the reliability of the TA indoor photovoltaic devices as a power source +for sensors, a sensor powering experiment was carried out. Rajan et al. developed a wearable +graphene-based textile temperature sensor with a 1 V requirement for operation47. The +graphene sensor is comprised of three layers of graphene and coated on a single polypropylene +textile fibre which is highly flexible. The graphene-based sensor shows a negative thermal +coefficient of resistance, indicating that its intrinsic resistance is reduced with increased +temperature. The output voltage from TA devices is tested in advance under two indoor +illumination levels of 220 lux and 1000 lux. In this case, two pixels of n-i-p TA devices on a +single substrate are connected in series to obtain a higher output voltage. Two illumination +conditions are tested: a) ambient indoor CFL light from the ceiling (220 lux); b) domestic LED +table lamp (1000 lux). The obtained output voltage from the two series connected photovoltaic +devices under the two indoor illumination conditions is 1.62 V and 1.86 V, respectively. Figure +7(a) shows the experimental setup. The Keithley source meter is only used to measure the +current across the graphene temperature sensor, no extra voltage from the Keithley source +meter is applied. Figure 7 (b) shows the resistance variation of the sensor, powered using TA +devices under the two selected indoor illumination conditions. The temperature sensor shows +a consistent trend in resistance reduction with an increase in temperature, similar to that +previously reported using a Keithley source meter as the voltage source47. These results +indicate that TA perovskite-based indoor photovoltaic devices are reliable for self-powering +the low-power sensors. + +22 + + +Figure 7. (a) Illustration of sensor powering with TA perovskite indoor photovoltaic device +powering a graphene-based textile temperature sensor. (b) Resistance variation of the +temperature sensor in two different illumination conditions. +3. Conclusion +The need for efficient indoor light harvesting was addressed by developing a +composition tuned wide bandgap triple anion perovskite CH3NH3PbI2.6(BrCl)0.2. The +corresponding indoor photovoltaic devices were capable of delivering a steady-state output +power of 75.4 µW/cm2 under 1000 lux warm white LED illumination and successfully self- +powered a textile integrated graphene based temperature sensor. Detailed microstructural and +optoelectronic investigations of the triple anion perovskite unravelled its excellent crystalline +quality, widened bandgap, less density of trap states and longer carrier lifetime, all contributing +positively to the enhanced photovoltaic properties. Our study revealed that to keep the Cl in +the triple anion perovskite and to reap the beneficial effects of high VOC and enhanced charge +carrier lifetime, the thermal annealing duration should be carefully optimized. Replacing the +Spiro-OMeTAD hole transporting layer with P3HT was found to reduce the J-V hysteresis +under indoor lighting conditions. The study shows the promise of simply processed triple anion +perovskites for indoor photovoltaic cells to sustainably power the IoT wireless sensor +components. + + +Keithley sMU for +current measurement +268 +1060 +(kOhm) +Measuring +266 +1050. +Resistance ( +Iluminating +1040 + 264 +1030. +262 +Series-connected +Textile-based +photovoltaic devices +temperature sensor +1020 +Heating +26023 + +Acknowledgements +LKJ acknowledges funding from UKRI-FLF through MR/T022094/1. LKJ also acknowledges, +Professor Iain Baikie for the work function and APS measurements and Professor Phil King +and Gordon Kentish, School of Physics and Astronomy, University of St Andrews for the XRD +measurements and would like to acknowledge (EPSRC): EP/T023449/1. This research used +resources of the Advanced Light Source, a U.S. DOE Office of Science User Facility under +contract no. DE-AC02-05CH11231. 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='uk Keywords: Internet of Things, WDX, transient photocurrent, chlorine incorporation, in situ GIWAXS Abstract: Indoor photovoltaics are receiving tremendous attention due to the continuous development of the Internet of Things (IoT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' Here we report a triple anion (TA) perovskite CH3NH3PbI2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='6(BrCl)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='2 with a tailored bandgap suitable for maximizing indoor light harvesting compared to methyl ammonium lead iodide CH3NH3PbI3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' The best-performing TA perovskite 2 indoor-photovoltaic device achieved a steady-state power conversion efficiency (PCE) of 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='1% with an output power density of ~ 75 µW/cm2 under 1000 lux indoor illumination (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='3 mW/cm2 irradiance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' This PCE is almost 40% higher than that of equivalent CH3NH3PbI3- based devices (PCE of 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='9%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' Longer carrier lifetime, reduced density of trap states and improved crystalline quality were achieved by the triple anion alloying method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' The decisive role of chlorine (Cl) in the better performance of TA-based indoor photovoltaic devices was further investigated by successively reducing the Cl content and correlating it with the corresponding photovoltaic device performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' Replacing the commonly used hole transporting layer of Spiro-MeOTAD with undoped P3HT was found to significantly reduce the current-voltage hysteresis under indoor lighting conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' A graphene-coated textile fiber-based temperature sensor was successfully powered by the triple anion perovskite indoor photovoltaic devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' The results from the present study demonstrate a novel route to maximize the PCE of halide perovskite indoor photovoltaic devices and their potential for application in the IoT industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' Introduction Indoor photovoltaic (indoor PV) technology is receiving rejuvenated research attention due to its potential for self-powering the innumerable wireless sensors in the huge technology field of the Internet of Things1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' More than half of these wireless sensors are going to be inside buildings due to the anticipated radical changes in the built environment to realize smart and energy-secure buildings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' The power requirements of the IoT components have been continuously decreasing in recent years: nowadays IoT wireless sensors need only a micro- to milliwatt range of power to operate, and efficient indoor photovoltaic cells are promising candidates to self-power them1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' This autonomous powering of the IoT wireless sensors would reduce the dependence of this emerging technology on batteries and make it more environmentally sustainable and widely deployable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' Among the various photovoltaic materials available today, hybrid halide perovskites are very promising for indoor light harvesting due to their various outstanding optoelectronic properties including tunable bandgap (≈1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='2–3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='1 eV)3, high absorption coefficient (absorption length of 200–300 nm)4, long carrier diffusion length (>1000 nm)5 and high defect tolerance6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' These promising properties have already triggered extensive research in perovskite solar cells and resulted in rapid development before their implementation in indoor PVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' 3 Developing efficient indoor PV starts with an understanding of the difference between indoor artificial light sources and outdoor sunlight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' These light sources differ in spectrum and illumination intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' As shown in Figure 1(a), the illumination spectrum of modern indoor artificial light is much narrower than that of the sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' It emits only in the visible spectral range whereas the solar spectrum spans from the near- UV to mid IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' The visible emission of the indoor light source means that the optimal bandgap for a single junction solar cell is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='9 eV (compared to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='4 eV for 1 sun illumination) to maximize the PCE of indoor PV7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' The wider bandgap of the photoactive layer can increase open circuit voltage (VOC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' In addition, the wider bandgap can also lead to the strong absorption of the narrow emission spectrum of the indoor light sources, increasing the short circuit current density (JSC)7,8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' A further important difference is that indoor light intensity is much lower than solar irradiance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' The standard irradiance level for sunlight is defined as AM 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='5G which represents 100 mW/cm2 (1 sun), while for indoor light, which is dominated by white light-emitting diodes (LEDs) and fluorescent lamps, the irradiance level is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='05–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='5 mW/cm2 and thus 100–1000 times lower than that of 1 sun illumination9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' This dramatically lower light intensity makes defect control an important topic for perovskite-based indoor PVs since the beneficial effect of trap filling at the higher excitation density is no longer present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' Therefore, there is greater possibility for trap-assisted recombination losses of the photogenerated charge carriers in the case of indoor PVs 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' In ABX3 perovskites, the valence band consists of a hybrid mixture of B site metal orbitals ns2 and X site halide orbitals np6, with the major contribution from the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' The conduction band is formed by a hybrid mixture of B site metal orbitals np6 and X site halide orbitals np6, with the major contribution from the former.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' The commonly used approach to widen the bandgap is to adjust the valence band-edge by compositional tuning of the halide ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' For the most widely investigated halide perovskite of methylammonium lead iodide (CH3NH3PbI3), which has a bandgap of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='56 eV, iodine ions locate at the halide (X-) site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' When iodine ions are substituted by halides with lower energy p orbitals, such as bromine ion and chlorine ion, the valence band is lowered significantly by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='60 eV11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' This makes mixed halide perovskite composition tuning a viable route to widen the bandgap in order to maximize the indoor light harvesting properties8,10,12,13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' Following this approach, iodide-bromide alloying has successfully shifted the bandgap from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='6 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='75 eV with a 40% Br ratio14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' However, it is reported that a larger amount of Br incorporation (greater than 20% halide mole fraction) into CH3NH3PbI3 results in phase segregation under illumination or during aging15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' Upon phase segregation, the perovskite phase is transformed into I-rich and Br-rich domains;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' the excited 4 electrons will relax down to the lower bandgap of the I-rich domains, making the bandgap shift ineffective and reducing the VOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' Hence, the addition of Cl to give the triple anion (TA) system is presented as an effective method to suppress phase segregation by modifying the morphology and surface passivation and to realize efficient bandgap tuning16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' The wide bandgap TA perovskites have been successfully used in perovskite-silicon tandem solar cells16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' However, little is known yet regarding their suitability for indoor photovoltaic applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' Our recent study on halide perovskite indoor PV has demonstrated the necessity of optimizing the device architecture and photoactive layers separately for indoor and 1 sun illumination conditions17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' found that mixed cation-triple anion perovskite [FA0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='963MA0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='037PbI2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='813Br0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='037Cl0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='15] is beneficial for suppressing ion migration and non- radiative recombination and reported a PCE of ~20% and output power density of 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='25 µW/cm2 (based on transient J-V measurements) under 800 lux LED illumination10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' Cheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' reported that triple-anion perovskite (CH3NH3PbI2−xBrClx) can have enhanced charge carrier lifetime and suppressed light-induced phase segregation up to 100 suns compared to CH3NH3PbI3 and CH3NH3PbI2Br, reporting a higher PCE of 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='2% under 1000 lux fluorescent light (275.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='4 µW/cm2)8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' These results indicate that triple anion alloying could be a promising method to develop efficient indoor PVs8,10,18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' However, the above reports on triple anion indoor PVs employ a two-step fabrication process for the incorporation of chloride ions (CH3NH3Cl, CH3NH3Br or HC(NH2)2I deposition after the CH3NH3PbI3 film formation) which increases the complexity of device fabrication, and they lack discussion of J-V hysteresis and the steady- state power output from these devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' Our recent study has shown that, under indoor lighting conditions, the J-V hysteresis effects can become more significant than for 1 sun illumination and hence steady-state power output measurements should be prioritized over the conventional transient J-V scan17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' In the present work, we report a facile single-step fabrication of TA halide perovskite composition of CH3NH3PbI2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='6(BrCl)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='2 with a tailored bandgap of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='69 eV and compare its indoor light harvesting and J-V hysteresis properties with that of CH3NH3PbI3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' To gain more insight into the benefits of the triple anion alloying method in indoor photovoltaics, we systematically investigated the microstructural, photophysical and optoelectronic properties of CH3NH3PbI2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='6(BrCl)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='2 films, partial heterostructures and completed devices and compared them to those of CH3NH3PbI3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' Our study revealed that the triple anion composition of CH3NH3PbI2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='6(BrCl)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='2 suffers significantly fewer defect-related recombination losses, exhibits 5 enhanced charge carrier lifetime, and possesses better crystalline properties in comparison to CH3NH3PbI3 resulting in improved power conversion efficiency and suppressed hysteresis effects under indoor lighting conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' The decisive role of Cl in the better performance of CH3NH3PbI2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='6(BrCl)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='2 based indoor photovoltaic devices is further verified by successively reducing the Cl content and correlating it with the corresponding photovoltaic device performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' Results and Discussion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' Microstructural characterization of triple anion perovskite films To obtain triple anion CH3NH3PbI2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='6(BrCl)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='2, equimolar amounts of PbBr2 and PbCl2 were added into the PbI2 stoichiometry-adjusted CH3NH3PbIx precursor solution (with the nominal perovskite composition then assumed to follow that of the precursor stoichiometry).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' The standard CH3NH3PbI3 perovskite is used as the control sample in the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' During spin coating, both films were washed by anti-solvent diethyl ether followed by a thermal annealing treatment for 2 minutes on a hotplate at 100 °C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' A detailed description of the preparation method is given in the experimental section in the supporting information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' The UV-vis spectroscopy measurement was performed initially on TA films and CH3NH3PbI3 control samples to characterize the bandgap properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' As shown in Figure 1 (b) the absorption edge for the TA perovskite composition is blue-shifted from ~775 nm (CH3NH3PbI3) to 750 nm (TA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' This corresponds to an increase of the bandgap from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='61 eV for CH3NH3PbI3 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='69 eV for the TA films (Figure S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' The valence band (VB) position was characterized by ambient photoemission spectroscopy (APS) as shown in Figure 1 (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' The TA films showed a VB edge of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='46 eV, deeper than that of the CH3NH3PbI3 VB at 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='39 eV as expected due to the incorporation of Br− and Cl− anions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' To understand the crystalline properties, X-ray diffraction (XRD) characterization was performed (Figure S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' Peaks at 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='18° and 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='6° can be indexed to the (110) and (220) diffraction peaks of the tetragonal perovskite phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' For both diffraction peaks, the TA sample shows significantly higher peak intensity indicating its enhanced crystallinity compared to CH3NH3PbI3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' Also, the TA sample revealed the existence of PbI2 with the appearance of a small peak at 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='65°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' This observation of PbI2 agrees with the previous research on perovskite compositions containing bromide and chloride anions16,19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' Figure 1 (d), shows a typical peak shifting from 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='11° for CH3NH3PbI3 to 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='20° for the TA sample as expected due to shrinkage 6 of the perovskite lattice due to incorporation of Cl− (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='81 Å) and Br− (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='96 Å) with smaller ionic radii compared to I− (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='20 Å).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='8 In addition, a new peak is observed at 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='49°, which is considered to be a characteristic peak of CH3NH3PbCl3 as per previous reports16,19,20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' The main challenge with the triple anion alloying compared to the iodide-bromide double halide system is to confirm the presence of Cl in the perovskite lattice, because it can volatilize during the thermal annealing process as CH3NH3Cl18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' The existence of the CH3NH3PbCl3 peak provides primary proof that Cl is incorporated and remains within the perovskite active layer instead of volatilizing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' Previous research has pointed out the difficulty in detecting chlorine within the perovskite layer after the thermal annealing process although it was present in the precursor solution21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' In the present study, the existence of chlorine in the CH3NH3PbI2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='6(BrCl)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='2 films were further confirmed using wavelength dispersive X-ray (WDX) spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' By contrasting the WDX results from the CH3NH3PbI3 and TA samples respectively, it is evident that Cl is incorporated within the TA samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' Figure 1 (e) compares the WDX counts from the Cl content of CH3NH3PbI3 and TA samples;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' the CH3NH3PbI3 sample has a bremsstrahlung (continuum) background of ~300 counts, while the TA sample shows a clear characteristic X- ray peak (~1600 counts) at the Cl Kα energy, confirming the presence of Cl within the perovskite active layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' Monitoring the WDX peak count rates over 10 minutes verified that there was minimal dissociation/volatilisation caused by the incident 8 keV electron beam [Figure S3 (a–c)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' This precaution was taken based on the previous studies, where it has been shown that under the high dose electron beam, methylamine and hydrogen iodide (hydrogen halide) can escape from the halide perovskite samples due to electron beam induced damage22– 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' Figure 1 (f) shows the SEM images of CH3NH3PbI3 and TA thin film samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' For both films, the surface morphology was compact and dense whereas, in the case of the TA sample, dark and bright contrast domains were observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' According to previous research, the brighter domains may be PbI225.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' Although the XRD spectra of the TA samples indicated the presence of PbI2, its presence has been further verified using low electron beam voltage (and hence surface-sensitive) cathodoluminescence (CL) spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' As shown in Figure 1(g), the CL emission spectrum of the TA sample shows two emission peaks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' one corresponding to its near band-edge emission and the other at ~2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='5 eV arising from the PbI225.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' Also, the near band-edge CL emission of the TA sample is blue-shifted compared to the control CH3NH3PbI3 sample as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' The peak of the near band-edge CL emission energy from both the CH3NH3PbI3 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='63 eV) and TA (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='73 eV) sample is found to be slightly higher than the band gap energy estimated using the UV-vis absorption spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' This could be due to the bandgap energy estimated from the absorption edge being lower than the actual bandgap due to the existence 7 of tail states in the CH3NH3PbI3 and TA films26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' Also, the UV-vis absorption measurements probe the full thickness (350 nm) of the sample, whereas the CL analysis depth is limited by the beam energy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' with the 5 keV electron beam used, the penetration depth and hence the signal generation depth is estimated as \uf0bb 240 nm using a Monte-Carlo simulation method27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' This would mean that any vertical compositional heterogeneity, as the authors have previously noted in all-inorganic mixed halide perovskites28, can also contribute to this slightly higher CL emission energy compared to the band gap energy estimated from the UV-vis measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' (a) Comparison of the 1 sun spectrum with the warm white LED indoor light source used in this study (b) Absorbance spectra of triple anion perovskite film and CH3NH3PbI3 control film from UV-Vis spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' (c) Ambient photoemission spectra of TA and CH3NH3PbI3 films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' (d) X-ray diffraction pattern of TA and CH3NH3PbI3 films (e) Wavelength- dispersive X-ray spectra of TA and CH3NH3PbI3 films in the region of the Cl Kα X-ray line showing incorporation of Cl in the TA film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' (f) Scanning electron microscopy images of TA and CH3NH3PbI3 films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' (g) Room temperature CL spectra of TA and CH3NH3PbI3 films showing the blue-shifted near bandage emission from the TA films and the appearance of PbI2 emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' (h) Time-resolved photoluminescence spectra of TA and CH3NH3PbI3 films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' Excitation was at 515 nm HoMQ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='46 eV CH,NH,Pbl3 t = 27 ns FA 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='00 um 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='00um8 The PbI2 induced by the triple anion alloying method and the incorporated Cl has been reported to effectively passivate the defects in the perovskite layer and reduce non-radiative recombination25,29–31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' We further tested this in our samples by making measurements of time- resolved photoluminescence (TRPL) measurements shown in Figure 1(h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' For the TRPL measurements, the perovskite layers were deposited onto bare ITO substrates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' The measured TRPL data clearly show a much slower decay for TA films than for CH3NH3PbI3 films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' The fitted decay time τ is 27 ns for TA films and 9 ns for CH3NH3PbI3 control films, indicating fewer trap states and non-radiative recombination losses for triple anion films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' After confirming the wider bandgap, better crystalline quality, compact surface morphology, enhanced PL lifetime and the presence of Cl in the CH3NH3PbI2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='6(BrCl)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='2 films, the photovoltaic properties of the TA films were characterized and compared to those of CH3NH3PbI3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' Photovoltaic properties To investigate the photovoltaic device performance of the TA perovskites, devices were fabricated in typical n-i-p planar architecture with a layer structure of glass/ITO/SnO2/perovskite/Spiro-OMeTAD/Au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' The J-V scan measurements were performed for the CH3NH3PbI3 and TA devices under indoor warm white LED illumination of 1000 lux (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='3 mW/cm2) and 1 sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' The J-V characteristics and the PCE distribution from these measurements are shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' The corresponding photovoltaic performance parameters are shown in Figure S4, Table 1 and Table S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' The box plots in Figures 2 and S4 present the distribution and average PCE of more than 20 photovoltaic devices and performance parameters of VOC, FF and JSC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' The J-V characteristics of the champion devices under indoor illumination are given in Figure 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' The TA device shows a maximum PCE of 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='1% for the forward scan and 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='6% for the reverse scan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' The CH3NH3PbI3 control devices present a forward scan PCE of 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='0% and reverse scan PCE of 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='1% for their best performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' As shown in the box plots in Figure 2(b) and Table 1, the average PCE of TA devices reaches 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='6% for the forward scan and 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='1% for the reverse scan while for CH3NH3PbI3 devices it is only 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='8% and 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='8% respectively, which shows a substantial enhancement of indoor light harvesting by the TA films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' From Figure 2 (a) & (c), and Table 1 it can be seen that a significant part of the enhancement in PCE of the TA device under indoor lighting is due to the higher VOC which is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='86 V and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='89 V for different scan directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' The corresponding VOC for the CH3NH3PbI3 control sample is only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='78 V and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='84 V, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' The fill factor of TA devices is also consistently improved particularly for the forward scan, the average FF of 9 TA devices is 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='2% while that of CH3NH3PbI3 devices is only 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' The significant improvement of FF indicates better charge extraction and collection in the TA films and can attribute to the better crystalline quality of TA films compared to CH3NH3PbI3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' On the other hand, the JSC of both types of devices are comparable under indoor illumination as shown in Figure S4 (c) & Table 1 (forward scan: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='15 mA/cm2 for CH3NH3PbI3 vs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='14 mA/cm2 for TA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' reverse scan: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='14 mA/cm2 for CH3NH3PbI3 vs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='13mA/cm2 for TA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' Figure 2 (f) shows the external quantum efficiency (EQE) spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' The blue shift of the TA absorption edge, compared to CH3NH3PbI3 is evident and supports the UV-vis absorption measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' Figure 2(f) also shows the emission spectrum of the indoor warm white LED source used in the present study to explore the indoor photovoltaic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' The larger bandgap of TA composition reduces the thermalization losses under the indoor light illumination, resulting in higher VOC without sacrificing JSC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' As for the device performance under 1 sun, from the J-V curves in Figure 2 (d), TA devices reach a maximum PCE of 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='6% for forward scan, and 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='5% for reverse scan, while the PCE of the CH3NH3PbI3 control device is only 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='8% for the forward scan and 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='4% for the reverse scan for its champion device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' From Figure 2 (e), it is noticed that the overall PCE of TA devices is slightly improved compared to CH3NH3PbI3 under 1 sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' A comparison of the photovoltaic performance parameters in Figure S4 (d) shows that the average VOC of TA devices is improved to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='06 V compared to that of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='9 V for CH3NH3PbI3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' This observation of higher VOC for TA devices is consistent with their larger bandgap compared to CH3NH3PbI3, as revealed by UV-vis spectroscopy and APS measurements [Figure 1 (b) & (c)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' The JSC from both types of devices are broadly comparable (forward scan: 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='0 mA/cm2 for TA vs 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='2 mA/cm2 for CH3NH3PbI3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' reverse scan: 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='9 mA/cm2 for TA vs 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='0 mA/cm2 for CH3NH3PbI3 from Figure S4 (f) and Table S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' The fill factors of these devices are also within a similar range, being 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='3% and 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='9% for forward, and 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='1% and 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='6% for reverse scans for CH3NH3PbI3 and TA devices, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' These results show that the triple anion alloying method is effective in boosting the VOC without compromising JSC and FF thus yielding higher PCE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' 10 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' (a) J-V curves of TA and CH3NH3PbI3 based devices under warm white LED illumination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' (b) The statistical distribution of PCE values of TA and CH3NH3PbI3 devices under warm white LED illumination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' (c) The distribution of VOC of two types of photovoltaic devices under warm white LED illumination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' (d) J-V curves of TA and CH3NH3PbI3 under 1 sun illumination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' (e) The statistical distribution of PCE values of TA and CH3NH3PbI3 devices under 1 sun illumination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' (f) EQE spectra of the TA and CH3NH3PbI3 devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' The irradiance spectrum of the warm white LED illumination used in the present study is also shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' Photovoltaic performance parameters of CH3NH3PbI3 and TA device under 1000 lux warm white LED illumination Device type Average PCE (%) Average FF (%) Average Jsc (mA/cm2) Average Voc (V) CH3NH3PbI3 FW 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='8 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='0 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='5 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='147 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='779 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='039 RV 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='8 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='7 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='4 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='131 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='844 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='017 TA FW 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='6 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='8 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='2 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='138 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='009 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='860 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='022 RV 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='1 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='3 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='4 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='129 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='892 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='036 FW RV FW RV CH3NH3PbI3 TA 0 4 8 12 16 20 PCE (%) 1 sun 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='2 25 20 15 10 5 0 5 10 Current Density (mA cm-2) Voltage (V) CH3NH3PbI3 TA 1 sun PCE CH3NH3PbI3 FW RV 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='8% 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='4% TA FW RV 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='6% 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='5% FW RV FW RV MAPI TA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='9 1 VOC (V) Warm white LED (a) (b) (c) (d) (e) FW RV FW RV MAPI TA 0 10 20 30 40 PCE (%) Warm white LED 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='8 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='6 Current Density (mA cm-2) Voltage (V) CH3NH3PbI3 TA Warm white LED PCE CH3NH3PbI3 FW RV 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='0% 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='1% TA FW RV 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='1% 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='6% (f) 400 500 600 700 800 900 0 20 40 60 80 100 MAPI TA Wavelength (nm) EQE (%) LED 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='5 Irradiance (mW/cm2/nm) 11 Since J-V hysteresis exists for the photovoltaic devices under both 1 sun and indoor illuminations, the PCE of the devices was also measured using the steady-state method of maximum power point tracking (MPPT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' Steady-state MPPT measurement holds the device at the highest power output point for a period, which simulates the scenario where the photovoltaic device used in the real industrial or domestic application of powering the load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' The PCE obtained from the MPPT measurements is shown in Figure 3 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' Under 1000 lux warm white LED illumination, the steady-state PCE of TA device reaches as high as 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='1%, which is significantly higher than 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='9% for CH3NH3PbI3 devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' Thus under 1000 lux illuminance conditions, the TA devices can deliver a steady-state power output of 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='4 µW/cm2 compared to the 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='7 µW/cm2 from the CH3NH3PbI3 devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' These results are summarized in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' Regarding the current IoT wireless protocols industry, the 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='4 µW/cm2 power density would allow a 20 cm2 TA perovskite indoor photovoltaic device to power most of the RFID, LoTA Backscatter, Passive Wi-Fi, BLE, ANT and ZigBee nodes2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' The excellent steady-state PCE of the TA device further proves the superiority of the triple anion alloying method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' The steady-state PCE of the devices under 1 sun illumination condition is shown in Figure 3 (b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' the steady-state PCE of TA devices is 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='5%, slightly higher (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='1%) than that of CH3NH3PbI3, which still emphasizes the better quality of TA devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' Summary of steady state PCE and output power density for CH3NH3PbI3 and TA devices To gain more insight into the enhanced photovoltaic properties of TA devices, light intensity-dependent J-V characterization, transient photocurrent (TPC) and photovoltage (TPV) measurements were carried out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' Under the low-intensity indoor lighting conditions Light source Device type Steady state PCE (%) Output Power Density (µW/cm2) Warm white LED (1000 lux) CH3NH3PbI3 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='9 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='7 TA 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='1 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='4 12 suppression of trap states is very crucial to maximize the power output and therefore non- radiative recombination losses were studied by plotting VOC vs light intensity (𝐿)32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' The relationship between light intensity (𝐿) and VOC is given as33: 𝑉𝑂𝐶 = 𝑛𝑖𝑑𝑘𝐵𝑇 𝑞 ln\u2061( 𝐼𝐿 𝐼0 + 1) = 𝑛𝑖𝑑𝑘𝐵𝑇 𝑞 ln𝐿, (1) where kB is the Boltzmann factor, nid is an ideality factor, 𝐼𝐿 is the total solar cell current under illumination (photocurrent), and I0 is the dark saturation current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' Since the current under illumination is much higher than the dark current and photocurrent is linearly related to light intensity, L can be approximated as the ratio of IL and I0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' In general, an ideality factor of 1 (or close to 1) indicates the dominant recombination mechanism is bimolecular (radiative) recombination, whereas values closer to 2 indicate pronounced trap-assisted Shockley-Read-Hall (SRH) recombination34,35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' From Figure 3(c), the ideality factor calculated for CH3NH3PbI3 is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' while for TA, it is only 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='41, indicating that the non-radiative recombination is significantly reduced for the TA devices, which is consistent with the TRPL measurement shown in Figure 1(h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' In addition to TRPL characterization of TA and CH3NH3PbI3 films on bare ITO substrates, TRPL was also carried out for perovskite films deposited on SnO2/ITO films (SnO2 is the ETL in the device architecture considered here).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' By taking the TRPL ratio of perovskite films on electron transport layer SnO2 to the perovskite films on bare ITO substrate, we account for the natural PL decay in perovskite films and get the decay which is caused solely by electron extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' The PL ratio shows very similar decay time of ~40 ns for both perovskites which indicates that the electron extraction rate is the same for CH3NH3PbI3 and TA (Figure S5)36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' Hence, the longer lifetime of free carriers from TA perovskite film provides more efficient charge extraction which significantly enhances FF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' TPV and TPC measurements were used to further investigate the improvements of carrier lifetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' The advantage of these measurements is that they can be made on completed devices under the conditions of practical photovoltaic operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' Figure 3(d) shows carrier lifetime variation as a function of VOC (light intensity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' The steep decrease of carrier lifetime at higher VOC is attributed to faster carrier recombination owing to higher light intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' The results in Figure 3(d) reveal that the carrier lifetime of TA devices is consistently higher than that of CH3NH3PbI3 devices, supporting their improved photovoltaic performance as shown in Figure 2, and implying their reduced trap states and better interface conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' The equation of carrier lifetime versus VOC used to extract 𝛽 is given as below (extracted from TPV measurements) 37: 13 𝜏 = 𝜏0𝑒−𝛽𝑉𝑂𝐶, (2) where τ is the carrier lifetime and β is the decay constant obtained by fitting the TPV data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' From Figure 3(d), β is 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='7 μs V-1 and 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='2 μs V-1 for TA and CH3NH3PbI3 devices, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' The enhanced voltage decay constant of TA devices also suggests a longer carrier lifetime and better device quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' Figure 3(e) shows the TPC charge extraction results for the devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' The equation used for the fitting is: 𝑛 = 𝑛0𝑒𝛾𝑉𝑂𝐶, (3) TA devices exhibit a higher γ parameter (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='5 cm-3V-1) for the rate of increase of charge density n compared to that of CH3NH3PbI3 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='4 cm-3V-1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' The γ parameter is the rate of increase of n with respect to bias which has a value of 19 V-1 for an ideal semiconductor (γ ≈ e/2kBT)37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' The larger deviation from ideality of the γ parameter of the CH3NH3PbI3 device indicates more non-radiative recombination, suggesting a higher density trap states for CH3NH3PbI3 devices compared to TA devices, and agrees with the findings of VOC vs light intensity measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' The higher γ parameter of TA devices implies better interfacial conditions which also agrees with the improved decay lifetime τ from the TRPL measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' The perturbed lifetime τ∆n can be related to the total charge density with a power dependence in η through the following equation37: 𝜏∆𝑛 = 𝜏∆𝑛0( 𝑛0 𝑛 )𝜂, (4) The exponent η can be obtained by fitting τ∆n vs the charge density 𝑛;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' it can also be obtained by rearranging equation (2) and (3) (η = β/γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' η for TA and CH3NH3PbI3 devices is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='68 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='44, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' The η exponent can be used to calculate the total lifetime for devices from the perturbed lifetime using equation (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' With the higher η value, the calculated total lifetime will be higher for TA devices with higher intrinsic perturbed lifetime, which further confirms reduced trap states from triple anion perovskites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' 𝜏𝑛 = (𝜂 + 1)𝜏∆𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' (5) 14 The reduced non-radiative recombination, prolonged lifetime and better interfacial properties can be linked to the improved device performance and emphasize the importance of controlling trap for maximizing the efficiency of indoor photovoltaic devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' (a) MPPT PCE comparison of TA and CH3NH3PbI3 devices under warm white LED illumination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' (b) MPPT PCE comparison of TA and CH3NH3PbI3 devices 1 sun illumination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' (c) VOC variation versus light intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' (d) Transient photovoltage characterization of TA and CH3NH3PbI3 devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' (e) Transient photocurrent characterization of TA and CH3NH3PbI3 devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' Role of Cl in the enhanced photovoltaic properties of TA devices To obtain deeper insight into the better indoor photovoltaic properties of the TA composition, the influence of the halide content needs to be investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' Since bromine is relatively stable, it is particularly important to investigate how the photovoltaic device properties are influenced by chlorine content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' Previously, thermal annealing process has been successfully used to vary the Cl content in a double halide CH3NH3PbI3−xClx perovskite layer19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' This was made possible because of the volatile nature of Cl, allowing the incorporated chlorine TA MPP PCE = 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='1% 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='41 KT/g TA MPP PCE = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='5% B= 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='7 μs/V : 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='5 cm3V-15 to be released in the form of CH3NH3Cl during thermal annealing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' In the present investigation, we used a similar thermal annealing method to control the content of chlorine in the TA perovskite layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' We selected the thermal annealing steps of 2 min, 10 min, 30 min, 45 min and 60 min with the same temperature 100 °C to investigate the effect of chlorine content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' We used WDX spectroscopy to estimate the Cl content in the resultant TA perovskite layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' The iodine and bromine contents are relatively stable during the different duration of the thermal annealing process as shown in Figure S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' Figure 4 (a) shows WDX spectra in the region of the chlorine Kα characteristic X-ray line, after background correcting by subtracting the corresponding spectrum for Cl-free CH3NH3PbI3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' This background correction accurately accounts for both the bremsstrahlung continuum radiation and the tail of the nearby Pb Mγ line at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='653 keV (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' 2e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' The resultant net X-ray peak intensity shows the trend of chlorine content decline on increasing the annealing duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' The 2 min-annealed sample has the highest chlorine content with a peak of over 1200 X-ray counts, reducing to a near-negligible signal after 60 min annealing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' In the absence of readily available WDX standards of a suitably close composition, it is adequate for the purposes of this work to estimate relative compositions based on the assumptions that (i) the 2-minute annealed TA sample is close to its nominal CH3NH3PbI2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='6Br0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='2Cl0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='2 stoichiometry, and (ii) any atomic number, X-ray absorption and secondary fluorescence (“ZAF”) effects are minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' The estimated Cl wt % is thus 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='2%, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='74 %, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='13 %, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='03 % and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='003% respectively for the 2-, 10-, 30-, 45- and 60-minute annealed TA samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' This WDX result matches the results from the Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' study, which constructs a Cl content gradient to investigate the effect of chlorine19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' It is noteworthy that, except for the 2 minute thermally annealed sample, the Cl content in different TA samples is relatively stable with respect to the WDX electron beam irradiation duration of 10 minutes, as shown in Figure S6(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' While some slight reduction in counts is seen in the Cl peak intensity (strongest, as expected, in the 2 minutes thermally annealed TA sample which retains the most Cl), the magnitude is not sufficient to have a significant effect on the measured WDX compositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' 16 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' (a) Background-corrected wavelength dispersive X-ray spectra of the Cl Kα line as a function of thermal annealing time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' (b) Absorbance spectra of triple anion incorporated perovskite film based on different thermal annealing times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' (c) X-ray diffraction pattern showing the effect of thermal annealing of the TA films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' (d) secondary electron (SE) images of annealed films as a function of different thermal annealing duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' The contrast domains represent PbI2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' (e) Cathodoluminescence spectra of TA films showing the enhanced CL emission with the increase in thermal annealing duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' Figure 4 (b) shows the UV-vis absorption spectra of the TA films with different Cl content, revealing a redshift with the decreasing chlorine content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' The absorption edge of the 2 min annealed sample is at ~1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='69 eV and is gradually shifted to ~1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='64 eV for the 60 min annealed sample, in which the chlorine content is negligibly small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' The redshift in the absorption edge with a longer annealing time is consistent with the WDX characterization that chlorine is constantly reduced in the thermal annealing process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' The crystalline properties of the TA perovskite layer as a function of thermal annealing (and hence as a function of chlorine content) were investigated by XRD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' The peaks at 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='18° and 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='6° which index to (110) and (220) planes of the tetragonal perovskite phase remain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' Another significant change can be 10min 30min 2 min 45 min 60 min 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='00 um 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='00 um 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='00um 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='00um 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='00 um17 noticed from the CH3NH3PbCl3 characteristic peak at 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='49°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' Though the CH3NH3PbCl3 peak can be identified from the 2 min-annealed samples (with the highest chlorine content as per the WDX data), it is then reduced dramatically in the 10 min-annealed sample and completely disappeared in other samples, which further implies the escape of chlorine during the thermal annealing process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' In addition, along with decreasing chlorine, the intensity of the PbI2 characteristic peak at 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='65° is enhanced as the thermal annealing time increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' Figure 4(d) shows the secondary electron SEM images of the TA films as a function of different thermal annealing duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' From Figure 4 (d), with longer thermal annealing time, the density of white domain-like features most likely related to (PbI2 phase) is increased, which is consistent with the XRD results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' For the 45 min and 60 min annealed samples, which only have a trace amount of chlorine, the PbI2 features dominate the film surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' The increased PbI2 further matches the CL emission results shown in Figure 4 (e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' Besides the TA perovskite near band-edge emission peak, the PbI2 peak at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='4 eV is significantly enhanced for samples annealed for 45 and 60 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' The removal of the chlorine content from the TA samples is further verified using grazing incidence wide-angle X-ray scattering (GIWAXS) experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' From Figure 5(a)-(e), the GIWAXS diffraction peaks located at q ~ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='0 and ~ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='0 Å-1 correspond to the (110) and (220) lattice planes, in line with the XRD peaks at 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='18° and 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='6°38,39as shown in Figure S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' The PbI2 diffraction peak at q ~ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='9 Å-1 appears with arc-like scattering with preferred out-of-plane orientation40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' The PbI2 peak intensity increases with the longer annealing time as evidenced by the XRD peaks at 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='65° also suggest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' Notably, the CH3NH3PbCl3 crystals exhibit preferred out-of-plane orientation for the 2 min and 10 min annealed TA films, as evidenced by the peak at q ~ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='1 Å-1, which further emphasizes the presence of chlorine within the TA samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' Consistent with the XRD and WDX results, the 2 min annealed films have the strongest diffraction intensity (at q ~ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='1 Å-1) corresponding to CH3NH3PbCl3 crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' The CH3NH3PbCl3 phase then gradually decreases and vanishes completely for the 30 minutes and longer thermally annealed samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' To probe the evolution of phases during thermal annealing, in-situ GIWAX measurements were performed based on prepared 2-minute annealed TA perovskite films as a function of thermal annealing time as shown in Figure 5(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' The PbI2 gradually increases with a longer annealing process while the CH3NH3PbCl3 phase gradually decreases and vanishes at around ~ 100 seconds of the 100 °C thermal annealing process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' 18 The evolution of the q position of the TA perovskite (110) peak during thermal annealing can give information about the changes in the composition of TA perovskite phase (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' incorporation or removal of Cl), as shown in Figure S7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' During the initial ramping of temperature from 25 °C to 100 °C, a change in the q position to lower values (increase in (110) d-spacing) can be attributed to lattice expansion dominated by thermal expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' The subsequent changes in q position occur at a constant temperature (100 °C) and hence can be attributed to changes in the composition of the TA perovskite phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' Upon reaching 100 °C, the evolution of the q position indicates a shrinkage of lattice constant for the initial 5 min of annealing followed by an expansion of lattice constant for a longer annealing time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' We recall that CH3NH3PbCl3 phase dissociates during the initial annealing time giving rise to the release of Cl ions that can be incorporated into the TA perovskite phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' Therefore, we hypothesize that the initial shrinkage of the lattice constant of TA perovskite phase results from the incorporation of smaller ions (such as Cl) into the TA perovskite phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' These Cl ions are perhaps supplied from CH3NH3PbCl3 phase dissociation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' However, for longer annealing times, some of these Cl leave the film due to the high volatility of Cl, in agreement with previous studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' These results reveal that careful optimization of annealing time can be used to control the Cl content in TA perovskite films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' The indoor photovoltaic properties of the devices with different thermal annealing times are characterized to investigate the effect of chlorine release and PbI2 growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' In terms of Figure 6, Figure S8 and Table S2, the optimized annealing time is 2 minutes, with these devices reaching the highest PCE of 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='6% and 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='1% for forward and reverse bias, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' The lowest PCE is obtained from 30 min-annealed devices (forward: 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='4%;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' reverse: 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='8%), followed by 60 min-annealed devices (forward: 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='0%;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' reverse: 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='9%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' Regarding the variation of photovoltaic parameters, 2 min-annealed samples hold the highest VOC of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='87 V and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='92 V among the five conditions, with VOC is continuously decreasing during the annealing process, indicating the gradual reduction of chlorine content and narrowing of the bandgap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' Notably, 2 min-annealed devices gave the highest FF of 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='2% and 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='4%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' This drop in FF with longer annealing suggests the worsening of interface conditions and transport properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' The 2 min-annealed TA samples show the highest MPPT PCE as well [Figure 6(b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' The increase in the PbI2 phase with an increase in thermal annealing time can hinder the charge 19 transport, deteriorating the PV properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' The thermal annealing study thus shows that the presence of chlorine in the TA films is contributing to better device properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' (a) – (e) Grazing incidence wide-angle X-ray scattering diffraction patterns of TA films thermally annealed at 100 °C for different durations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' With a longer thermal annealing time, the PbI2 diffraction peaks increase whereas the CH3NH3PbCl3 diffraction peaks decrease (f) False-color plot of in-situ GIWAXS pattern during the thermal annealing process for the pre-prepared 2 min-annealed TA film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' 2 min 10 min 30 min (220) CH,NH,PbCl3 (110) Pbl2 45 min 60 min CH3NH3PbC13 Pbl220 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' (a) The statistics of PCE of devices as a function of different thermal annealing time under 1000 lux warm white LED illumination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' The 2 min-annealed samples hold the highest average PCE of 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='6% and 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' (b) The comparison of MPPT PCE of devices as a function of different thermal annealing time 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' Hysteresis properties Our recent investigation has shown that compared to 1 sun illumination, the halide perovskite photovoltaic devices demonstrate a completely different J-V hysteresis behaviour under indoor lighting conditions, depending on the selection of the device architecture and the photoactive layers17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' Addressing J-V hysteresis is a critical aspect since reliable PCE/power output is required to self-power an external circuit using a photovoltaic device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' Compared to CH3NH3PbI3, TA devices show slightly greater J-V hysteresis under 1 sun illumination, [Figure 2(e)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' However, under indoor lighting illumination, J-V hysteresis of TA devices is suppressed in comparison to the control CH3NH3PbI3 devices [Figure 2(b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' The more pronounced J-V hysteresis of TA devices under 1 sun compared to indoor illumination can be related to the higher possibility of light-induced ion migration effects under 1 sun light intensity and the presence of different types of halide ions in the TA composition41–45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' Previously we have shown that the SnO2/perovskite interface can contribute to J-V hysteresis in halide perovskite indoor photovoltaic devices17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' However, the origin of the hysteresis effect can also be due to the interfacial defects existing at the perovskite/Spiro-OMeTAD interface46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' In this case, we employed P3HT as the hole transport layer to replace Spiro-OMeTAD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' Under indoor lighting, the PCE of P3HT-based devices reached 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='4% and 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='6% for forward and reverse bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' FW RV FW RV FW RV FW RV FW RV 2 min 10 min 30 min 45 min 60 min 10 20 30 40 PCE (%) Average forward PCE Average reverse PCE 0 20 40 60 80 100 120 140 160 180 0 5 10 15 20 25 30 PCE (%) Time (s) 2 min 10 min 30 min 45 min 60 min (a) (b) 21 Notably,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' the hysteresis of P3HT-based devices is reduced drastically compared to Spiro- OMeTAD based devices,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' as shown in Figure S9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' indicating that the Spiro-OMeTAD interface as well as the SnO2/perovskite interface are contributing to the J-V hysteresis17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' However, the photovoltaic performances of P3HT-based devices are consistently lower than Spiro-OMeTAD devices and one contributing factor could be the low conductivity of the undoped P3HT films used as HTL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' More optimization is needed at the charge transporting layer/perovskite buried interfaces to eliminate the J-V hysteresis without compromising the PCE of the TA based indoor photovoltaic devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' Powering a temperature sensor To investigate the reliability of the TA indoor photovoltaic devices as a power source for sensors, a sensor powering experiment was carried out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' Rajan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' developed a wearable graphene-based textile temperature sensor with a 1 V requirement for operation47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' The graphene sensor is comprised of three layers of graphene and coated on a single polypropylene textile fibre which is highly flexible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' The graphene-based sensor shows a negative thermal coefficient of resistance, indicating that its intrinsic resistance is reduced with increased temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' The output voltage from TA devices is tested in advance under two indoor illumination levels of 220 lux and 1000 lux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' In this case, two pixels of n-i-p TA devices on a single substrate are connected in series to obtain a higher output voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' Two illumination conditions are tested: a) ambient indoor CFL light from the ceiling (220 lux);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' b) domestic LED table lamp (1000 lux).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' The obtained output voltage from the two series connected photovoltaic devices under the two indoor illumination conditions is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='62 V and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='86 V, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' Figure 7(a) shows the experimental setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' The Keithley source meter is only used to measure the current across the graphene temperature sensor, no extra voltage from the Keithley source meter is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' Figure 7 (b) shows the resistance variation of the sensor, powered using TA devices under the two selected indoor illumination conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' The temperature sensor shows a consistent trend in resistance reduction with an increase in temperature, similar to that previously reported using a Keithley source meter as the voltage source47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' These results indicate that TA perovskite-based indoor photovoltaic devices are reliable for self-powering the low-power sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' 22 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' (a) Illustration of sensor powering with TA perovskite indoor photovoltaic device powering a graphene-based textile temperature sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' (b) Resistance variation of the temperature sensor in two different illumination conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' Conclusion The need for efficient indoor light harvesting was addressed by developing a composition tuned wide bandgap triple anion perovskite CH3NH3PbI2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='6(BrCl)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' The corresponding indoor photovoltaic devices were capable of delivering a steady-state output power of 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='4 µW/cm2 under 1000 lux warm white LED illumination and successfully self- powered a textile integrated graphene based temperature sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' Detailed microstructural and optoelectronic investigations of the triple anion perovskite unravelled its excellent crystalline quality, widened bandgap, less density of trap states and longer carrier lifetime, all contributing positively to the enhanced photovoltaic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' Our study revealed that to keep the Cl in the triple anion perovskite and to reap the beneficial effects of high VOC and enhanced charge carrier lifetime, the thermal annealing duration should be carefully optimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' Replacing the Spiro-OMeTAD hole transporting layer with P3HT was found to reduce the J-V hysteresis under indoor lighting conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' The study shows the promise of simply processed triple anion perovskites for indoor photovoltaic cells to sustainably power the IoT wireless sensor components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' Keithley sMU for current measurement 268 1060 (kOhm) Measuring 266 1050.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' Resistance ( Iluminating 1040 264 1030.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' 262 Series-connected Textile-based photovoltaic devices temperature sensor 1020 Heating 26023 Acknowledgements LKJ acknowledges funding from UKRI-FLF through MR/T022094/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' LKJ also acknowledges, Professor Iain Baikie for the work function and APS measurements and Professor Phil King and Gordon Kentish, School of Physics and Astronomy, University of St Andrews for the XRD measurements and would like to acknowledge (EPSRC): EP/T023449/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' This research used resources of the Advanced Light Source, a U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' DOE Office of Science User Facility under contract no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' DE-AC02-05CH11231.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' Work was performed at beamline 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='2, beamline scientist Nobumichi Tamura.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' Conflict of Interest The authors declare no conflict of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' Data accessibility The research data underpinning this publication can be accessed at https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content='17630/93ab4fa0-34e3-45a9-aa48-032dec3f675e [REF] Reference 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' Jagadamma, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' & Wang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' Wide-Bandgap Halide Perovskites for Indoor Photovoltaics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' Front.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' 11, 2404–2413 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' Rajan, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' Low Operating Voltage Carbon-Graphene Hybrid E-textile for Temperature Sensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' ACS Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} +page_content=' Interfaces 12, 29861–29867 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFST4oBgHgl3EQfUTgK/content/2301.13772v1.pdf'} diff --git a/ztE2T4oBgHgl3EQf4Qjg/content/tmp_files/2301.04180v1.pdf.txt b/ztE2T4oBgHgl3EQf4Qjg/content/tmp_files/2301.04180v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..4ded507a9d0ce29d05b86647f0066e265744e5bc --- /dev/null +++ b/ztE2T4oBgHgl3EQf4Qjg/content/tmp_files/2301.04180v1.pdf.txt @@ -0,0 +1,742 @@ +A Holographic Principle for Non-Relativistic Quantum Mechanics +Russell B. Thompson∗ +Department of Physics & Astronomy and Waterloo Institute for Nanotechnology, +University of Waterloo, 200 University Avenue West, +Waterloo, Ontario, Canada N2L 3G1 +(Dated: January 12, 2023) +Abstract +The quantum-classical isomorphism for self-consistent field theory, which allows quantum par- +ticles in space-time to be represented as classical one-dimensional threads embedded in a five +dimensional thermal-space-time, is summarized and used to explain a selection of quantum phe- +nomena. Introduced by Feynman, and used for modern quantum simulations, the isomorphism, +when phrased in a field-theoretic way, has been shown to be the same as quantum density func- +tional theory, the theorems of which guarantee equivalent predictions with non-relativistic quantum +mechanics. If the Feynman dimension is considered to be real, there is a duality between classical +threads in five dimensions and quantum particles in four dimensions. Using the 5D picture, intuitive +explanations are given for quantum phenomena including the uncertainty principle, tunnelling, ge- +ometric phase, and interference effects. Advantages of the 5D picture are presented, which include +fewer postulates, no measurement problem, and the need for only classical concepts in the higher +dimensional space. Limitations of the approach such as the interpretation of entanglement and +spin are discussed. +∗ thompson@uwaterloo.ca +1 +arXiv:2301.04180v1 [quant-ph] 10 Jan 2023 + +I. +INTRODUCTION +A holographic principle is an equivalence between the different mathematical descriptions +of a system in a volume and on the surface of the volume. The term “holographic” is used +because all the information about the system in the volume is contained on the surface, +analogous to true optical holograms that encode information about a three-dimensional +object on a flat surface. For a general holographic principle, the volume does not need +to be three dimensional; a d dimensional volume would contain the same information as +the d − 1 dimensional surface. A famous example of a holographic principle is the anti-de +Sitter/conformal field theory (AdS/CFT) correspondence [1]. +Richard Feynman introduced a mathematical trick by which quantum mechanical many- +body problems can be solved using classical statistical mechanics by treating the inverse of +the thermal energy in the partition function as an imaginary time dimension [2–5]. This +allows quantum simulations, such as path integral Monte Carlo [6] or centroid [7, 8] and ring +polymer [9] molecular dynamics, to be solved classically using an extra, fictitious, dimension. +This has been called the quantum-classical isomorphism [10]. +In addition to simulations, it has been shown that polymer self-consistent field theory +(SCFT) also obeys the quantum-classical isomorphism, and is, under the right conditions, +equivalent to quantum density functional theory (DFT) [11, 12]. This allows one to treat the +thermal dimension introduced by Feynman as more than just mathematics, and consider it a +physical dimension giving a 5-dimensional thermal-space-time. A major motivation for doing +this is that the theorems of DFT guarantee the equivalence between the DFT formalism and +the predictions of quantum mechanics (QM) [13–15], allowing the explanation of all non- +relativistic QM with fewer postulates [12, 16] that use only classical concepts, albeit in a +higher dimensional classical space. +SCFT represents quantum particles as classical one-dimensional threads embedded in the +5D thermal-space-time, and through the theorems of DFT, this model must make all the +same predictions as QM [11, 12, 16]. It represents a holographic principle between QM in +4D (the surface) and polymer-like threads in 5D (the volume) which captures the wave- +particle duality of QM. The 5D SCFT picture has fewer postulates than 4D wave function +QM [12, 16], and does not suffer the same pathologies as wave function QM. +For example, since SCFT does not use wave functions, the concept of superposition is +2 + +not necessary, and there is no wave function collapse. From the SCFT perspective, the +measurement problem in 4D is an artifact of a projection onto the 4D surface which hides +the physical 5D thread nature of quantum particles. There is no measurement problem or +superposition in the 5D SCFT quantum particle picture any more than there is in SCFT +for real polymers, which uses the same mathematics. +The origins of randomness in QM is also trivial in the 5D picture. SCFT is a statistical +mechanics theory, and so in 5D, randomness has the same origins as in classical statistical +mechanics. Specifically, there is ignorance about the conformations of the threads in the +system. SCFT thus adheres to the ensemble interpretation of QM [17], in which QM only +makes predictions about ensembles of systems. +SCFT mathematics predict, using a 5D classical framework without wave functions, the +stability and shell structure of atoms [11, 16, 18], spontaneous spherical symmetry breaking +in atoms [19], molecular bonding [20], and even the spontaneous emergence of classical +electromagnetism [12]. In this paper, many of these features of SCFT, as presented at the +15th Biennial Quantum Structure Conference, are reviewed and other topics which arose +in the questions and discussions following the presentation are explained, such as quantum +statistics, the uncertainty principle, quantum kinetic energy, tunnelling, the double slit +experiment, geometric phase including the Aharonov-Bohm effect, entanglement, exchange, +and the Pauli exclusion principle. First principles derivations, numerical methods and more +detailed discussions of SCFT applied to quantum systems can be found in references [11, +12, 16, 18–20]. +II. +STATIC PROPERTIES +Consider a single quantum mechanical particle of mass m at a temperature T. +The +quantum statistical mechanical partition function for this system is +Q = +� +i +e−βEi +(1) +where β = 1/kBT, kB is Boltzmann’s constant and Ei are the allowed energy states. Feyn- +man showed that (1) can be exactly rephrased as the path integral [3] +Q = +� +dr0 +� +tr +exp +� +− +� β +0 +� +m +2ℏ2 +�dr +ds +�2 ++ V (r(s)) +� +ds +� +Dr +(2) +3 + +where ℏ is Planck’s reduced constant and V is the potential. The integral +� +tr is taken over +all paths such that r(0) = r(β) ≡ r0, that is, closed ring paths. The partition function (2) +is identical with the classical partition function for a ring polymer in the Gaussian thread +model of polymer self-consistent field theory (SCFT) [21]. A ring polymer is represented in a +coarse-grained way in SCFT as a mathematical contour r(s), embedded in three dimensional +space R3. In (2) however, s refers to the inverse thermal energy β, and so is an independent +dimension. Since SCFT is a classical statistical mechanical theory, one can choose to picture +quantum particles as classical one-dimensional threads in a space with an extra thermal +dimension – a five-dimensional thermal-space-time. +For many-body systems, SCFT uses a mean field approximation for a single particle +subject to a field due to all other particles. For quantum systems, correlations and exchange +effects can be included through the addition of an exchange-correlation term, just as in +quantum DFT. In fact, as shown in appendix B of reference [11] and appendix C of reference +[19], polymer SCFT can be shown to be equivalent to Kohn-Sham DFT assuming perfect +enforcement of the Pauli exclusion principle. Through the theorems of DFT [13, 14], this +means that all predictions of SCFT, again assuming the Pauli exclusion principle, must +be consistent with those of non-relativistic quantum mechanics. In reality, approximations +must always be made, and so discrepancies may arise, but in the non-interacting static case, +one may expect agreement. +For example, agreement should be perfect for particle-in-a-box situations. As expected, +the SCFT solution of the hydrogen atom agrees exactly with the analytical quantum results +[16, 18]. If correlations are ignored, the helium atom solved with SCFT agrees exactly with +Hartree-Fock wave function theory [18]. Other quantum phenomena that can be explained +intuitively in the static, non-interacting situation include the uncertainty principle, quantum +kinetic energy, tunnelling and the measurement problem (collapse of the wave function). The +equations of SCFT, as applied to quantum particles in the static case, are now reviewed in +order to explain each of these phenomena. Complete derivational details can be found in +reference [11]. +4 + +A. +Summary of SCFT Equations +The main governing equation in SCFT is +∂q(r0, r, s) +∂s += ℏ2 +2m∇2q(r0, r, s) − w(r, β)q(r0, r, s) +(3) +subject to the initial condition +q(r0, r, 0) = δ(r − r0). +(4) +This gives the unnormalized probability q(r0, r, s) that a particle at position r0 at high +temperature (s = 0) will be found at position r at a temperature corresponding to s = β. +Note that q(r0, r, s) is a completely real and positive definite quantity that depends on +the field w(r, β) which quantifies the interactions due to all other quantum particles in the +system and external potentials. For N particles in the system, the single particle density at +position r is then +n(r, β) = +N +Q(β)q(r, r, β) +(5) +where the normalization factor +Q(β) = +� +q(r, r, β)dr +(6) +is the partition function for a single particle subject to the field w(r, β). Equation (5) shows +that it is sufficient to consider ring polymers only, since only paths that start at r and return +to r contribute to the quantum particle density at the position r. That is, the density gives +the probability of a particle at position r at high temperature (s = 0) returning to position +r at a temperature corresponding to β – see the black contours in figure 1(a). An intuitive +consideration of paths in 5D allows one to include some quantum correlations by considering +paths beyond rings. It will be shown that these also reduce to ring configurations, and these +many-body paths can lead to a second order transition as Feynman found for superfluid +helium [2, 3]. +To see that many-body non-ring paths are equivalent to rings, consider two particles at +positions r and r′, respectively, both at high temperature (s = 0). One can ask what the +probability is of still finding two particles at positions r and r′ at a temperature corresponding +to β. The various paths the two particles can take in order to contribute to the two-particle +density can be seen in figure 1(b). +This argument can be extended to any number of +5 + +FIG. 1. Schematic of threads in thermal-space. x-axis is space and y-axis is inverse thermal energy +β. (a) Contours that start from r and return to r, forming rings, are shown in black; those that do +not return to the starting position r, forming open threads, are shown in grey. (b) Two starting +positions r and r′ and some contours that terminate at r and r′. These include both rings and +cross-paths that start at r and end at r′ and vice versa. +particles, but is here confined to the pair density for simplicity. Following figure 1(b), the +unnormalized pair density will be +n(r, r′, β) ∝ q(r, r, β)q(r′, r′, β) + q(r, r′, β)q(r′, r, β). +(7) +Integrating (7) over all possible pairs of particles should give the total number of pairs +� � +n(r, r′, β)drdr′ = N(N − 1) +2 +(8) +which allows the proportionality constant for (7) to be set to +k = N(N − 1) +2Q(2)(β) +(9) +6 + +s=0 +S=0 +r +r +(a) +(b)where +Q(2)(β) = +� � +[q(r, r, β)q(r′, r′, β) + q(r, r′, β)q(r′, r, β)] drdr′ += +� +q(r, r, β)dr +� +q(r′, r′, β)dr′ + +� +dr +�� +dr′q(r, r′, β)q(r′, r, β) +� += Q(β)2 + Q(2β). +(10) +The last line of (10) follows from (6) and the fact that +� +dr′q(r, r′, β)q(r′, r, β) = q(r, r, 2β). +(11) +Equation (11) shows that these cross-paths are equivalent to ring paths of twice the contour +length, as first pointed out by Feynman [3]. Equations (7)-(11) allows one to write the pair +density as +n(r, r′, β) = +N(N − 1) +2 [Q(β)2 + Q(2β)] [q(r, r, β)q(r′, r′, β) + q(r, r′, β)q(r′, r, β)] . +(12) +One gets the single-particle density by integrating over one of the position variables according +to +n(r, β) = +2 +(N − 1) +� +n(r, r′, β)dr′. +(13) +The pre-factor in (13) is to switch from counting pairs to counting singlets. This gives +n(r, β) = +N +[Q(β)2 + Q(2β)] [q(r, r, β)Q(β) + q(r, r, 2β)] . +(14) +In the limit of low temperatures β → ∞, numerical calculations show that the density +saturates with respect to β and so stops changing with β. In particular, it is found that +lim +β→∞ Q(2β) = Q(β)2 +(15) +lim +β→∞ q(r, r, 2β) = q(r, r, β)Q(β). +(16) +Substituting (15) and (16) into (14) immediately gives the ring polymer density expression +(5). For some exotic systems, such as the superfluid helium studied by Feynman [2, 3], +there is a second order transition due to the correlations in (14). Since the helium system is +bosonic, one should also include three and higher-body terms in (14). For fermions, due to +the Pauli exclusion principle, the maximum number of particles in any state will be two, and +so (14) would not require any higher terms, and one would not expect to see a transition.1 +1 It may be possible to show in future work that the superconducting transition is related to quantum +correlations in (14), but for now this is speculation. +7 + +For example, even using a temperature on the order of the surface of the sun, one gets less +than a one one-thousandth percent change in the SCFT calculated helium atomic binding +energy due to non-ring (longer ring) paths. In order to intuitively picture phenomena using +the 5D holographic principle, equation (3) for open threads will be used instead of rings +since this will give qualitatively equivalent results. Several archetypal quantum phenomena +are now examined using the open thread picture. +B. +Wave functions, quantum statistics and the measurement problem +The non-ring paths shown in figure 1(a) by the grey contours and expressed in equation +(14) correspond exactly to matrix elements in 4D wave function QM arising from symmet- +ric statistics [3]. Anti-symmetric statistics requires replacing the plus sign in equation (7) +with a minus sign, as should be done when replacing Feynman’s discussion of bosons with +anti-symmetric wave functions for fermions [3]. In 5D, the minus sign in (7) causes the math- +ematics to break down at zero temperature because the partition function corresponding to +equation (10) becomes equal to zero. This makes sense for two reasons: First, the propa- +gator q(r, r, β) can be expanded in terms of complex orbitals that correspond to quantum +mechanical states [19]. Since fermions cannot multiply occupy a single state (ignoring spin), +the number of microstates available for such a configuration should be zero, as is found. In +other words, from the 5D perspective, Pauli exclusion is the cause of anti-symmetric wave +function statistics and not the effect. Second, the 5D holographic statement of QM does +not use wave functions (or for that matter, quantum numbers or superpositions), and so +concepts of bosons and fermions, which are based on wave function statistics, are super- +fluous. The theorems of DFT guarantee that all predictions of polymer SCFT applied to +quantum particles must be the same as those of non-relativistic QM, and particles will turn +out to behave in 4D as bosons or fermions based on their intrinsic and interaction properties. +Threads in 5D need not be assigned a priori to either group to perform SCFT calculations. +Likewise, since there are no wave functions, one can see from equation (5) for the density, +for example, that there is no wave function collapse during calculations. The measurement +problem is an artifact of doing calculations in 4D for a system that can be rigorously viewed +as projected out of 5D. +8 + +C. +Quantum kinetic energy and the uncertainty principle +An extended object like a Gaussian thread has properties which arise from its internal +degrees of freedom. Thermodynamically, these aspects are captured by the conformational +entropy, which counts the number of configurations available to a thread when subject to a +field w(r, β) at a temperature T. In SCFT, the contribution of the conformational entropy +to the free energy of a system is given by [11] +Fconf = − 1 +β +� +drn(r) [ln q(r, r, β) + βw(r)] . +(17) +The complete SCFT free energy expression for a quantum mechanical system is given in +reference [11]2 and accounts for all thermodynamic quantities except the quantum kinetic +energy, while having the extra term (17) for conformational degrees of freedom of a classical +thread in 5D. Thus, since the theorems of DFT guarantee that the SCFT expression must +give the same results as QM, and exactly correct results for systems like the hydrogen +atom and the Hartree-Fock helium atom are obtained, one is forced to identify the 5D +conformational entropy as the 4D quantum kinetic energy. +The thread-like internal degrees of freedom are also the 5D expression of the 4D un- +certainty principle [11, 16]. The derivation of the 5D governing diffusion equation (3) in +appendix A of reference [11] depends crucially on the uncertainty principle. In terms of +position and momentum, if these quantities commute, then (3) collapses to the thermody- +namics of a point-like particle subject to the field w(r, β) rather than a random walk. In +other words, equation (3), which expresses the Gaussian thread nature of quantum particles +in 5D, is equivalent to including the uncertainty principle in 4D. There is a close connection +therefore between the uncertainty principle, quantum kinetic energy and conformational +entropy which the holographic principle makes clear. +D. +Stability of atoms +The conformational entropy of SCFT threads also explains the stability of atoms. Clas- +sically, it is energetically favourable for electrons to be attracted into the Coulomb potential +of the ionic core, collapsing the electron density into a spike at the nucleus. The conforma- +tional entropy, given by equation (17) however, would become enormous because, even at +2 Equation (17) of reference [11] contains typos which are corrected here. +9 + +zero temperature, the thread would be confined to a single microstate. SCFT reveals that +the experimentally observed electron density is a result of the frustration between energy +and conformational entropy in the 5D statistical mechanical picture. +E. +Tunnelling +Consider the operator on the right hand side of the SCFT diffusion equation (3), which +is +H ≡ ℏ2 +2m∇2 − w(r, β). +(18) +This obeys an eigenvalue equation +Hφi(r) = Eiφi(r) +(19) +where φi(r) and Ei are the eigenfunctions and eigenvalues, respectively. Note that whereas +q(r0, r, s) was completely real and positive definite, φi(r) can be complex. Equation (19) is a +one-particle time-independent Schr¨odinger equation, and is called the Kohn-Sham equation +in the context of DFT. The density expression (5) can likewise be shown to become identical +to the Kohn-Sham density expression, assuming perfect enforcement of the Pauli exclusion +principle – see appendix B of reference [11] and appendix C of reference [19]. +For the +case of a single quantum particle, the SCFT – Kohn-Sham DFT duality holds without +conditions. Thus, the SCFT equations formed around (3) will make all the same predictions +as static quantum mechanics, including tunnelling. This is not surprising since ring polymer +formalisms are used in quantum simulations to study tunnelling [22]. +Equation (3) can therefore be used to qualitatively understand tunnelling in the holo- +graphic 5D context. Figure 2(a) shows a typical tunnelling situation with a barrier of finite +energy E. A classical particle with insufficient kinetic energy cannot penetrate the barrier. +For a 5D thread described by (3), w(r, β) = E in the range x > 0 represents an energy +penalty that discourages the contour from entering that region, but does not forbid it – +there is a competing entropy benefit which lowers the free energy due to increased confor- +mations when the thread enters the region – see figure 2(b). At s = β, there is thus a +non-zero probability of finding the particle in the barrier. As long as the initial position of +the thread at s = 0 is outside the barrier, the extra thermal dimension provides a classical +trajectory through which to enter the barrier zone. +10 + +FIG. 2. Schematic of static tunnelling in the holographic picture. (a) A classical particle, shown +by a dot, cannot enter the energy region E if it has too low a kinetic energy. (b) A thread in +thermal-space starts at the same position as the classical particle, where it also cannot enter the +region E. As the contour extends from the initial point according to (3), the barrier E discourages, +but does not forbid entry. +III. +DYNAMIC PROPERTIES +It is not immediately obvious how to write down the mathematical formulation for the +dynamics of quantum particles in 5D as classical thermal threads. This is not surprising, +since writing a dynamical version of SCFT for actual polymers is similarly difficult and +one is forced to use approximations. However, just as it is known that chemical polymers +continue to be extended one dimensional objects even though one cannot write exact SCFT +mathematical equations for their dynamics, so it can be shown that quantum particles can +continue to be represented as 1D thermal-world-lines even without giving explicit dynamic +11 + +(a) +E +X=0 +(q) +s=βSCFT equations. +To see this, the argument of reference [12] is followed, considering again the eigenvalue +equation of the operator on the right hand side of (3). As mentioned, the Kohn-Sham equa- +tion (19) is a one-particle time-independent Schr¨odinger equation. In 4D wave function QM, +non-relativistic dynamics are commonly included through postulating the time-dependent +Schr¨odinger equation. It is therefore convenient to do a similar thing with the Kohn-Sham +DFT expression (19), that is, it is generalized and dynamics are postulated to be given by +the equation +iℏ ∂ +∂tφi(r, t) = Hφi(r, t). +(20) +Together with the corresponding density and field expression – see reference [12] – equation +(20) is identical to the formulas of time-dependent DFT (TDDFT). A result of TDDFT +is that the time-dependent density n(r, t) is proportional to a real quantity which is the +time-dependent version of q(r, r, β) [23]: +n(r, t) ∝ q(r, r, β, t). +(21) +Just as q(r, r, β) is the mathematical expression of thermal-world-line statistics in the static +case, (21) demonstrates that the classical thread picture of quantum particles in 5D is +consistent with quantum dynamics, even if on a practical level, one uses the equations +of TDDFT, including (20), as a black-box to perform calculations in terms of complex +eigenfunctions. +The consistency of the 5D polymer thread model with TDDFT means that, through +the Runge-Gross theorem [15], all dynamic predictions of the 5D thread picture must be +consistent with those of the dynamic 4D wave function picture. Similar to the static case, +one expects to be able to use ring polymers to effectively describe quantum particles, but +for explanatory purposes, open threads based on equation (3) will be used instead. The +holographic principle allows one to consider a number of dynamic effects in non-relativistic +QM. +A. +The double slit experiment +A quasi-static approximation can be adopted in order to intuitively understand inter- +ference in the double slit experiment. The static governing equation (3) requires that if a +12 + +FIG. 3. A schematic of the double slit experiment. (a) A classical particle, denoted as a dot and +with a velocity v, is shown incident on the double slit at a position such that it cannot pass through +either slit; it will therefore be blocked from being detected on the final screen (not shown). (b) +A thread in thermal-space with s = 0 at the same position as the classical particle in (a). The +probability of the particle being found at this position for s = β is not 1; there is a non-zero +probability that the other end of the contour will be found at a position that would be impossible +for a classical point particle. +classical particle is at a position r at high temperature, then in the absence of other par- +ticles or external influences, the probability of finding the particle at a temperature where +quantum effects are important will be a Gaussian distribution centred on r. Microscopically, +this means that each particle thread will explore various thermal paths and might be found +at a position that would be impossible classically by wrapping around infinite potential ob- +structions in the thermal dimension– see figure 3. As discussed in references [11] and [12], +13 + +y +(a) +(b) +1the static non-locality of threads means that the density of particles predicted at the screen +will be different from that expected classically, but without a further dynamic aspect, it +does not yet agree with experimental predictions. It is known from TDDFT and the Runge- +Gross theorem that 5D classical threads must be able to reproduce interference phenomena +– this is consistent with equation (20) which has an oscillatory nature. In other words, the +propagator q(r, r, β, t) for the probability of a thread returning to its starting point must +change in time commensurate with the de Broglie wavelength, or the function q(r0, r, β, t) +for the probability of a thread ending at a different position, for a fixed r0 and t, will show +the de Broglie wavelength. Much like an oscillating pendulum has a non-uniform probability +of being found at various points along its period, so a pulsating 5D thread at an instant +of time should oscillate with its high temperature initial point r0 at s = 0 fixed and other +points s free to vibrate according to (20) – see figure 1(a). By construction then, a classical +5D thread must agree with experimental double slit results since the time-dependent Kohn- +Sham equation (20) is identical with the time-dependent Schr¨odinger equation for a single +particle, while still being consistent with classical 5D thread statistics through (21). +B. +Geometric phase and the Aharonov-Bohm effect +When a quantum particle is transported through a closed loop in a parameter space, a +phase change results due to the non-flat geometry of the space [24]. This phase, known as the +geometric phase or Berry’s phase, is detectable in experiments, a famous example being the +Aharonov-Bohm effect [25, 26], but is not unique to quantum systems. It is seen in classical +phenomena such as parallel transport and the precession of classical pendulums, which also +experience phase changes when transported through closed loops in non-flat geometries. +The 5D holographic principle in which quantum particles are classical threads oscillating in +thermal-space-time, as described in the previous section, immediately allows a 5D classical +interpretation of quantum geometric phase phenomena, such as the Aharonov-Bohm effect, +in terms of classical geometric phase changes in systems like pendulums. +14 + +FIG. 4. +Three examples, (a)-(c), of 5D threads moving in opposite directions with the same +conformations. The conformations change as the particles travel, but each thread in a pair will +maintain identical conformations at any given time. +C. +Entanglement +Imagine a quantum particle which is split into two daughter particles which travel off +in opposite directions. The 5D holographic principle view of this is shown in figures 4 (a)- +(c). +In each case, the two daughter threads can be considered to be entangled because +their internal degrees of freedom (the conformational shapes they have) are correlated (they +have the same shape as each other). With no further communication between them, any +experiments done on one will be correlated with the other, to a larger extent than would +be expected for classical point particles. +This can be quantitatively seen with SCFT if +measurements of positions and momenta for pairs of entangled particles at distant locations +are considered. +15 + +(a) +V +(q) +(c)The Fourier transform of the governing thread equation (3) gives a relationship between +momentum and position – see appendix A of reference [11]. Suppose Alice and Bob each +perform a series of momentum or position measurements on opposite sides of the exper- +imental apparatus. +Given that the threads emitted in the two directions have identical +conformations,3 if Bob on his side of the apparatus happens to have chosen to measure +the same quantity as Alice, they will completely agree on the results they find. However, +if Alice measures momentum and Bob measures position, after they have collected enough +data, it will seem as if the wave function of the entire system collapses whenever one of them +takes a measurement only on their own side, even though the 5D picture does not use wave +functions. +To see this, suppose Alice performs a series of momentum measurements on her side +of the experimental setup, and she makes note of all momenta that fall within a narrow +range – that is she notes all particles that have a chosen momentum. If Bob happens to +have measured position for those pairs, when they compare their results, Alice will have +one value of momentum for all her particles, but Bob will have essentially random values of +position. That is because although each particle in a pair has the same thread conformation +as its partner, separate pairs will have completely different conformations between them – +see figure 4. Since the polymeric threads are not point particles, the classical relationship +between momentum and position is not valid. From the Fourier transform of equation (3), in +the absence of a field w(r, β), a known single momentum will give a distribution of positions +– the actual position found for a single experiment will depend on the specific conformation +of the polymer. The quantitative values of the correlations for 5D classical thread pairs +must agree with those of 4D quantum mechanics from the theorems of DFT. +This guess for the mechanism of entanglement in 5D is very speculative and includes only +the maximally and minimally correlated situations, which Bell comments are “... the only +features ... commonly used in verbal discussions of this problem” [27]. He notes that these +features are easily explained in terms of local hidden variables. In order to completely satisfy +Bell-type theorems with hidden variables, intuitively unacceptable non-local variables are +needed. What is unacceptable in 4D may be more palatable in higher dimensions, and an +extended mechanism involving cross-paths, equivalent to a single double-length contour that +3 The conformations would change as the particles travel, but they would change in exactly the same way, +so they would always have the same conformation as each other at any given time. +16 + +must pass through both Alice and Bob’s measuring devices, offers an intriguing mechanism +not restricted by the speed of light since the imaginary time plane is not Lorentzian. +D. +Other features +The holographic principle for QM works in a 5D thermal-space-time. It has also been +shown that it obeys a cylinder condition; specifically, in the classical limit, the thermal +dimension still exists, but no longer has any quantum effect on predictions – see appendix +C of reference [11]. Five dimensions subject to a cylinder condition in the classical limit are +the assumptions of Kaluza theory [28, 29], and it follows that the postulates necessary for +the polymer thread picture of QM include those necessary to derive electromagnetism from +the structure of general relativity by using five dimensions instead of four [12]. Thus, the 5D +holographic principle of QM, in addition to having fewer postulates than 4D wave function +theory, and avoiding various QM pathologies, leads directly to other physical phenomena +outside of QM without additional assumptions [12]. +There are many other quantum phenomena that one could attempt to intuitively ex- +plain, that have not yet been addressed. For example, the physical origins of spin in 5D, +interaction-free measurement (the Elitzur-Vaidman bomb experiment) [30], other interfer- +ometric predictions of QM [31, 32], non-statistical tests of entanglement, such as Green- +berger–Horne–Zeilinger states, etc. While the holographic principle will not always be as +easy to apply as in the examples listed in this paper, the theorems of DFT guarantee that the +5D classical thread model will make exactly the same predictions as standard non-relativistic +QM. +IV. +THE PAULI EXCLUSION PRINCIPLE +As has been mentioned, SCFT can be shown to be equivalent to quantum DFT assuming +enforcement of the Pauli exclusion principle [11]. This is what allows the use of the theorems +of DFT to show the holographic principle connecting 5D polymer SCFT with 4D wave +function QM. It is therefore necessary to postulate the nature of the exclusion principle in +the classical 5D picture, in addition to the two other assumptions, namely that quantum +particles are classical threads in 5D, and that these threads vibrate according to (20) [12]. +17 + +Quantum threads in 5D are postulated to obey excluded volume – just as the trajectories +of classical particles do not allow multiple particles to be in the same place at the same +time, so it is assumed that threads cannot be in the same place at the same imaginary time +(same value of β). This property for classical threads in 5D maps onto the Pauli exclusion +principle for particles in 4D. +There are several reasons for assuming this excluded volume: First, it is already an +accepted feature of the quantum-classical isomorphism that gives correct results. Feynman +used the excluded volume of threads in imaginary time to justify removing trajectories in +his study of the λ-transition [3]. He did not identify this with the exclusion principle – he +was working with bosons – but following his example, any massive particle would have to +have this excluded volume feature, including electrons. The mystery in terms of the electron +is why do up to two of these fermions, assuming opposite spins, not feel excluded volume +in the 5D space? This question has not been answered, but this feature can be accepted +as a property of electrons since, although quantum particles are given structure as one- +dimensional threads, this does not say anything about the cross-sectional structure of each +thread – just like in 4D QM, electrons have no internal structure. Using 5D excluded volume +in this way for electron calculations produces correct shell structure of the atoms [16, 18] +and correct molecular bonding [20], at least within the context of the approximations used +to implement the excluded volume. +A second reason that supports 5D classical thread excluded volume is that it gives the +correct scaling behaviour in the uniform limit for both the quantum kinetic energy and Dirac +exchange. Thomas [33] and Fermi [34] found that the quantum kinetic energy of the high +density uniform electron gas should scale with the density of the gas n as n5/3. Dirac [35] +found a correction term due to exchange effects that scales as n4/3. If techniques of polymer +scaling theory as given by de Gennes [36] are used, assuming excluded volume between +threads in 5D gives both the correct Thomas-Fermi expression [16, 18] and the correct Dirac +exchange term [18]. +Third, the 5D excluded volume picture approaches satisfying conditions necessary for +the Pauli field as described by Levy and Ou-Yang [37]. +To date, the excluded volume +model has only been implemented on a coarse approximate level [18, 19], but even this +approximate version satisfies four of five conditions, with the fifth being violated by an +amount commensurate with the scale of the approximation – more details can be found in +18 + +references [18] and [19]. More work is needed to further test the postulate relating excluded +volume in 5D to the Pauli exclusion principle in 4D. +V. +CONCLUSIONS AND FUTURE WORK +A classical thread model for quantum particles in 5D can produce all the same predictions +as 4D quantum mechanics, but with fewer postulates. +These axioms are: 1. +Quantum +particles are classical threads in 5D (which should be formally treated as rings, although +that condition has been relaxed in this work for illustrative purposes). 2. Time evolution is +governed by the time-dependent Kohn-Sham equation, which can be shown to be consistent +with the statistics of 5D classical threads. 3. Pauli exclusion is enforced through higher +dimensional excluded volume. +The duality between 5D classical threads and 4D quantum wave functions defines a holo- +graphic principle. Advantages of the 5D perspective, in addition to having fewer postulates, +include: a realist model which is, in the context of higher dimensions, deterministic; no +measurement problem; an explanation of randomness through the ensemble interpretation +utilizing the internal conformational degrees of freedom of 5D threads within classical sta- +tistical mechanics; an interpretation of quantum statistics, exchange and the Pauli exclusion +principle; intuitive explanations for quantum phenomena such as the Aharonov-Bohm effect, +the uncertainty principle, quantum kinetic energy, tunnelling and the double slit experiment. +Many issues remain unexplored however. Spin is easily included in 5D SCFT calcula- +tions, but no physically intuitive picture for it has been provided yet – for example, no +explanation is given as to why two electrons of opposite spins do not feel excluded volume +in 5D. Non-statistical tests of entanglement, such as Greenberger–Horne–Zeilinger states, +should be considered. A relativistic version of the principle is required, and connections +with thermal quantum field theory, related to Matsubara imaginary time, should be ex- +plored [38]. Justifications that the Pauli exclusion principle is excluded volume in 5D have +been listed, but further evidence is necessary. Improved approximations of the 5D excluded +volume in SCFT calculations could show whether the shell structure of atoms approaches +chemical accuracy, or diverges from experiment. Likewise, more complex SCFT molecular +calculations and time-dependent SCFT problems could either support or contradict the 5D +picture. One advantage of 5D SCFT is that it is readily compared to experiments, and +19 + +although inevitable approximations in numerical calculations are a confounding factor, this +limitation is no different from any other quantum calculation. If nothing else, the SCFT +holographic principle is a useful quantum calculational tool, but its implications for quantum +foundations are much more interesting. +ACKNOWLEDGEMENTS +The author acknowledges helpful discussions with many of the participants of the 15th +Biennial Quantum Structure Conference, with P. A. LeMaitre for suggesting changes to +the manuscript, and with M. W. Matsen who pointed out the relationship (11) between +cross-path contours and longer single rings. +[1] Juan Maldacena, “The large N limit of superconformal field theories and supergravity,” Ad- +vances in Theoretical and Mathematical Physics 2, 231–252 (1998). +[2] Richard P. Feynman, “The λ-transition in liquid helium,” Physical Review 90, 1116–1117 +(1953). +[3] Richard P. Feynman, “Atomic theory of the λ-transition in helium,” Physical Review 91, +1291–1301 (1953). +[4] Richard P. Feynman, “Atomic theory of liquid helium near absolute zero,” Physical Review +91, 1301–1308 (1953). +[5] Richard P. 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Dirac, “Note on exchange phenomena in the Thomas atom,” Proceedings of the +Cambridge Philosophical Society 26, 376–385 (1930). +[36] Pierre-Gilles de Gennes, Scaling Concepts in Polymer Physics (Cornell University Press, +Ithaca NY, 1979). +22 + +[37] M. Levy and H. Ou-Yang, “Exact properties of the Pauli potential for the square root of the +electron density and the kinetic energy functional,” Phys. Rev. A 38, 625–629 (1988). +[38] Massimo Blasone, Giuseppe Vitiello, and Petr Jizba, Quantum Field Theory and its Macro- +scopic Manifestations (Imperial College Press, London, 2011). +23 + diff --git a/ztE2T4oBgHgl3EQf4Qjg/content/tmp_files/load_file.txt b/ztE2T4oBgHgl3EQf4Qjg/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d1bdc5369d4618ece06bef8727d509bfd81c732f --- /dev/null +++ b/ztE2T4oBgHgl3EQf4Qjg/content/tmp_files/load_file.txt @@ -0,0 +1,394 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf,len=393 +page_content='A Holographic Principle for Non-Relativistic Quantum Mechanics Russell B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' Thompson∗ Department of Physics & Astronomy and Waterloo Institute for Nanotechnology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' University of Waterloo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' 200 University Avenue West,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' Waterloo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' Ontario,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' Canada N2L 3G1 (Dated: January 12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' 2023) Abstract The quantum-classical isomorphism for self-consistent field theory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' which allows quantum par- ticles in space-time to be represented as classical one-dimensional threads embedded in a five dimensional thermal-space-time,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' is summarized and used to explain a selection of quantum phe- nomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' Introduced by Feynman, and used for modern quantum simulations, the isomorphism, when phrased in a field-theoretic way, has been shown to be the same as quantum density func- tional theory, the theorems of which guarantee equivalent predictions with non-relativistic quantum mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' If the Feynman dimension is considered to be real, there is a duality between classical threads in five dimensions and quantum particles in four dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' Using the 5D picture, intuitive explanations are given for quantum phenomena including the uncertainty principle, tunnelling, ge- ometric phase, and interference effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' Advantages of the 5D picture are presented, which include fewer postulates, no measurement problem, and the need for only classical concepts in the higher dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' Limitations of the approach such as the interpretation of entanglement and spin are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' ∗ thompson@uwaterloo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content='ca 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content='04180v1 [quant-ph] 10 Jan 2023 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' INTRODUCTION A holographic principle is an equivalence between the different mathematical descriptions of a system in a volume and on the surface of the volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' The term “holographic” is used because all the information about the system in the volume is contained on the surface, analogous to true optical holograms that encode information about a three-dimensional object on a flat surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' For a general holographic principle, the volume does not need to be three dimensional;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' a d dimensional volume would contain the same information as the d − 1 dimensional surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' A famous example of a holographic principle is the anti-de Sitter/conformal field theory (AdS/CFT) correspondence [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' Richard Feynman introduced a mathematical trick by which quantum mechanical many- body problems can be solved using classical statistical mechanics by treating the inverse of the thermal energy in the partition function as an imaginary time dimension [2–5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' This allows quantum simulations, such as path integral Monte Carlo [6] or centroid [7, 8] and ring polymer [9] molecular dynamics, to be solved classically using an extra, fictitious, dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' This has been called the quantum-classical isomorphism [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' In addition to simulations, it has been shown that polymer self-consistent field theory (SCFT) also obeys the quantum-classical isomorphism, and is, under the right conditions, equivalent to quantum density functional theory (DFT) [11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' This allows one to treat the thermal dimension introduced by Feynman as more than just mathematics, and consider it a physical dimension giving a 5-dimensional thermal-space-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' A major motivation for doing this is that the theorems of DFT guarantee the equivalence between the DFT formalism and the predictions of quantum mechanics (QM) [13–15], allowing the explanation of all non- relativistic QM with fewer postulates [12, 16] that use only classical concepts, albeit in a higher dimensional classical space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' SCFT represents quantum particles as classical one-dimensional threads embedded in the 5D thermal-space-time, and through the theorems of DFT, this model must make all the same predictions as QM [11, 12, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' It represents a holographic principle between QM in 4D (the surface) and polymer-like threads in 5D (the volume) which captures the wave- particle duality of QM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' The 5D SCFT picture has fewer postulates than 4D wave function QM [12, 16], and does not suffer the same pathologies as wave function QM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' For example, since SCFT does not use wave functions, the concept of superposition is 2 not necessary, and there is no wave function collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' From the SCFT perspective, the measurement problem in 4D is an artifact of a projection onto the 4D surface which hides the physical 5D thread nature of quantum particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' There is no measurement problem or superposition in the 5D SCFT quantum particle picture any more than there is in SCFT for real polymers, which uses the same mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' The origins of randomness in QM is also trivial in the 5D picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' SCFT is a statistical mechanics theory, and so in 5D, randomness has the same origins as in classical statistical mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' Specifically, there is ignorance about the conformations of the threads in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' SCFT thus adheres to the ensemble interpretation of QM [17], in which QM only makes predictions about ensembles of systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' SCFT mathematics predict, using a 5D classical framework without wave functions, the stability and shell structure of atoms [11, 16, 18], spontaneous spherical symmetry breaking in atoms [19], molecular bonding [20], and even the spontaneous emergence of classical electromagnetism [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' In this paper, many of these features of SCFT, as presented at the 15th Biennial Quantum Structure Conference, are reviewed and other topics which arose in the questions and discussions following the presentation are explained, such as quantum statistics, the uncertainty principle, quantum kinetic energy, tunnelling, the double slit experiment, geometric phase including the Aharonov-Bohm effect, entanglement, exchange, and the Pauli exclusion principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' First principles derivations, numerical methods and more detailed discussions of SCFT applied to quantum systems can be found in references [11, 12, 16, 18–20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' STATIC PROPERTIES Consider a single quantum mechanical particle of mass m at a temperature T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' The quantum statistical mechanical partition function for this system is Q = � i e−βEi (1) where β = 1/kBT, kB is Boltzmann’s constant and Ei are the allowed energy states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' Feyn- man showed that (1) can be exactly rephrased as the path integral [3] Q = � dr0 � tr exp � − � β 0 � m 2ℏ2 �dr ds �2 + V (r(s)) � ds � Dr (2) 3 where ℏ is Planck’s reduced constant and V is the potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' The integral � tr is taken over all paths such that r(0) = r(β) ≡ r0, that is, closed ring paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' The partition function (2) is identical with the classical partition function for a ring polymer in the Gaussian thread model of polymer self-consistent field theory (SCFT) [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' A ring polymer is represented in a coarse-grained way in SCFT as a mathematical contour r(s), embedded in three dimensional space R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' In (2) however, s refers to the inverse thermal energy β, and so is an independent dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' Since SCFT is a classical statistical mechanical theory, one can choose to picture quantum particles as classical one-dimensional threads in a space with an extra thermal dimension – a five-dimensional thermal-space-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' For many-body systems, SCFT uses a mean field approximation for a single particle subject to a field due to all other particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' For quantum systems, correlations and exchange effects can be included through the addition of an exchange-correlation term, just as in quantum DFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' In fact, as shown in appendix B of reference [11] and appendix C of reference [19], polymer SCFT can be shown to be equivalent to Kohn-Sham DFT assuming perfect enforcement of the Pauli exclusion principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' Through the theorems of DFT [13, 14], this means that all predictions of SCFT, again assuming the Pauli exclusion principle, must be consistent with those of non-relativistic quantum mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' In reality, approximations must always be made, and so discrepancies may arise, but in the non-interacting static case, one may expect agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' For example, agreement should be perfect for particle-in-a-box situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' As expected, the SCFT solution of the hydrogen atom agrees exactly with the analytical quantum results [16, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' If correlations are ignored, the helium atom solved with SCFT agrees exactly with Hartree-Fock wave function theory [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' Other quantum phenomena that can be explained intuitively in the static, non-interacting situation include the uncertainty principle, quantum kinetic energy, tunnelling and the measurement problem (collapse of the wave function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' The equations of SCFT, as applied to quantum particles in the static case, are now reviewed in order to explain each of these phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' Complete derivational details can be found in reference [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' 4 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' Summary of SCFT Equations The main governing equation in SCFT is ∂q(r0, r, s) ∂s = ℏ2 2m∇2q(r0, r, s) − w(r, β)q(r0, r, s) (3) subject to the initial condition q(r0, r, 0) = δ(r − r0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' (4) This gives the unnormalized probability q(r0, r, s) that a particle at position r0 at high temperature (s = 0) will be found at position r at a temperature corresponding to s = β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' Note that q(r0, r, s) is a completely real and positive definite quantity that depends on the field w(r, β) which quantifies the interactions due to all other quantum particles in the system and external potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' For N particles in the system, the single particle density at position r is then n(r, β) = N Q(β)q(r, r, β) (5) where the normalization factor Q(β) = � q(r, r, β)dr (6) is the partition function for a single particle subject to the field w(r, β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' Equation (5) shows that it is sufficient to consider ring polymers only, since only paths that start at r and return to r contribute to the quantum particle density at the position r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' That is, the density gives the probability of a particle at position r at high temperature (s = 0) returning to position r at a temperature corresponding to β – see the black contours in figure 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' An intuitive consideration of paths in 5D allows one to include some quantum correlations by considering paths beyond rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' It will be shown that these also reduce to ring configurations, and these many-body paths can lead to a second order transition as Feynman found for superfluid helium [2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' To see that many-body non-ring paths are equivalent to rings, consider two particles at positions r and r′, respectively, both at high temperature (s = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' One can ask what the probability is of still finding two particles at positions r and r′ at a temperature corresponding to β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' The various paths the two particles can take in order to contribute to the two-particle density can be seen in figure 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' This argument can be extended to any number of 5 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' Schematic of threads in thermal-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' x-axis is space and y-axis is inverse thermal energy β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' (a) Contours that start from r and return to r, forming rings, are shown in black;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' those that do not return to the starting position r, forming open threads, are shown in grey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' (b) Two starting positions r and r′ and some contours that terminate at r and r′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' These include both rings and cross-paths that start at r and end at r′ and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' particles, but is here confined to the pair density for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' Following figure 1(b), the unnormalized pair density will be n(r, r′, β) ∝ q(r, r, β)q(r′, r′, β) + q(r, r′, β)q(r′, r, β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' (7) Integrating (7) over all possible pairs of particles should give the total number of pairs � � n(r, r′, β)drdr′ = N(N − 1) 2 (8) which allows the proportionality constant for (7) to be set to k = N(N − 1) 2Q(2)(β) (9) 6 s=0 S=0 r r (a) (b)where Q(2)(β) = � � [q(r, r, β)q(r′, r′, β) + q(r, r′, β)q(r′, r, β)] drdr′ = � q(r, r, β)dr � q(r′, r′, β)dr′ + � dr �� dr′q(r, r′, β)q(r′, r, β) � = Q(β)2 + Q(2β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' (10) The last line of (10) follows from (6) and the fact that � dr′q(r, r′, β)q(r′, r, β) = q(r, r, 2β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' (11) Equation (11) shows that these cross-paths are equivalent to ring paths of twice the contour length, as first pointed out by Feynman [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' Equations (7)-(11) allows one to write the pair density as n(r, r′, β) = N(N − 1) 2 [Q(β)2 + Q(2β)] [q(r, r, β)q(r′, r′, β) + q(r, r′, β)q(r′, r, β)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' (12) One gets the single-particle density by integrating over one of the position variables according to n(r, β) = 2 (N − 1) � n(r, r′, β)dr′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' (13) The pre-factor in (13) is to switch from counting pairs to counting singlets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' This gives n(r, β) = N [Q(β)2 + Q(2β)] [q(r, r, β)Q(β) + q(r, r, 2β)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' (14) In the limit of low temperatures β → ∞, numerical calculations show that the density saturates with respect to β and so stops changing with β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' In particular, it is found that lim β→∞ Q(2β) = Q(β)2 (15) lim β→∞ q(r, r, 2β) = q(r, r, β)Q(β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' (16) Substituting (15) and (16) into (14) immediately gives the ring polymer density expression (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' For some exotic systems, such as the superfluid helium studied by Feynman [2, 3], there is a second order transition due to the correlations in (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' Since the helium system is bosonic, one should also include three and higher-body terms in (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' For fermions, due to the Pauli exclusion principle, the maximum number of particles in any state will be two, and so (14) would not require any higher terms, and one would not expect to see a transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content='1 1 It may be possible to show in future work that the superconducting transition is related to quantum correlations in (14), but for now this is speculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' 7 For example, even using a temperature on the order of the surface of the sun, one gets less than a one one-thousandth percent change in the SCFT calculated helium atomic binding energy due to non-ring (longer ring) paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' In order to intuitively picture phenomena using the 5D holographic principle, equation (3) for open threads will be used instead of rings since this will give qualitatively equivalent results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' Several archetypal quantum phenomena are now examined using the open thread picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' Wave functions, quantum statistics and the measurement problem The non-ring paths shown in figure 1(a) by the grey contours and expressed in equation (14) correspond exactly to matrix elements in 4D wave function QM arising from symmet- ric statistics [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' Anti-symmetric statistics requires replacing the plus sign in equation (7) with a minus sign, as should be done when replacing Feynman’s discussion of bosons with anti-symmetric wave functions for fermions [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' In 5D, the minus sign in (7) causes the math- ematics to break down at zero temperature because the partition function corresponding to equation (10) becomes equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' This makes sense for two reasons: First, the propa- gator q(r, r, β) can be expanded in terms of complex orbitals that correspond to quantum mechanical states [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' Since fermions cannot multiply occupy a single state (ignoring spin), the number of microstates available for such a configuration should be zero, as is found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' In other words, from the 5D perspective, Pauli exclusion is the cause of anti-symmetric wave function statistics and not the effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' Second, the 5D holographic statement of QM does not use wave functions (or for that matter, quantum numbers or superpositions), and so concepts of bosons and fermions, which are based on wave function statistics, are super- fluous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' The theorems of DFT guarantee that all predictions of polymer SCFT applied to quantum particles must be the same as those of non-relativistic QM, and particles will turn out to behave in 4D as bosons or fermions based on their intrinsic and interaction properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' Threads in 5D need not be assigned a priori to either group to perform SCFT calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' Likewise, since there are no wave functions, one can see from equation (5) for the density, for example, that there is no wave function collapse during calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' The measurement problem is an artifact of doing calculations in 4D for a system that can be rigorously viewed as projected out of 5D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' 8 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' Quantum kinetic energy and the uncertainty principle An extended object like a Gaussian thread has properties which arise from its internal degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' Thermodynamically, these aspects are captured by the conformational entropy, which counts the number of configurations available to a thread when subject to a field w(r, β) at a temperature T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' In SCFT, the contribution of the conformational entropy to the free energy of a system is given by [11] Fconf = − 1 β � drn(r) [ln q(r, r, β) + βw(r)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' (17) The complete SCFT free energy expression for a quantum mechanical system is given in reference [11]2 and accounts for all thermodynamic quantities except the quantum kinetic energy, while having the extra term (17) for conformational degrees of freedom of a classical thread in 5D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' Thus, since the theorems of DFT guarantee that the SCFT expression must give the same results as QM, and exactly correct results for systems like the hydrogen atom and the Hartree-Fock helium atom are obtained, one is forced to identify the 5D conformational entropy as the 4D quantum kinetic energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' The thread-like internal degrees of freedom are also the 5D expression of the 4D un- certainty principle [11, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' The derivation of the 5D governing diffusion equation (3) in appendix A of reference [11] depends crucially on the uncertainty principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' In terms of position and momentum, if these quantities commute, then (3) collapses to the thermody- namics of a point-like particle subject to the field w(r, β) rather than a random walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' In other words, equation (3), which expresses the Gaussian thread nature of quantum particles in 5D, is equivalent to including the uncertainty principle in 4D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' There is a close connection therefore between the uncertainty principle, quantum kinetic energy and conformational entropy which the holographic principle makes clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' Stability of atoms The conformational entropy of SCFT threads also explains the stability of atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' Clas- sically, it is energetically favourable for electrons to be attracted into the Coulomb potential of the ionic core, collapsing the electron density into a spike at the nucleus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' The conforma- tional entropy, given by equation (17) however, would become enormous because, even at 2 Equation (17) of reference [11] contains typos which are corrected here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' 9 zero temperature, the thread would be confined to a single microstate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' SCFT reveals that the experimentally observed electron density is a result of the frustration between energy and conformational entropy in the 5D statistical mechanical picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' Tunnelling Consider the operator on the right hand side of the SCFT diffusion equation (3), which is H ≡ ℏ2 2m∇2 − w(r, β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' (18) This obeys an eigenvalue equation Hφi(r) = Eiφi(r) (19) where φi(r) and Ei are the eigenfunctions and eigenvalues, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' Note that whereas q(r0, r, s) was completely real and positive definite, φi(r) can be complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' Equation (19) is a one-particle time-independent Schr¨odinger equation, and is called the Kohn-Sham equation in the context of DFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' The density expression (5) can likewise be shown to become identical to the Kohn-Sham density expression, assuming perfect enforcement of the Pauli exclusion principle – see appendix B of reference [11] and appendix C of reference [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' For the case of a single quantum particle, the SCFT – Kohn-Sham DFT duality holds without conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' Thus, the SCFT equations formed around (3) will make all the same predictions as static quantum mechanics, including tunnelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' This is not surprising since ring polymer formalisms are used in quantum simulations to study tunnelling [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' Equation (3) can therefore be used to qualitatively understand tunnelling in the holo- graphic 5D context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' Figure 2(a) shows a typical tunnelling situation with a barrier of finite energy E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' A classical particle with insufficient kinetic energy cannot penetrate the barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' For a 5D thread described by (3), w(r, β) = E in the range x > 0 represents an energy penalty that discourages the contour from entering that region, but does not forbid it – there is a competing entropy benefit which lowers the free energy due to increased confor- mations when the thread enters the region – see figure 2(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' At s = β, there is thus a non-zero probability of finding the particle in the barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' As long as the initial position of the thread at s = 0 is outside the barrier, the extra thermal dimension provides a classical trajectory through which to enter the barrier zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' 10 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' Schematic of static tunnelling in the holographic picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' (a) A classical particle, shown by a dot, cannot enter the energy region E if it has too low a kinetic energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' (b) A thread in thermal-space starts at the same position as the classical particle, where it also cannot enter the region E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' As the contour extends from the initial point according to (3), the barrier E discourages, but does not forbid entry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' DYNAMIC PROPERTIES It is not immediately obvious how to write down the mathematical formulation for the dynamics of quantum particles in 5D as classical thermal threads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' This is not surprising, since writing a dynamical version of SCFT for actual polymers is similarly difficult and one is forced to use approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' However, just as it is known that chemical polymers continue to be extended one dimensional objects even though one cannot write exact SCFT mathematical equations for their dynamics, so it can be shown that quantum particles can continue to be represented as 1D thermal-world-lines even without giving explicit dynamic 11 (a) E X=0 (q) s=βSCFT equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' To see this, the argument of reference [12] is followed, considering again the eigenvalue equation of the operator on the right hand side of (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' As mentioned, the Kohn-Sham equa- tion (19) is a one-particle time-independent Schr¨odinger equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' In 4D wave function QM, non-relativistic dynamics are commonly included through postulating the time-dependent Schr¨odinger equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' It is therefore convenient to do a similar thing with the Kohn-Sham DFT expression (19), that is, it is generalized and dynamics are postulated to be given by the equation iℏ ∂ ∂tφi(r, t) = Hφi(r, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' (20) Together with the corresponding density and field expression – see reference [12] – equation (20) is identical to the formulas of time-dependent DFT (TDDFT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' A result of TDDFT is that the time-dependent density n(r, t) is proportional to a real quantity which is the time-dependent version of q(r, r, β) [23]: n(r, t) ∝ q(r, r, β, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' (21) Just as q(r, r, β) is the mathematical expression of thermal-world-line statistics in the static case, (21) demonstrates that the classical thread picture of quantum particles in 5D is consistent with quantum dynamics, even if on a practical level, one uses the equations of TDDFT, including (20), as a black-box to perform calculations in terms of complex eigenfunctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' The consistency of the 5D polymer thread model with TDDFT means that, through the Runge-Gross theorem [15], all dynamic predictions of the 5D thread picture must be consistent with those of the dynamic 4D wave function picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' Similar to the static case, one expects to be able to use ring polymers to effectively describe quantum particles, but for explanatory purposes, open threads based on equation (3) will be used instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' The holographic principle allows one to consider a number of dynamic effects in non-relativistic QM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' The double slit experiment A quasi-static approximation can be adopted in order to intuitively understand inter- ference in the double slit experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' The static governing equation (3) requires that if a 12 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' A schematic of the double slit experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' (a) A classical particle, denoted as a dot and with a velocity v, is shown incident on the double slit at a position such that it cannot pass through either slit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' it will therefore be blocked from being detected on the final screen (not shown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' (b) A thread in thermal-space with s = 0 at the same position as the classical particle in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' The probability of the particle being found at this position for s = β is not 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' there is a non-zero probability that the other end of the contour will be found at a position that would be impossible for a classical point particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' classical particle is at a position r at high temperature, then in the absence of other par- ticles or external influences, the probability of finding the particle at a temperature where quantum effects are important will be a Gaussian distribution centred on r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' Microscopically, this means that each particle thread will explore various thermal paths and might be found at a position that would be impossible classically by wrapping around infinite potential ob- structions in the thermal dimension– see figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' As discussed in references [11] and [12], 13 y (a) (b) 1the static non-locality of threads means that the density of particles predicted at the screen will be different from that expected classically, but without a further dynamic aspect, it does not yet agree with experimental predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' It is known from TDDFT and the Runge- Gross theorem that 5D classical threads must be able to reproduce interference phenomena – this is consistent with equation (20) which has an oscillatory nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' In other words, the propagator q(r, r, β, t) for the probability of a thread returning to its starting point must change in time commensurate with the de Broglie wavelength, or the function q(r0, r, β, t) for the probability of a thread ending at a different position, for a fixed r0 and t, will show the de Broglie wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' Much like an oscillating pendulum has a non-uniform probability of being found at various points along its period, so a pulsating 5D thread at an instant of time should oscillate with its high temperature initial point r0 at s = 0 fixed and other points s free to vibrate according to (20) – see figure 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' By construction then, a classical 5D thread must agree with experimental double slit results since the time-dependent Kohn- Sham equation (20) is identical with the time-dependent Schr¨odinger equation for a single particle, while still being consistent with classical 5D thread statistics through (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' Geometric phase and the Aharonov-Bohm effect When a quantum particle is transported through a closed loop in a parameter space, a phase change results due to the non-flat geometry of the space [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' This phase, known as the geometric phase or Berry’s phase, is detectable in experiments, a famous example being the Aharonov-Bohm effect [25, 26], but is not unique to quantum systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' It is seen in classical phenomena such as parallel transport and the precession of classical pendulums, which also experience phase changes when transported through closed loops in non-flat geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' The 5D holographic principle in which quantum particles are classical threads oscillating in thermal-space-time, as described in the previous section, immediately allows a 5D classical interpretation of quantum geometric phase phenomena, such as the Aharonov-Bohm effect, in terms of classical geometric phase changes in systems like pendulums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' 14 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' Three examples, (a)-(c), of 5D threads moving in opposite directions with the same conformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' The conformations change as the particles travel, but each thread in a pair will maintain identical conformations at any given time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' Entanglement Imagine a quantum particle which is split into two daughter particles which travel off in opposite directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' The 5D holographic principle view of this is shown in figures 4 (a)- (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' In each case, the two daughter threads can be considered to be entangled because their internal degrees of freedom (the conformational shapes they have) are correlated (they have the same shape as each other).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' With no further communication between them, any experiments done on one will be correlated with the other, to a larger extent than would be expected for classical point particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' This can be quantitatively seen with SCFT if measurements of positions and momenta for pairs of entangled particles at distant locations are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' 15 (a) V (q) (c)The Fourier transform of the governing thread equation (3) gives a relationship between momentum and position – see appendix A of reference [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' Suppose Alice and Bob each perform a series of momentum or position measurements on opposite sides of the exper- imental apparatus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' Given that the threads emitted in the two directions have identical conformations,3 if Bob on his side of the apparatus happens to have chosen to measure the same quantity as Alice, they will completely agree on the results they find.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' However, if Alice measures momentum and Bob measures position, after they have collected enough data, it will seem as if the wave function of the entire system collapses whenever one of them takes a measurement only on their own side, even though the 5D picture does not use wave functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' To see this, suppose Alice performs a series of momentum measurements on her side of the experimental setup, and she makes note of all momenta that fall within a narrow range – that is she notes all particles that have a chosen momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' If Bob happens to have measured position for those pairs, when they compare their results, Alice will have one value of momentum for all her particles, but Bob will have essentially random values of position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' That is because although each particle in a pair has the same thread conformation as its partner, separate pairs will have completely different conformations between them – see figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' Since the polymeric threads are not point particles, the classical relationship between momentum and position is not valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' From the Fourier transform of equation (3), in the absence of a field w(r, β), a known single momentum will give a distribution of positions – the actual position found for a single experiment will depend on the specific conformation of the polymer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' The quantitative values of the correlations for 5D classical thread pairs must agree with those of 4D quantum mechanics from the theorems of DFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' This guess for the mechanism of entanglement in 5D is very speculative and includes only the maximally and minimally correlated situations, which Bell comments are “.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' the only features .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' commonly used in verbal discussions of this problem” [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' He notes that these features are easily explained in terms of local hidden variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' In order to completely satisfy Bell-type theorems with hidden variables, intuitively unacceptable non-local variables are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' What is unacceptable in 4D may be more palatable in higher dimensions, and an extended mechanism involving cross-paths, equivalent to a single double-length contour that 3 The conformations would change as the particles travel, but they would change in exactly the same way, so they would always have the same conformation as each other at any given time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' 16 must pass through both Alice and Bob’s measuring devices, offers an intriguing mechanism not restricted by the speed of light since the imaginary time plane is not Lorentzian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' Other features The holographic principle for QM works in a 5D thermal-space-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' It has also been shown that it obeys a cylinder condition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' specifically, in the classical limit, the thermal dimension still exists, but no longer has any quantum effect on predictions – see appendix C of reference [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' Five dimensions subject to a cylinder condition in the classical limit are the assumptions of Kaluza theory [28, 29], and it follows that the postulates necessary for the polymer thread picture of QM include those necessary to derive electromagnetism from the structure of general relativity by using five dimensions instead of four [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' Thus, the 5D holographic principle of QM, in addition to having fewer postulates than 4D wave function theory, and avoiding various QM pathologies, leads directly to other physical phenomena outside of QM without additional assumptions [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' There are many other quantum phenomena that one could attempt to intuitively ex- plain, that have not yet been addressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' For example, the physical origins of spin in 5D, interaction-free measurement (the Elitzur-Vaidman bomb experiment) [30], other interfer- ometric predictions of QM [31, 32], non-statistical tests of entanglement, such as Green- berger–Horne–Zeilinger states, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' While the holographic principle will not always be as easy to apply as in the examples listed in this paper, the theorems of DFT guarantee that the 5D classical thread model will make exactly the same predictions as standard non-relativistic QM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' THE PAULI EXCLUSION PRINCIPLE As has been mentioned, SCFT can be shown to be equivalent to quantum DFT assuming enforcement of the Pauli exclusion principle [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' This is what allows the use of the theorems of DFT to show the holographic principle connecting 5D polymer SCFT with 4D wave function QM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' It is therefore necessary to postulate the nature of the exclusion principle in the classical 5D picture, in addition to the two other assumptions, namely that quantum particles are classical threads in 5D, and that these threads vibrate according to (20) [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' 17 Quantum threads in 5D are postulated to obey excluded volume – just as the trajectories of classical particles do not allow multiple particles to be in the same place at the same time, so it is assumed that threads cannot be in the same place at the same imaginary time (same value of β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' This property for classical threads in 5D maps onto the Pauli exclusion principle for particles in 4D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' There are several reasons for assuming this excluded volume: First, it is already an accepted feature of the quantum-classical isomorphism that gives correct results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' Feynman used the excluded volume of threads in imaginary time to justify removing trajectories in his study of the λ-transition [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' He did not identify this with the exclusion principle – he was working with bosons – but following his example, any massive particle would have to have this excluded volume feature, including electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' The mystery in terms of the electron is why do up to two of these fermions, assuming opposite spins, not feel excluded volume in the 5D space?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' This question has not been answered, but this feature can be accepted as a property of electrons since, although quantum particles are given structure as one- dimensional threads, this does not say anything about the cross-sectional structure of each thread – just like in 4D QM, electrons have no internal structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' Using 5D excluded volume in this way for electron calculations produces correct shell structure of the atoms [16, 18] and correct molecular bonding [20], at least within the context of the approximations used to implement the excluded volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' A second reason that supports 5D classical thread excluded volume is that it gives the correct scaling behaviour in the uniform limit for both the quantum kinetic energy and Dirac exchange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' Thomas [33] and Fermi [34] found that the quantum kinetic energy of the high density uniform electron gas should scale with the density of the gas n as n5/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' Dirac [35] found a correction term due to exchange effects that scales as n4/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' If techniques of polymer scaling theory as given by de Gennes [36] are used, assuming excluded volume between threads in 5D gives both the correct Thomas-Fermi expression [16, 18] and the correct Dirac exchange term [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' Third, the 5D excluded volume picture approaches satisfying conditions necessary for the Pauli field as described by Levy and Ou-Yang [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' To date, the excluded volume model has only been implemented on a coarse approximate level [18, 19], but even this approximate version satisfies four of five conditions, with the fifth being violated by an amount commensurate with the scale of the approximation – more details can be found in 18 references [18] and [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' More work is needed to further test the postulate relating excluded volume in 5D to the Pauli exclusion principle in 4D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' CONCLUSIONS AND FUTURE WORK A classical thread model for quantum particles in 5D can produce all the same predictions as 4D quantum mechanics, but with fewer postulates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' These axioms are: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' Quantum particles are classical threads in 5D (which should be formally treated as rings, although that condition has been relaxed in this work for illustrative purposes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' Time evolution is governed by the time-dependent Kohn-Sham equation, which can be shown to be consistent with the statistics of 5D classical threads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' Pauli exclusion is enforced through higher dimensional excluded volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' The duality between 5D classical threads and 4D quantum wave functions defines a holo- graphic principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' Advantages of the 5D perspective, in addition to having fewer postulates, include: a realist model which is, in the context of higher dimensions, deterministic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' no measurement problem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' an explanation of randomness through the ensemble interpretation utilizing the internal conformational degrees of freedom of 5D threads within classical sta- tistical mechanics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' an interpretation of quantum statistics, exchange and the Pauli exclusion principle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' intuitive explanations for quantum phenomena such as the Aharonov-Bohm effect, the uncertainty principle, quantum kinetic energy, tunnelling and the double slit experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' Many issues remain unexplored however.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' Spin is easily included in 5D SCFT calcula- tions, but no physically intuitive picture for it has been provided yet – for example, no explanation is given as to why two electrons of opposite spins do not feel excluded volume in 5D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' Non-statistical tests of entanglement, such as Greenberger–Horne–Zeilinger states, should be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' A relativistic version of the principle is required, and connections with thermal quantum field theory, related to Matsubara imaginary time, should be ex- plored [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' Justifications that the Pauli exclusion principle is excluded volume in 5D have been listed, but further evidence is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' Improved approximations of the 5D excluded volume in SCFT calculations could show whether the shell structure of atoms approaches chemical accuracy, or diverges from experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' Likewise, more complex SCFT molecular calculations and time-dependent SCFT problems could either support or contradict the 5D picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' One advantage of 5D SCFT is that it is readily compared to experiments, and 19 although inevitable approximations in numerical calculations are a confounding factor, this limitation is no different from any other quantum calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' If nothing else, the SCFT holographic principle is a useful quantum calculational tool, but its implications for quantum foundations are much more interesting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' ACKNOWLEDGEMENTS The author acknowledges helpful discussions with many of the participants of the 15th Biennial Quantum Structure Conference, with P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' LeMaitre for suggesting changes to the manuscript, and with M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' Matsen who pointed out the relationship (11) between cross-path contours and longer single rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQf4Qjg/content/2301.04180v1.pdf'} +page_content=' [1] Juan Maldacena, “The large N limit of superconformal field theories and supergravity,” 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