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+arXiv:2301.01011v1  [math.SG]  3 Jan 2023
+GEOMETRIC QUANTIZATIONS ASSOCIATED TO MIXED
+POLARIZATIONS ON K¨AHLER MANIFOLDS WITH T-SYMMETRY
+NAICHUNG CONAN LEUNG, AND DAN WANG
+Abstract. 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, ��, 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: [email protected]
+The Institute of Mathematical Sciences and Department of Mathematics, The Chinese
+University of Hong Kong, Shatin, Hong Kong
+Email address: [email protected]
+

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+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 =
+<mask>;, 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), ��BERT 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 <mask>, 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 <mask> d;
+2. AST Nodes 
+Masking
+a = b + c; 
+return a == d;
+a = <mask> + 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 <mask>.
+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:
+• <mask> = a + b
+• res <mask>= a + b
+• res = <mask> + b
+• res = a <mask> b
+• res = a + <mask>
+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 + <mask>
+res + 0
+identifiers
+res + 10
+<mask> + 10
+a + 10
+binary expressions
+a && b
+a <mask> b
+a || b
+unary expressions
+--a
+<mask>a
+++a
+assignments
+sum += current
+sum <mask>= current
+sum -= current
+object fields
+node.next
+node.<mask>
+node.prev
+method calls
+list.add(node)
+list.<mask>(node)
+list.push(node)
+array access
+arr[index + 1]
+arr[<mask>]
+arr[index]
+static type references
+Math.random() * 10
+<mask>.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.
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+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]. 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: [email protected], 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
+

1dE2T4oBgHgl3EQfNQZD/content/tmp_files/load_file.txt

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+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': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'}
+page_content='Application Progress  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'}
+page_content='50000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'}
+page_content='100%  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'}
+page_content='70 GB  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'}
+page_content='40000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'}
+page_content='60 GB  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'}
+page_content='80%  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'}
+page_content='30000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'}
+page_content='50 GB  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'}
+page_content='20000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'}
+page_content='60%  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'}
+page_content='40 GB  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'}
+page_content='10000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'}
+page_content='30 GB  ' 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='20 GB  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'}
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+page_content='10 GB  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'}
+page_content='map_in_progress  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'}
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+page_content='reduce_in_progress  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'}
+page_content='reduce_in_progress  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'}
+page_content='reduce_in_progress  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'}
+page_content='0 B  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'}
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+page_content='median objmem  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'}
+page_content='reducer_completed  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'}
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+page_content='min objmem  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'}
+page_content='max objmem  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'}
+page_content='min workmem  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'}
+page_content='median cpu - min cpu - max cpu  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'}
+page_content='max workmem  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'}
+page_content='merge_in_progress  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'}
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+page_content='NVMe Disk I/0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'}
+page_content='Network I/0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'}
+page_content='Disk Usage  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'}
+page_content='7 GB/s  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'}
+page_content='3 GB/s  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'}
+page_content='100%  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'}
+page_content='6 GB/s  ' 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='50 GB/s  80%  5 GB/s  2 GB/s  4 GB/s  60%  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'}
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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'}
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+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'}
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+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'}
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+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': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'}
+page_content=' http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'}
+page_content='ordinal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'}
+page_content='com/  gensort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'}
+page_content='html  [6] Mehul A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'}
+page_content=' Shah, Amiato, and Chris Nyberg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'}
+page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'}
+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': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'}
+page_content=' Ray v2 Architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'}
+page_content=' Anyscale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'}
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+page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfNQZD/content/2301.03734v1.pdf'}
+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'}
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+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
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+Does Gaia Play Dice? : 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. 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: [email protected]
+†E-mail: [email protected]
+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. Biology
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+[3] Timothy M Lenton, Stuart J Daines, James G Dyke, Arwen E Nicholson, David M Wilkinson,
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+visited on 1/1/2023.
+15
+

6NE0T4oBgHgl3EQfvwGn/content/tmp_files/load_file.txt

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+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': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'}
+page_content=' Kenworthy, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'}
+page_content=' Albrecht, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'}
+page_content=' Anglada-Escud´e, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'}
+page_content=' Baraffe, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'}
+page_content=' Bau-  doz, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'}
+page_content=' Benz, 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=' 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': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'}
+page_content=' Ruane, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'}
+page_content=' Selsis, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'}
+page_content=' Sozzetti, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'}
+page_content=' Stam, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'}
+page_content=' Stark, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'}
+page_content=' Vigan, and Pieter de Visser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'}
+page_content='  Detecting life outside our solar system with a large high-contrast-imaging mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'}
+page_content=' Experimental  Astronomy, October 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE0T4oBgHgl3EQfvwGn/content/2301.02623v1.pdf'}
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+ANNA: Abstractive Text-to-Image Synthesis with Filtered News Captions
+Aashish Anantha Ramakrishnan
+Sharon X. 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
+
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+

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+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: [email protected]
+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
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+x[pc]
+6
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+y[pc]
+(a)
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+30
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+0
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+(e)
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+(f)
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+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.
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+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(°

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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: [email protected]
+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. [13].
+It can be shown that the Airy function solutions of PII (7.2) satisfy first-order
+equation of the form (7.1) for n odd, whereas the Airy function solutions of P34 (7.3) satisfy a first-order
+equation of the form (7.1) for n even.
+Acknowledgements
+I thank Clare Dunning and Steffen Krusch for helpful comments and illuminating discussions.
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+18
+

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+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
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+DEPARTMENT OF MATHEMATICS, GRADUATE SCHOOL OF NATURAL SCIENCES PUSAN NATIONAL UNIVER-
+SITY, BUSAN 46241, REPUBLIC OF KOREA
+Email address: [email protected]
+UNIVERSITY OF SOUTH FLORIDA, TAMPA, FLORIDA, USA
+Email address: [email protected]
+DEPARTMENT OF MATHEMATICS, DALIAN UNIVERSITY OF TECHNOLOGY, CHINA
+Email address: [email protected]
+15
+

D9E2T4oBgHgl3EQfSgcI/content/tmp_files/load_file.txt

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+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': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'}
+page_content=' Table of Unoriented Dichromatic Singular Links  ACKNOWLEDGEMENT  Mohamed Elhamdadi was partially supported by Simons Foundation collaboration grant 712462.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'}
+page_content='  14  REFERENCES  [1] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'}
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+page_content='  [2] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'}
+page_content=' Bataineh and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'}
+page_content=' Saidi, Involutory quandles and dichromatic links, Symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'}
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+page_content=' Knot Theory Ramifications 30 (2021), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'}
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+page_content=' 2150070, 15, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'}
+page_content='1142/S021821652150070X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'}
+page_content=' MR4358333  [4] Jose Ceniceros, Indu R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'}
+page_content=' Churchill, and Mohamed Elhamdadi, Singquandle shadows and singular knot invariants,  Canad.' 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=' Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'}
+page_content=' 65 (2022), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'}
+page_content=' 3, 770–787, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'}
+page_content='4153/S0008439521000837.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf'}
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@@ -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: [email protected]
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+

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+Detecting the heterodyning of gravitational waves
+Jakob Stegmann∗ and Sander M. 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
+�j
+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<T
+DECIGO BNSS
+LISA BWDS8
+FIG. 5. Cumulative SNR of a background gravitational wave
+signal as a function of the n loudest binary white dwarfs
+(BWDs) in the set of carrier signals. Stars at the end of each
+line indicate the total number N of binaries in each model. In
+any model several 102 to 103 of the loudest BWDs are enough
+to achieve sensitivities similar to the entire sample.
+This sensitivity could potentially enable the detection
+of very massive SMBBHs with a chirp mass of several
+1010 M⊙ at a distance of D = 10 Mpc, if such systems
+exist.
+Single super-massive BHs of several ∼ 1010 M⊙
+would be close to theoretical mass upper limits above
+which they cannot grow through luminous gas accretion
+[55], and so far candidates have only been observed at
+distances of more than several ∼ 100 Mpc [e.g., 56, 57].
+Our results show that an even better sensitivity could
+be achieved using GW signals from compact binary stars
+detectable with next-generation GW detectors that op-
+erate in the dHz regime. In particular, using signals of
+binary neutron stars expected to be detected with DE-
+CIGO would yield a sensitivity competitive with that of
+current pulsar timing arrays.
+Our results show that future detectors designed to
+detect GW signals in a higher frequency range could be
+used to indirectly probe GWs down to the frequencies
+given by the inverse instrument lifetime. Conveniently,
+this could be achieved without modification of the
+detector designs and with the same data outputs. This
+method could therefore prove a valuable tool in the
+exploration of the gravitational-wave spectrum and the
+development of gravitational-wave astronomy in general.
+ACKNOWLEDGEMENTS
+We thank Vivien Raymond, Bangalore Sathyaprakash,
+Fabio Antonini, Hartmut Grote, Guido M¨uller, Antoine
+Petiteau, Martin Hewitson, Lucio Mayer, and Valeriya
+Korol for helpful input and discussions.
+DATA AVAILABILITY
+The data underlying this article will be shared on rea-
+sonable request to the authors.
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